the graph regularity method - video lectures lecture 1.pdf · the graph regularity method jacob fox...
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![Page 1: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/1.jpg)
The Graph Regularity Method
Jacob FoxStanford University
Marston Morse Lecture Series
Institute for Advanced Studies
October 24, 2016
![Page 2: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/2.jpg)
Graphs
Definition (graph)
A graph G = (V ,E ) has a vertex set V and
a set E of edges, which are pairs of vertices.
Examples:
The Internet with computers connected by links
The World Wide Web with webpages and hyperlinks
Social networks like Facebook with users and friendships
Chemical networks with atoms and chemical bonds
Biological networks like the brain with neurons and synapses
Engineered networks like chips with transistors and wires
From Large Networks and Graph Limits by Lovasz
![Page 3: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/3.jpg)
Graphs
Definition (graph)
A graph G = (V ,E ) has a vertex set V and
a set E of edges, which are pairs of vertices.
Examples:
The Internet with computers connected by links
The World Wide Web with webpages and hyperlinks
Social networks like Facebook with users and friendships
Chemical networks with atoms and chemical bonds
Biological networks like the brain with neurons and synapses
Engineered networks like chips with transistors and wires
From Large Networks and Graph Limits by Lovasz
![Page 4: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/4.jpg)
Graphs
Definition (graph)
A graph G = (V ,E ) has a vertex set V and
a set E of edges, which are pairs of vertices.
Examples:
The Internet with computers connected by links
The World Wide Web with webpages and hyperlinks
Social networks like Facebook with users and friendships
Chemical networks with atoms and chemical bonds
Biological networks like the brain with neurons and synapses
Engineered networks like chips with transistors and wires
From Large Networks and Graph Limits by Lovasz
![Page 5: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/5.jpg)
Graphs
Definition (graph)
A graph G = (V ,E ) has a vertex set V and
a set E of edges, which are pairs of vertices.
Examples:
The Internet with computers connected by links
The World Wide Web with webpages and hyperlinks
Social networks like Facebook with users and friendships
Chemical networks with atoms and chemical bonds
Biological networks like the brain with neurons and synapses
Engineered networks like chips with transistors and wires
From Large Networks and Graph Limits by Lovasz
![Page 6: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/6.jpg)
Graphs
Definition (graph)
A graph G = (V ,E ) has a vertex set V and
a set E of edges, which are pairs of vertices.
Examples:
The Internet with computers connected by links
The World Wide Web with webpages and hyperlinks
Social networks like Facebook with users and friendships
Chemical networks with atoms and chemical bonds
Biological networks like the brain with neurons and synapses
Engineered networks like chips with transistors and wires
From Large Networks and Graph Limits by Lovasz
![Page 7: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/7.jpg)
Graphs
Definition (graph)
A graph G = (V ,E ) has a vertex set V and
a set E of edges, which are pairs of vertices.
Examples:
The Internet with computers connected by links
The World Wide Web with webpages and hyperlinks
Social networks like Facebook with users and friendships
Chemical networks with atoms and chemical bonds
Biological networks like the brain with neurons and synapses
Engineered networks like chips with transistors and wires
From Large Networks and Graph Limits by Lovasz
![Page 8: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/8.jpg)
Graphs
Definition (graph)
A graph G = (V ,E ) has a vertex set V and
a set E of edges, which are pairs of vertices.
Examples:
The Internet with computers connected by links
The World Wide Web with webpages and hyperlinks
Social networks like Facebook with users and friendships
Chemical networks with atoms and chemical bonds
Biological networks like the brain with neurons and synapses
Engineered networks like chips with transistors and wires
From Large Networks and Graph Limits by Lovasz
![Page 9: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/9.jpg)
Graphs
Definition (graph)
A graph G = (V ,E ) has a vertex set V and
a set E of edges, which are pairs of vertices.
Examples:
The Internet with computers connected by links
The World Wide Web with webpages and hyperlinks
Social networks like Facebook with users and friendships
Chemical networks with atoms and chemical bonds
Biological networks like the brain with neurons and synapses
Engineered networks like chips with transistors and wires
From Large Networks and Graph Limits by Lovasz
![Page 10: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/10.jpg)
Szemeredi’s Regularity Lemma
Szemeredi’s regularity lemma
Roughly speaking, every graph can be
partitioned into a bounded number of roughly
equally-sized parts so that the graph is
random-like between almost all pairs of parts.
Rough structural result for all graphs.
One of the most powerful tool in combinatorics.
![Page 11: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/11.jpg)
Szemeredi’s Regularity Lemma
Szemeredi’s regularity lemma
Roughly speaking, every graph can be
partitioned into a bounded number of roughly
equally-sized parts so that the graph is
random-like between almost all pairs of parts.
Rough structural result for all graphs.
One of the most powerful tool in combinatorics.
![Page 12: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/12.jpg)
Szemeredi’s Regularity Lemma
Szemeredi’s regularity lemma
Roughly speaking, every graph can be
partitioned into a bounded number of roughly
equally-sized parts so that the graph is
random-like between almost all pairs of parts.
Rough structural result for all graphs.
One of the most powerful tool in combinatorics.
![Page 13: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/13.jpg)
Regularity
Let X and Y be vertex subsets of a graph G .
e(X ,Y ) = number of pairs in X × Y that are edges.
Definition (density)
d(X ,Y ) =e(X ,Y )
|X | |Y |
Definition (irregularity)
irreg(X ,Y ) is the maximum over all
A ⊂ X and B ⊂ Y of∣∣∣e(A,B)− d(X ,Y )|A||B|∣∣∣
A B
X Y
(X ,Y ) is ε-regular if irreg(X ,Y ) ≤ ε|X ||Y |.
![Page 14: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/14.jpg)
Regularity
Let X and Y be vertex subsets of a graph G .
e(X ,Y ) = number of pairs in X × Y that are edges.
Definition (density)
d(X ,Y ) =e(X ,Y )
|X | |Y |
Definition (irregularity)
irreg(X ,Y ) is the maximum over all
A ⊂ X and B ⊂ Y of∣∣∣e(A,B)− d(X ,Y )|A||B|∣∣∣
A B
X Y
(X ,Y ) is ε-regular if irreg(X ,Y ) ≤ ε|X ||Y |.
![Page 15: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/15.jpg)
Regularity
Let X and Y be vertex subsets of a graph G .
e(X ,Y ) = number of pairs in X × Y that are edges.
Definition (density)
d(X ,Y ) =e(X ,Y )
|X | |Y |
Definition (irregularity)
irreg(X ,Y ) is the maximum over all
A ⊂ X and B ⊂ Y of∣∣∣e(A,B)− d(X ,Y )|A||B|∣∣∣
A B
X Y
(X ,Y ) is ε-regular if irreg(X ,Y ) ≤ ε|X ||Y |.
![Page 16: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/16.jpg)
Regularity
Let X and Y be vertex subsets of a graph G .
e(X ,Y ) = number of pairs in X × Y that are edges.
Definition (density)
d(X ,Y ) =e(X ,Y )
|X | |Y |
Definition (irregularity)
irreg(X ,Y ) is the maximum over all
A ⊂ X and B ⊂ Y of∣∣∣e(A,B)− d(X ,Y )|A||B|∣∣∣
A B
X Y
(X ,Y ) is ε-regular if irreg(X ,Y ) ≤ ε|X ||Y |.
![Page 17: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/17.jpg)
Regularity
Let X and Y be vertex subsets of a graph G .
e(X ,Y ) = number of pairs in X × Y that are edges.
Definition (density)
d(X ,Y ) =e(X ,Y )
|X | |Y |
Definition (irregularity)
irreg(X ,Y ) is the maximum over all
A ⊂ X and B ⊂ Y of∣∣∣e(A,B)− d(X ,Y )|A||B|∣∣∣
A B
X Y
(X ,Y ) is ε-regular if irreg(X ,Y ) ≤ ε|X ||Y |.
![Page 18: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/18.jpg)
Szemeredi’s regularity lemma
Definition (Partition irregularity)
The irregularity of a vertex partition P of a graph G = (V ,E ) is
irreg(P) :=∑
X ,Y∈Pirreg(X ,Y )
Partition P is ε-regular if irreg(P) ≤ ε|V |2.
Szemeredi’s regularity lemma
For every ε > 0, there is an M(ε) so that every graph G has an
ε-regular vertex partition P with at most M(ε) parts.
![Page 19: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/19.jpg)
Szemeredi’s regularity lemma
Definition (Partition irregularity)
The irregularity of a vertex partition P of a graph G = (V ,E ) is
irreg(P) :=∑
X ,Y∈Pirreg(X ,Y )
Partition P is ε-regular if irreg(P) ≤ ε|V |2.
Szemeredi’s regularity lemma
For every ε > 0, there is an M(ε) so that every graph G has an
ε-regular vertex partition P with at most M(ε) parts.
![Page 20: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/20.jpg)
Szemeredi’s regularity lemma
Definition (Partition irregularity)
The irregularity of a vertex partition P of a graph G = (V ,E ) is
irreg(P) :=∑
X ,Y∈Pirreg(X ,Y )
Partition P is ε-regular if irreg(P) ≤ ε|V |2.
Szemeredi’s regularity lemma
For every ε > 0, there is an M(ε) so that every graph G has an
ε-regular vertex partition P with at most M(ε) parts.
![Page 21: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/21.jpg)
Number of parts
Szemeredi’s regularity lemma
Every graph has an ε-regular partition into at most M(ε) parts.
Question
How big is M(ε)?
Tower function T (n) is given by T (1) = 2 and T (n) = 2T (n−1).
The proof of the regularity lemma gives M(ε) ≤ T (ε−2).
Gowers proved M(ε) ≥ T (ε−c) for some constant c > 0.
Question (Gowers 1997)
Determine the order of the tower height of M(ε).
Theorem (F.-L. M. Lovasz 2016+)
M(ε) = T (Θ(ε−2))
![Page 22: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/22.jpg)
Number of parts
Szemeredi’s regularity lemma
Every graph has an ε-regular partition into at most M(ε) parts.
Question
How big is M(ε)?
Tower function T (n) is given by T (1) = 2 and T (n) = 2T (n−1).
The proof of the regularity lemma gives M(ε) ≤ T (ε−2).
Gowers proved M(ε) ≥ T (ε−c) for some constant c > 0.
Question (Gowers 1997)
Determine the order of the tower height of M(ε).
Theorem (F.-L. M. Lovasz 2016+)
M(ε) = T (Θ(ε−2))
![Page 23: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/23.jpg)
Number of parts
Szemeredi’s regularity lemma
Every graph has an ε-regular partition into at most M(ε) parts.
Question
How big is M(ε)?
Tower function T (n) is given by T (1) = 2 and T (n) = 2T (n−1).
The proof of the regularity lemma gives M(ε) ≤ T (ε−2).
Gowers proved M(ε) ≥ T (ε−c) for some constant c > 0.
Question (Gowers 1997)
Determine the order of the tower height of M(ε).
Theorem (F.-L. M. Lovasz 2016+)
M(ε) = T (Θ(ε−2))
![Page 24: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/24.jpg)
Number of parts
Szemeredi’s regularity lemma
Every graph has an ε-regular partition into at most M(ε) parts.
Question
How big is M(ε)?
Tower function T (n) is given by T (1) = 2 and T (n) = 2T (n−1).
The proof of the regularity lemma gives M(ε) ≤ T (ε−2).
Gowers proved M(ε) ≥ T (ε−c) for some constant c > 0.
Question (Gowers 1997)
Determine the order of the tower height of M(ε).
Theorem (F.-L. M. Lovasz 2016+)
M(ε) = T (Θ(ε−2))
![Page 25: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/25.jpg)
Number of parts
Szemeredi’s regularity lemma
Every graph has an ε-regular partition into at most M(ε) parts.
Question
How big is M(ε)?
Tower function T (n) is given by T (1) = 2 and T (n) = 2T (n−1).
The proof of the regularity lemma gives M(ε) ≤ T (ε−2).
Gowers proved M(ε) ≥ T (ε−c) for some constant c > 0.
Question (Gowers 1997)
Determine the order of the tower height of M(ε).
Theorem (F.-L. M. Lovasz 2016+)
M(ε) = T (Θ(ε−2))
![Page 26: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/26.jpg)
Number of parts
Szemeredi’s regularity lemma
Every graph has an ε-regular partition into at most M(ε) parts.
Question
How big is M(ε)?
Tower function T (n) is given by T (1) = 2 and T (n) = 2T (n−1).
The proof of the regularity lemma gives M(ε) ≤ T (ε−2).
Gowers proved M(ε) ≥ T (ε−c) for some constant c > 0.
Question (Gowers 1997)
Determine the order of the tower height of M(ε).
Theorem (F.-L. M. Lovasz 2016+)
M(ε) = T (Θ(ε−2))
![Page 27: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/27.jpg)
Number of parts
Szemeredi’s regularity lemma
Every graph has an ε-regular partition into at most M(ε) parts.
Question
How big is M(ε)?
Tower function T (n) is given by T (1) = 2 and T (n) = 2T (n−1).
The proof of the regularity lemma gives M(ε) ≤ T (ε−2).
Gowers proved M(ε) ≥ T (ε−c) for some constant c > 0.
Question (Gowers 1997)
Determine the order of the tower height of M(ε).
Theorem (F.-L. M. Lovasz 2016+)
M(ε) = T (Θ(ε−2))
![Page 28: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/28.jpg)
Szemeredi’s regularity lemma: proof idea
Definition (Mean square density)
For a vertex partition P : V = V1 ∪ . . . ∪ Vk ,
q(P) =∑i ,j
pipjd(Vi ,Vj)2
where pi = |Vi ||V | .
Properties:
0 ≤ q(P) ≤ 1.
If P ′ is a refinement of P, then q(P ′) ≥ q(P).
Claim
If P with |P| = k is not ε-regular, then there is a refinement P ′
into at most k2k+1 parts such that q(P ′) ≥ q(P) + ε2.
At most ε−2 iterations before obtaining an ε-regular partition.
![Page 29: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/29.jpg)
Szemeredi’s regularity lemma: proof idea
Definition (Mean square density)
For a vertex partition P : V = V1 ∪ . . . ∪ Vk ,
q(P) =∑i ,j
pipjd(Vi ,Vj)2
where pi = |Vi ||V | .
Properties:
0 ≤ q(P) ≤ 1.
If P ′ is a refinement of P, then q(P ′) ≥ q(P).
Claim
If P with |P| = k is not ε-regular, then there is a refinement P ′
into at most k2k+1 parts such that q(P ′) ≥ q(P) + ε2.
At most ε−2 iterations before obtaining an ε-regular partition.
![Page 30: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/30.jpg)
Szemeredi’s regularity lemma: proof idea
Definition (Mean square density)
For a vertex partition P : V = V1 ∪ . . . ∪ Vk ,
q(P) =∑i ,j
pipjd(Vi ,Vj)2
where pi = |Vi ||V | .
Properties:
0 ≤ q(P) ≤ 1.
If P ′ is a refinement of P, then q(P ′) ≥ q(P).
Claim
If P with |P| = k is not ε-regular, then there is a refinement P ′
into at most k2k+1 parts such that q(P ′) ≥ q(P) + ε2.
At most ε−2 iterations before obtaining an ε-regular partition.
![Page 31: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/31.jpg)
Szemeredi’s regularity lemma: proof idea
Definition (Mean square density)
For a vertex partition P : V = V1 ∪ . . . ∪ Vk ,
q(P) =∑i ,j
pipjd(Vi ,Vj)2
where pi = |Vi ||V | .
Properties:
0 ≤ q(P) ≤ 1.
If P ′ is a refinement of P, then q(P ′) ≥ q(P).
Claim
If P with |P| = k is not ε-regular, then there is a refinement P ′
into at most k2k+1 parts such that q(P ′) ≥ q(P) + ε2.
At most ε−2 iterations before obtaining an ε-regular partition.
![Page 32: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/32.jpg)
Regularity method
Regularity method
1 Apply Szemeredi’s regularity lemma.
2 Use a counting lemma for embedding small graphs.
Triangle counting lemma
If each pair of parts is ε-regular, the number
of triangles across the three parts is
≈ d(X ,Y )d(X ,Z )d(Y ,Z ) |X | |Y | |Z | .
X
Y Z
![Page 33: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/33.jpg)
Regularity method
Regularity method
1 Apply Szemeredi’s regularity lemma.
2 Use a counting lemma for embedding small graphs.
Triangle counting lemma
If each pair of parts is ε-regular, the number
of triangles across the three parts is
≈ d(X ,Y )d(X ,Z )d(Y ,Z ) |X | |Y | |Z | .
X
Y Z
![Page 34: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/34.jpg)
Triangle removal lemma
Triangle removal lemma (Ruzsa-Szemeredi 1976)
For any ε > 0 there is a δ > 0 such that every n-vertex graph with
≤ δn3 triangles can be made triangle-free by removing εn2 edges.
Has many applications in extremal graph theory, additive numbertheory, theoretical computer science, and discrete geometry.
Proof idea:
1 Apply Szemeredi’s regularity lemma.
2 Delete edges between pairs which are irregular or sparse.
3 If there is a remaining triangle, then its edges go between
pairs which are both dense and regular. The counting lemma
then implies that there are more than δn3 triangles.
![Page 35: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/35.jpg)
Triangle removal lemma
Triangle removal lemma (Ruzsa-Szemeredi 1976)
For any ε > 0 there is a δ > 0 such that every n-vertex graph with
≤ δn3 triangles can be made triangle-free by removing εn2 edges.
Has many applications in extremal graph theory, additive numbertheory, theoretical computer science, and discrete geometry.
Proof idea:
1 Apply Szemeredi’s regularity lemma.
2 Delete edges between pairs which are irregular or sparse.
3 If there is a remaining triangle, then its edges go between
pairs which are both dense and regular. The counting lemma
then implies that there are more than δn3 triangles.
![Page 36: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/36.jpg)
Triangle removal lemma
Triangle removal lemma (Ruzsa-Szemeredi 1976)
For any ε > 0 there is a δ > 0 such that every n-vertex graph with
≤ δn3 triangles can be made triangle-free by removing εn2 edges.
Has many applications in extremal graph theory, additive numbertheory, theoretical computer science, and discrete geometry.
Proof idea:
1 Apply Szemeredi’s regularity lemma.
2 Delete edges between pairs which are irregular or sparse.
3 If there is a remaining triangle, then its edges go between
pairs which are both dense and regular. The counting lemma
then implies that there are more than δn3 triangles.
![Page 37: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/37.jpg)
Triangle removal lemma
Triangle removal lemma (Ruzsa-Szemeredi 1976)
For any ε > 0 there is a δ > 0 such that every n-vertex graph with
≤ δn3 triangles can be made triangle-free by removing εn2 edges.
Has many applications in extremal graph theory, additive numbertheory, theoretical computer science, and discrete geometry.
Proof idea:
1 Apply Szemeredi’s regularity lemma.
2 Delete edges between pairs which are irregular or sparse.
3 If there is a remaining triangle, then its edges go between
pairs which are both dense and regular. The counting lemma
then implies that there are more than δn3 triangles.
![Page 38: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/38.jpg)
Triangle removal lemma
Triangle removal lemma (Ruzsa-Szemeredi 1976)
For any ε > 0 there is a δ > 0 such that every n-vertex graph with
≤ δn3 triangles can be made triangle-free by removing εn2 edges.
Has many applications in extremal graph theory, additive numbertheory, theoretical computer science, and discrete geometry.
Proof idea:
1 Apply Szemeredi’s regularity lemma.
2 Delete edges between pairs which are irregular or sparse.
3 If there is a remaining triangle, then its edges go between
pairs which are both dense and regular. The counting lemma
then implies that there are more than δn3 triangles.
![Page 39: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/39.jpg)
Triangle removal lemma
Triangle removal lemma (Ruzsa-Szemeredi 1976)
For any ε > 0 there is a δ > 0 such that every n-vertex graph with
≤ δn3 triangles can be made triangle-free by removing εn2 edges.
Problem (Erdos, Alon, Gowers, Tao)
Find a new proof which gives a better bound.
Theorem (F. 2011)
We may take δ−1 to be a tower of twos of height log ε−1.
Best known lower bound on δ−1 is only ε−c log ε−1.
![Page 40: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/40.jpg)
Triangle removal lemma
Triangle removal lemma (Ruzsa-Szemeredi 1976)
For any ε > 0 there is a δ > 0 such that every n-vertex graph with
≤ δn3 triangles can be made triangle-free by removing εn2 edges.
Problem (Erdos, Alon, Gowers, Tao)
Find a new proof which gives a better bound.
Theorem (F. 2011)
We may take δ−1 to be a tower of twos of height log ε−1.
Best known lower bound on δ−1 is only ε−c log ε−1.
![Page 41: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/41.jpg)
Triangle removal lemma
Triangle removal lemma (Ruzsa-Szemeredi 1976)
For any ε > 0 there is a δ > 0 such that every n-vertex graph with
≤ δn3 triangles can be made triangle-free by removing εn2 edges.
Problem (Erdos, Alon, Gowers, Tao)
Find a new proof which gives a better bound.
Theorem (F. 2011)
We may take δ−1 to be a tower of twos of height log ε−1.
Best known lower bound on δ−1 is only ε−c log ε−1.
![Page 42: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/42.jpg)
Triangle removal lemma
Triangle removal lemma (Ruzsa-Szemeredi 1976)
For any ε > 0 there is a δ > 0 such that every n-vertex graph with
≤ δn3 triangles can be made triangle-free by removing εn2 edges.
Problem (Erdos, Alon, Gowers, Tao)
Find a new proof which gives a better bound.
Theorem (F. 2011)
We may take δ−1 to be a tower of twos of height log ε−1.
Best known lower bound on δ−1 is only ε−c log ε−1.
![Page 43: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/43.jpg)
Triangle removal lemma: new proof idea
V1
V2 V3
Key lemma
If there are at most α|V1||V2||V3| triangles across V1,V2,V3,
then
there are 1 ≤ i < j ≤ 3 and equitable partitions Qi of Vi and Qj of
Vj each with at most 2α−O(1)
parts such that there are at least1
10 |Qi ||Qj | pairs (X ,Y ) ∈ Qi × Qj with d(X ,Y ) < 2α1/3.
Definition: Mean entropy density
For a vertex partition P : V = V1 ∪ . . . ∪ Vk ,
h(P) =∑i ,j
pipjd(Vi ,Vj) log d(Vi ,Vj)
where pi = |Vi ||V | .
![Page 44: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/44.jpg)
Triangle removal lemma: new proof idea
V1
V2 V3
Key lemma
If there are at most α|V1||V2||V3| triangles across V1,V2,V3, then
there are 1 ≤ i < j ≤ 3 and equitable partitions Qi of Vi and Qj of
Vj each with at most 2α−O(1)
parts such that there are at least1
10 |Qi ||Qj | pairs (X ,Y ) ∈ Qi × Qj with d(X ,Y ) < 2α1/3.
Definition: Mean entropy density
For a vertex partition P : V = V1 ∪ . . . ∪ Vk ,
h(P) =∑i ,j
pipjd(Vi ,Vj) log d(Vi ,Vj)
where pi = |Vi ||V | .
![Page 45: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/45.jpg)
Triangle removal lemma: new proof idea
V1
V2 V3
Key lemma
If there are at most α|V1||V2||V3| triangles across V1,V2,V3, then
there are 1 ≤ i < j ≤ 3 and equitable partitions Qi of Vi and Qj of
Vj each with at most 2α−O(1)
parts such that there are at least1
10 |Qi ||Qj | pairs (X ,Y ) ∈ Qi × Qj with d(X ,Y ) < 2α1/3.
Definition: Mean entropy density
For a vertex partition P : V = V1 ∪ . . . ∪ Vk ,
h(P) =∑i ,j
pipjd(Vi ,Vj) log d(Vi ,Vj)
where pi = |Vi ||V | .
![Page 46: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/46.jpg)
Property Testing
Goal
Quickly distinguish between objects that have a property from
objects that are far from having that property.
Introduced by Rubinfeld and Sudan in 1996 and investigated byGoldreich, Goldwasser, and Ron.
Testable graph properties
A graph property P is testable if for each ε > 0 there is a
randomized algorithm which in constant time (depending on ε and
P) distinguishes with probability at least 2/3 between graphs
having P and graphs which are ε-far from having P.
![Page 47: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/47.jpg)
Property Testing
Goal
Quickly distinguish between objects that have a property from
objects that are far from having that property.
Introduced by Rubinfeld and Sudan in 1996 and investigated byGoldreich, Goldwasser, and Ron.
Testable graph properties
A graph property P is testable if for each ε > 0 there is a
randomized algorithm which in constant time (depending on ε and
P) distinguishes with probability at least 2/3 between graphs
having P and graphs which are ε-far from having P.
![Page 48: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/48.jpg)
Property Testing
Goal
Quickly distinguish between objects that have a property from
objects that are far from having that property.
Introduced by Rubinfeld and Sudan in 1996 and investigated byGoldreich, Goldwasser, and Ron.
Testable graph properties
A graph property P is testable if for each ε > 0 there is a
randomized algorithm which in constant time (depending on ε and
P) distinguishes with probability at least 2/3 between graphs
having P and graphs which are ε-far from having P.
![Page 49: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/49.jpg)
Property Testing
Goal
Quickly distinguish between objects that have a property from
objects that are far from having that property.
Introduced by Rubinfeld and Sudan in 1996 and investigated byGoldreich, Goldwasser, and Ron.
Testable graph properties
A graph property P is testable if for each ε > 0 there is a
randomized algorithm which in constant time (depending on ε and
P) distinguishes with probability at least 2/3 between graphs
having P and graphs which are ε-far from having P.
![Page 50: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/50.jpg)
What is testable?
Claim
Triangle-freeness is testable.
Proof: Pick 1/δ random triples of vertices.Accept if no triple of vertices makes a triangle; reject otherwise.If the graph is triangle-free, then the algorithm accepts.If the graph is ε-far from being triangle-free, then it accepts withprobability ≤ (1− 6δ)1/δ ≤ e−6 < .01.
Further, H-freeness is testable (by the graph removal lemma).
Theorem (Alon-Fischer-Krivelevich-Szegedy 2000)
Induced H-freeness is testable.
Theorem (Alon-Shapira 2008)
Every hereditary graph property is testable.
![Page 51: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/51.jpg)
What is testable?
Claim
Triangle-freeness is testable.
Proof: Pick 1/δ random triples of vertices.
Accept if no triple of vertices makes a triangle; reject otherwise.If the graph is triangle-free, then the algorithm accepts.If the graph is ε-far from being triangle-free, then it accepts withprobability ≤ (1− 6δ)1/δ ≤ e−6 < .01.
Further, H-freeness is testable (by the graph removal lemma).
Theorem (Alon-Fischer-Krivelevich-Szegedy 2000)
Induced H-freeness is testable.
Theorem (Alon-Shapira 2008)
Every hereditary graph property is testable.
![Page 52: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/52.jpg)
What is testable?
Claim
Triangle-freeness is testable.
Proof: Pick 1/δ random triples of vertices.Accept if no triple of vertices makes a triangle; reject otherwise.
If the graph is triangle-free, then the algorithm accepts.If the graph is ε-far from being triangle-free, then it accepts withprobability ≤ (1− 6δ)1/δ ≤ e−6 < .01.
Further, H-freeness is testable (by the graph removal lemma).
Theorem (Alon-Fischer-Krivelevich-Szegedy 2000)
Induced H-freeness is testable.
Theorem (Alon-Shapira 2008)
Every hereditary graph property is testable.
![Page 53: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/53.jpg)
What is testable?
Claim
Triangle-freeness is testable.
Proof: Pick 1/δ random triples of vertices.Accept if no triple of vertices makes a triangle; reject otherwise.If the graph is triangle-free, then the algorithm accepts.
If the graph is ε-far from being triangle-free, then it accepts withprobability ≤ (1− 6δ)1/δ ≤ e−6 < .01.
Further, H-freeness is testable (by the graph removal lemma).
Theorem (Alon-Fischer-Krivelevich-Szegedy 2000)
Induced H-freeness is testable.
Theorem (Alon-Shapira 2008)
Every hereditary graph property is testable.
![Page 54: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/54.jpg)
What is testable?
Claim
Triangle-freeness is testable.
Proof: Pick 1/δ random triples of vertices.Accept if no triple of vertices makes a triangle; reject otherwise.If the graph is triangle-free, then the algorithm accepts.If the graph is ε-far from being triangle-free, then it accepts withprobability ≤ (1− 6δ)1/δ ≤ e−6 < .01.
Further, H-freeness is testable (by the graph removal lemma).
Theorem (Alon-Fischer-Krivelevich-Szegedy 2000)
Induced H-freeness is testable.
Theorem (Alon-Shapira 2008)
Every hereditary graph property is testable.
![Page 55: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/55.jpg)
What is testable?
Claim
Triangle-freeness is testable.
Proof: Pick 1/δ random triples of vertices.Accept if no triple of vertices makes a triangle; reject otherwise.If the graph is triangle-free, then the algorithm accepts.If the graph is ε-far from being triangle-free, then it accepts withprobability ≤ (1− 6δ)1/δ ≤ e−6 < .01.
Further, H-freeness is testable (by the graph removal lemma).
Theorem (Alon-Fischer-Krivelevich-Szegedy 2000)
Induced H-freeness is testable.
Theorem (Alon-Shapira 2008)
Every hereditary graph property is testable.
![Page 56: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/56.jpg)
What is testable?
Claim
Triangle-freeness is testable.
Proof: Pick 1/δ random triples of vertices.Accept if no triple of vertices makes a triangle; reject otherwise.If the graph is triangle-free, then the algorithm accepts.If the graph is ε-far from being triangle-free, then it accepts withprobability ≤ (1− 6δ)1/δ ≤ e−6 < .01.
Further, H-freeness is testable (by the graph removal lemma).
Theorem (Alon-Fischer-Krivelevich-Szegedy 2000)
Induced H-freeness is testable.
Theorem (Alon-Shapira 2008)
Every hereditary graph property is testable.
![Page 57: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/57.jpg)
What is testable?
Claim
Triangle-freeness is testable.
Proof: Pick 1/δ random triples of vertices.Accept if no triple of vertices makes a triangle; reject otherwise.If the graph is triangle-free, then the algorithm accepts.If the graph is ε-far from being triangle-free, then it accepts withprobability ≤ (1− 6δ)1/δ ≤ e−6 < .01.
Further, H-freeness is testable (by the graph removal lemma).
Theorem (Alon-Fischer-Krivelevich-Szegedy 2000)
Induced H-freeness is testable.
Theorem (Alon-Shapira 2008)
Every hereditary graph property is testable.
![Page 58: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/58.jpg)
Induced graph removal lemma
Lemma (Alon-Fischer-Krivelevich-Szegedy 2000)
For each ε > 0 and graph H on h vertices there is δ > 0 such that
every graph on n vertices with at most δnh induced copies of H
can be made induced H-free by adding or removing εn2 edges.
Proof developed the strong regularity lemma, which is proved byrepeated use of Szemeredi’s regularity lemma and gives awowzer-type bound.
Wowzer function is given by W (1) = 2 and W (n) = T (W (n− 1)).
Problem (Alon)
Estimate the dependence of δ on ε.
In particular, can the wowzer-type bound be improved?
![Page 59: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/59.jpg)
Induced graph removal lemma
Lemma (Alon-Fischer-Krivelevich-Szegedy 2000)
For each ε > 0 and graph H on h vertices there is δ > 0 such that
every graph on n vertices with at most δnh induced copies of H
can be made induced H-free by adding or removing εn2 edges.
Proof developed the strong regularity lemma,
which is proved byrepeated use of Szemeredi’s regularity lemma and gives awowzer-type bound.
Wowzer function is given by W (1) = 2 and W (n) = T (W (n− 1)).
Problem (Alon)
Estimate the dependence of δ on ε.
In particular, can the wowzer-type bound be improved?
![Page 60: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/60.jpg)
Induced graph removal lemma
Lemma (Alon-Fischer-Krivelevich-Szegedy 2000)
For each ε > 0 and graph H on h vertices there is δ > 0 such that
every graph on n vertices with at most δnh induced copies of H
can be made induced H-free by adding or removing εn2 edges.
Proof developed the strong regularity lemma, which is proved byrepeated use of Szemeredi’s regularity lemma and gives awowzer-type bound.
Wowzer function is given by W (1) = 2 and W (n) = T (W (n− 1)).
Problem (Alon)
Estimate the dependence of δ on ε.
In particular, can the wowzer-type bound be improved?
![Page 61: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/61.jpg)
Induced graph removal lemma
Lemma (Alon-Fischer-Krivelevich-Szegedy 2000)
For each ε > 0 and graph H on h vertices there is δ > 0 such that
every graph on n vertices with at most δnh induced copies of H
can be made induced H-free by adding or removing εn2 edges.
Proof developed the strong regularity lemma, which is proved byrepeated use of Szemeredi’s regularity lemma and gives awowzer-type bound.
Wowzer function is given by W (1) = 2 and W (n) = T (W (n− 1)).
Problem (Alon)
Estimate the dependence of δ on ε.
In particular, can the wowzer-type bound be improved?
![Page 62: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/62.jpg)
Induced graph removal lemma
Lemma (Alon-Fischer-Krivelevich-Szegedy 2000)
For each ε > 0 and graph H on h vertices there is δ > 0 such that
every graph on n vertices with at most δnh induced copies of H
can be made induced H-free by adding or removing εn2 edges.
Proof developed the strong regularity lemma, which is proved byrepeated use of Szemeredi’s regularity lemma and gives awowzer-type bound.
Wowzer function is given by W (1) = 2 and W (n) = T (W (n− 1)).
Problem (Alon)
Estimate the dependence of δ on ε.
In particular, can the wowzer-type bound be improved?
![Page 63: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/63.jpg)
Induced graph removal lemma
Lemma (Alon-Fischer-Krivelevich-Szegedy 2000)
For each ε > 0 and graph H on h vertices there is δ > 0 such that
every graph on n vertices with at most δnh induced copies of H
can be made induced H-free by adding or removing εn2 edges.
Proof developed the strong regularity lemma, which is proved byrepeated use of Szemeredi’s regularity lemma and gives awowzer-type bound.
Wowzer function is given by W (1) = 2 and W (n) = T (W (n− 1)).
Problem (Alon)
Estimate the dependence of δ on ε.
In particular, can the wowzer-type bound be improved?
![Page 64: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/64.jpg)
Wowzer and Tower
Wowzer function is given by W (1) = 2 and W (n) = T (W (n− 1)).
Theorem (Conlon-F. 2012)
W (ε−Ω(1)) parts are needed in the strong regularity lemma.
Lemma (Alon-Fischer-Krivelevich-Szegedy 2000)
For each ε > 0 and graph H on h vertices there is δ > 0 such that
every graph on n vertices with at most δnh induced copies of H
can be made H-free by adding or removing εn2 edges.
Theorem (Conlon-F. 2012)
We can take δ−1 = T (ε−O(1)) in the induced graph removal lemma.
Further, a tower-type bound holds for testing hereditary properties.
![Page 65: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/65.jpg)
Wowzer and Tower
Wowzer function is given by W (1) = 2 and W (n) = T (W (n− 1)).
Theorem (Conlon-F. 2012)
W (ε−Ω(1)) parts are needed in the strong regularity lemma.
Lemma (Alon-Fischer-Krivelevich-Szegedy 2000)
For each ε > 0 and graph H on h vertices there is δ > 0 such that
every graph on n vertices with at most δnh induced copies of H
can be made H-free by adding or removing εn2 edges.
Theorem (Conlon-F. 2012)
We can take δ−1 = T (ε−O(1)) in the induced graph removal lemma.
Further, a tower-type bound holds for testing hereditary properties.
![Page 66: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/66.jpg)
Wowzer and Tower
Wowzer function is given by W (1) = 2 and W (n) = T (W (n− 1)).
Theorem (Conlon-F. 2012)
W (ε−Ω(1)) parts are needed in the strong regularity lemma.
Lemma (Alon-Fischer-Krivelevich-Szegedy 2000)
For each ε > 0 and graph H on h vertices there is δ > 0 such that
every graph on n vertices with at most δnh induced copies of H
can be made H-free by adding or removing εn2 edges.
Theorem (Conlon-F. 2012)
We can take δ−1 = T (ε−O(1)) in the induced graph removal lemma.
Further, a tower-type bound holds for testing hereditary properties.
![Page 67: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/67.jpg)
Wowzer and Tower
Wowzer function is given by W (1) = 2 and W (n) = T (W (n− 1)).
Theorem (Conlon-F. 2012)
W (ε−Ω(1)) parts are needed in the strong regularity lemma.
Lemma (Alon-Fischer-Krivelevich-Szegedy 2000)
For each ε > 0 and graph H on h vertices there is δ > 0 such that
every graph on n vertices with at most δnh induced copies of H
can be made H-free by adding or removing εn2 edges.
Theorem (Conlon-F. 2012)
We can take δ−1 = T (ε−O(1)) in the induced graph removal lemma.
Further, a tower-type bound holds for testing hereditary properties.
![Page 68: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/68.jpg)
Wowzer and Tower
Wowzer function is given by W (1) = 2 and W (n) = T (W (n− 1)).
Theorem (Conlon-F. 2012)
W (ε−Ω(1)) parts are needed in the strong regularity lemma.
Lemma (Alon-Fischer-Krivelevich-Szegedy 2000)
For each ε > 0 and graph H on h vertices there is δ > 0 such that
every graph on n vertices with at most δnh induced copies of H
can be made H-free by adding or removing εn2 edges.
Theorem (Conlon-F. 2012)
We can take δ−1 = T (ε−O(1)) in the induced graph removal lemma.
Further, a tower-type bound holds for testing hereditary properties.
![Page 69: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/69.jpg)
Algorithmic regularity lemma
Question
Can the regularity lemma be made algorithmic?
Importance lies in that it would give approximation algorithms fora large class of computationally hard problems.
Bottleneck: Determine if a pair of parts X ,Y is ε-regular, and, ifnot, find X ′ ⊂ X , Y ′ ⊂ Y realizing this.
Theorem (Alon-Duke-Lefmann-Rodl-Yuster)
Yes!
Up to changing ε, checking regularity is the same as showing thedensity of C4 with edges across X × Y is ≈ d(X ,Y )4.
Theorem (Kohayakawa-Rodl-Thoma)
Can be done in O(n2) time.
![Page 70: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/70.jpg)
Algorithmic regularity lemma
Question
Can the regularity lemma be made algorithmic?
Importance lies in that it would give approximation algorithms fora large class of computationally hard problems.
Bottleneck: Determine if a pair of parts X ,Y is ε-regular, and, ifnot, find X ′ ⊂ X , Y ′ ⊂ Y realizing this.
Theorem (Alon-Duke-Lefmann-Rodl-Yuster)
Yes!
Up to changing ε, checking regularity is the same as showing thedensity of C4 with edges across X × Y is ≈ d(X ,Y )4.
Theorem (Kohayakawa-Rodl-Thoma)
Can be done in O(n2) time.
![Page 71: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/71.jpg)
Algorithmic regularity lemma
Question
Can the regularity lemma be made algorithmic?
Importance lies in that it would give approximation algorithms fora large class of computationally hard problems.
Bottleneck: Determine if a pair of parts X ,Y is ε-regular, and, ifnot, find X ′ ⊂ X , Y ′ ⊂ Y realizing this.
Theorem (Alon-Duke-Lefmann-Rodl-Yuster)
Yes!
Up to changing ε, checking regularity is the same as showing thedensity of C4 with edges across X × Y is ≈ d(X ,Y )4.
Theorem (Kohayakawa-Rodl-Thoma)
Can be done in O(n2) time.
![Page 72: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/72.jpg)
Algorithmic regularity lemma
Question
Can the regularity lemma be made algorithmic?
Importance lies in that it would give approximation algorithms fora large class of computationally hard problems.
Bottleneck: Determine if a pair of parts X ,Y is ε-regular, and, ifnot, find X ′ ⊂ X , Y ′ ⊂ Y realizing this.
Theorem (Alon-Duke-Lefmann-Rodl-Yuster)
Yes!
Up to changing ε, checking regularity is the same as showing thedensity of C4 with edges across X × Y is ≈ d(X ,Y )4.
Theorem (Kohayakawa-Rodl-Thoma)
Can be done in O(n2) time.
![Page 73: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/73.jpg)
Algorithmic regularity lemma
Question
Can the regularity lemma be made algorithmic?
Importance lies in that it would give approximation algorithms fora large class of computationally hard problems.
Bottleneck: Determine if a pair of parts X ,Y is ε-regular, and, ifnot, find X ′ ⊂ X , Y ′ ⊂ Y realizing this.
Theorem (Alon-Duke-Lefmann-Rodl-Yuster)
Yes!
Up to changing ε, checking regularity is the same as showing thedensity of C4 with edges across X × Y is ≈ d(X ,Y )4.
Theorem (Kohayakawa-Rodl-Thoma)
Can be done in O(n2) time.
![Page 74: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/74.jpg)
Algorithmic regularity lemma
Question
Can the regularity lemma be made algorithmic?
Importance lies in that it would give approximation algorithms fora large class of computationally hard problems.
Bottleneck: Determine if a pair of parts X ,Y is ε-regular, and, ifnot, find X ′ ⊂ X , Y ′ ⊂ Y realizing this.
Theorem (Alon-Duke-Lefmann-Rodl-Yuster)
Yes!
Up to changing ε, checking regularity is the same as showing thedensity of C4 with edges across X × Y is ≈ d(X ,Y )4.
Theorem (Kohayakawa-Rodl-Thoma)
Can be done in O(n2) time.
![Page 75: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/75.jpg)
Counting cliques
Algorithmic problem
Count the number of cliques of order k in a graph on n vertices.
Zuckerman showed NP-hard to approximate the size of the largestclique within a factor n1−ε.
How fast can we approximate the count within an additive εnk?
![Page 76: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/76.jpg)
Counting cliques
Algorithmic problem
Count the number of cliques of order k in a graph on n vertices.
Zuckerman showed NP-hard to approximate the size of the largestclique within a factor n1−ε.
How fast can we approximate the count within an additive εnk?
![Page 77: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/77.jpg)
Counting cliques
Algorithmic problem
Count the number of cliques of order k in a graph on n vertices.
Zuckerman showed NP-hard to approximate the size of the largestclique within a factor n1−ε.
How fast can we approximate the count within an additive εnk?
![Page 78: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/78.jpg)
Counting cliques
Algorithmic problem
Count the number of cliques of order k in a graph on n vertices
within an additive εnk .
A simple randomized algorithm gives 99% certainty:
Sample 10/ε2 random k-sets of vertices.
What about deterministic algorithms?
Can use the regularity method!
![Page 79: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/79.jpg)
Counting cliques
Algorithmic problem
Count the number of cliques of order k in a graph on n vertices
within an additive εnk .
A simple randomized algorithm gives 99% certainty:
Sample 10/ε2 random k-sets of vertices.
What about deterministic algorithms?
Can use the regularity method!
![Page 80: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/80.jpg)
Counting cliques
Algorithmic problem
Count the number of cliques of order k in a graph on n vertices
within an additive εnk .
A simple randomized algorithm gives 99% certainty:
Sample 10/ε2 random k-sets of vertices.
What about deterministic algorithms?
Can use the regularity method!
![Page 81: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/81.jpg)
Counting cliques
Algorithmic problem
Count the number of cliques of order k in a graph on n vertices
within an additive εnk .
A simple randomized algorithm gives 99% certainty:
Sample 10/ε2 random k-sets of vertices.
What about deterministic algorithms?
Can use the regularity method!
![Page 82: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/82.jpg)
Counting cliques
Algorithmic problem
Count the number of cliques of order k in a graph on n vertices
within an additive εnk .
A simple randomized algorithm gives 99% certainty:
Sample 10/ε2 random k-sets of vertices.
What about deterministic algorithms?
Can use the regularity method!
![Page 83: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/83.jpg)
Counting cliques
Algorithmic problem
Count the number of cliques of order k in a graph on n vertices
within an additive εnk deterministically.
Duke, Lefmann, Rodl 1996: Can be done in time 2(k/ε)O(1)nω.
Theorem (F.-L. M. Lovasz-Zhao 2016)
Can be done in time O(ε−k2n + nω).
Corollary
We can approximate the count of K1000 in a graph on n vertices
within an additive n1000−10−6in time O(n2.4).
![Page 84: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/84.jpg)
Counting cliques
Algorithmic problem
Count the number of cliques of order k in a graph on n vertices
within an additive εnk deterministically.
Duke, Lefmann, Rodl 1996: Can be done in time 2(k/ε)O(1)nω.
Theorem (F.-L. M. Lovasz-Zhao 2016)
Can be done in time O(ε−k2n + nω).
Corollary
We can approximate the count of K1000 in a graph on n vertices
within an additive n1000−10−6in time O(n2.4).
![Page 85: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/85.jpg)
Counting cliques
Algorithmic problem
Count the number of cliques of order k in a graph on n vertices
within an additive εnk deterministically.
Duke, Lefmann, Rodl 1996: Can be done in time 2(k/ε)O(1)nω.
Theorem (F.-L. M. Lovasz-Zhao 2016)
Can be done in time O(ε−k2n + nω).
Corollary
We can approximate the count of K1000 in a graph on n vertices
within an additive n1000−10−6in time O(n2.4).
![Page 86: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/86.jpg)
Counting cliques
Algorithmic problem
Count the number of cliques of order k in a graph on n vertices
within an additive εnk deterministically.
Duke, Lefmann, Rodl 1996: Can be done in time 2(k/ε)O(1)nω.
Theorem (F.-L. M. Lovasz-Zhao 2016)
Can be done in time O(ε−k2n + nω).
Corollary
We can approximate the count of K1000 in a graph on n vertices
within an additive n1000−10−6in time O(n2.4).
![Page 87: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/87.jpg)
Regularity method for sparse graphs?
The original regularity method is only useful for dense graphs.
Kohayakawa, Rodl, and Scott proved a sparse regularity lemma.
Main Problem
Prove a counting lemma in sparse graphs.
Caveat: A general counting lemma in sparse graphs cannot hold.
How about within subgraphs of sparse pseudorandom graphs?
Theorem (Conlon-F.-Zhao)
A sparse counting lemma in graphs and hypergraphs.
![Page 88: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/88.jpg)
Regularity method for sparse graphs?
The original regularity method is only useful for dense graphs.
Kohayakawa, Rodl, and Scott proved a sparse regularity lemma.
Main Problem
Prove a counting lemma in sparse graphs.
Caveat: A general counting lemma in sparse graphs cannot hold.
How about within subgraphs of sparse pseudorandom graphs?
Theorem (Conlon-F.-Zhao)
A sparse counting lemma in graphs and hypergraphs.
![Page 89: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/89.jpg)
Regularity method for sparse graphs?
The original regularity method is only useful for dense graphs.
Kohayakawa, Rodl, and Scott proved a sparse regularity lemma.
Main Problem
Prove a counting lemma in sparse graphs.
Caveat: A general counting lemma in sparse graphs cannot hold.
How about within subgraphs of sparse pseudorandom graphs?
Theorem (Conlon-F.-Zhao)
A sparse counting lemma in graphs and hypergraphs.
![Page 90: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/90.jpg)
Regularity method for sparse graphs?
The original regularity method is only useful for dense graphs.
Kohayakawa, Rodl, and Scott proved a sparse regularity lemma.
Main Problem
Prove a counting lemma in sparse graphs.
Caveat: A general counting lemma in sparse graphs cannot hold.
How about within subgraphs of sparse pseudorandom graphs?
Theorem (Conlon-F.-Zhao)
A sparse counting lemma in graphs and hypergraphs.
![Page 91: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/91.jpg)
Regularity method for sparse graphs?
The original regularity method is only useful for dense graphs.
Kohayakawa, Rodl, and Scott proved a sparse regularity lemma.
Main Problem
Prove a counting lemma in sparse graphs.
Caveat: A general counting lemma in sparse graphs cannot hold.
How about within subgraphs of sparse pseudorandom graphs?
Theorem (Conlon-F.-Zhao)
A sparse counting lemma in graphs and hypergraphs.
![Page 92: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/92.jpg)
Regularity method for sparse graphs?
The original regularity method is only useful for dense graphs.
Kohayakawa, Rodl, and Scott proved a sparse regularity lemma.
Main Problem
Prove a counting lemma in sparse graphs.
Caveat: A general counting lemma in sparse graphs cannot hold.
How about within subgraphs of sparse pseudorandom graphs?
Theorem (Conlon-F.-Zhao)
A sparse counting lemma in graphs and hypergraphs.
![Page 93: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/93.jpg)
Regularity method for sparse graphs?
The original regularity method is only useful for dense graphs.
Kohayakawa, Rodl, and Scott proved a sparse regularity lemma.
Main Problem
Prove a counting lemma in sparse graphs.
Caveat: A general counting lemma in sparse graphs cannot hold.
How about within subgraphs of sparse pseudorandom graphs?
Theorem (Conlon-F.-Zhao)
A sparse counting lemma in graphs and hypergraphs.
![Page 94: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/94.jpg)
Green–Tao Theorem (2008)
The primes contain arbitrarily long arithmetic progressions.
Examples:
3, 5, 7
5, 11, 17, 23, 29
7, 37, 67, 97, 127, 157
Longest known: 26 terms
![Page 95: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/95.jpg)
Green–Tao Theorem (2008)
The primes contain arbitrarily long arithmetic progressions (AP).
Szemeredi’s Theorem (1975)
Every positive density subset of contains arbitrarily long APs.
(upper) density of A ⊂ is lim supN→∞
|A ∩ [N]|N
[N] := 1, 2, . . . ,N
P = prime numbers
Prime number theorem:|P ∩ [N]|
N∼ 1
logN
![Page 96: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/96.jpg)
Proof strategy of the Green–Tao theorem
Step 1:
Relative Szemeredi theorem (informally)
If S ⊂ satisfies certain pseudorandomness conditions, then every
subset of S of relative positive density contains long APs.
Step 2:Construct a superset of the primes satisfying these conditions.
P = prime numbers, Q = “almost primes”
P ⊆ Q with relative positive density, i.e.,|P ∩ [N]||Q ∩ [N]|
> δ
![Page 97: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/97.jpg)
Proof strategy of the Green–Tao theorem
Step 1:
Relative Szemeredi theorem (informally)
If S ⊂ satisfies certain pseudorandomness conditions, then every
subset of S of relative positive density contains long APs.
Step 2:Construct a superset of the primes satisfying these conditions.
P = prime numbers, Q = “almost primes”
P ⊆ Q with relative positive density, i.e.,|P ∩ [N]||Q ∩ [N]|
> δ
![Page 98: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/98.jpg)
Proof strategy of the Green–Tao theorem
Step 1:
Relative Szemeredi theorem (informally)
If S ⊂ satisfies certain pseudorandomness conditions, then every
subset of S of relative positive density contains long APs.
Step 2:Construct a superset of the primes satisfying these conditions.
P = prime numbers, Q = “almost primes”
P ⊆ Q with relative positive density, i.e.,|P ∩ [N]||Q ∩ [N]|
> δ
![Page 99: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/99.jpg)
Relative Szemeredi theorem
Relative Szemeredi theorem (informally)
If S ⊂ satisfies certain pseudorandomness conditions, then every
subset of S of positive density contains long APs.
What pseudorandomness conditions?
Green–Tao:1 Linear forms condition
2 Correlation condition
← no longer needed
A natural question (asked by Green, Gowers, . . . )
Does a relative Szemeredi theorem hold with weaker and more
natural hypotheses?
Our main result
Yes! A weak linear forms condition suffices.
![Page 100: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/100.jpg)
Relative Szemeredi theorem
Relative Szemeredi theorem (informally)
If S ⊂ satisfies certain pseudorandomness conditions, then every
subset of S of positive density contains long APs.
What pseudorandomness conditions?
Green–Tao:1 Linear forms condition
2 Correlation condition
← no longer needed
A natural question (asked by Green, Gowers, . . . )
Does a relative Szemeredi theorem hold with weaker and more
natural hypotheses?
Our main result
Yes! A weak linear forms condition suffices.
![Page 101: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/101.jpg)
Relative Szemeredi theorem
Relative Szemeredi theorem (informally)
If S ⊂ satisfies certain pseudorandomness conditions, then every
subset of S of positive density contains long APs.
What pseudorandomness conditions?
Green–Tao:1 Linear forms condition
2 Correlation condition ← no longer needed
A natural question (asked by Green, Gowers, . . . )
Does a relative Szemeredi theorem hold with weaker and more
natural hypotheses?
Our main result
Yes! A weak linear forms condition suffices.
![Page 102: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/102.jpg)
Roth’s theorem
Roth’s theorem (1952)
If A ⊆ [N] is 3-AP-free, then |A| = o(N).
[N] := 1, 2, . . . ,N
3-AP = 3-term arithmetic progression
It’ll be easier (and equivalent) to work in ZN := Z/NZ.
![Page 103: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/103.jpg)
Proof of Roth’s theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Given A, construct tripartitegraph GA with vertex setsX = Y = Z = ZN .
Triangle xyz in GA ⇐⇒2x + y , x − z , −y − 2z ∈ A
It’s a 3-AP with diff −x − y − z
GA
X
Y Z
No triangles? Only triangles ←→ trivial 3-APs with diff 0.Every edge of the graph is contained in exactly one triangle(the one with x + y + z = 0).
![Page 104: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/104.jpg)
Proof of Roth’s theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Given A, construct tripartitegraph GA with vertex setsX = Y = Z = ZN .
Triangle xyz in GA ⇐⇒2x + y , x − z , −y − 2z ∈ A
It’s a 3-AP with diff −x − y − z
GA
X
Y Z
x
y
x ∼ y iff
2x + y ∈ A
No triangles? Only triangles ←→ trivial 3-APs with diff 0.Every edge of the graph is contained in exactly one triangle(the one with x + y + z = 0).
![Page 105: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/105.jpg)
Proof of Roth’s theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Given A, construct tripartitegraph GA with vertex setsX = Y = Z = ZN .
Triangle xyz in GA ⇐⇒2x + y , x − z , −y − 2z ∈ A
It’s a 3-AP with diff −x − y − z
GA
X
Y Z
x
z
x ∼ z iff
x − z ∈ A
No triangles? Only triangles ←→ trivial 3-APs with diff 0.Every edge of the graph is contained in exactly one triangle(the one with x + y + z = 0).
![Page 106: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/106.jpg)
Proof of Roth’s theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Given A, construct tripartitegraph GA with vertex setsX = Y = Z = ZN .
Triangle xyz in GA ⇐⇒2x + y , x − z , −y − 2z ∈ A
It’s a 3-AP with diff −x − y − z
GA
X
Y Z
y z
y ∼ z iff
−y − 2z ∈ A
No triangles? Only triangles ←→ trivial 3-APs with diff 0.Every edge of the graph is contained in exactly one triangle(the one with x + y + z = 0).
![Page 107: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/107.jpg)
Proof of Roth’s theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Given A, construct tripartitegraph GA with vertex setsX = Y = Z = ZN .
Triangle xyz in GA ⇐⇒2x + y , x − z , −y − 2z ∈ A
It’s a 3-AP with diff −x − y − z
GA
X
Y Z
x
y z
x ∼ y iff
2x + y ∈ A
x ∼ z iff
x − z ∈ A
y ∼ z iff
−y − 2z ∈ A
No triangles? Only triangles ←→ trivial 3-APs with diff 0.Every edge of the graph is contained in exactly one triangle(the one with x + y + z = 0).
![Page 108: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/108.jpg)
Proof of Roth’s theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Given A, construct tripartitegraph GA with vertex setsX = Y = Z = ZN .
Triangle xyz in GA ⇐⇒2x + y , x − z , −y − 2z ∈ A
It’s a 3-AP with diff −x − y − z
GA
X
Y Z
x
y z
x ∼ y iff
2x + y ∈ A
x ∼ z iff
x − z ∈ A
y ∼ z iff
−y − 2z ∈ A
No triangles? Only triangles ←→ trivial 3-APs with diff 0.Every edge of the graph is contained in exactly one triangle(the one with x + y + z = 0).
![Page 109: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/109.jpg)
Proof of Roth’s theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Given A, construct tripartitegraph GA with vertex setsX = Y = Z = ZN .
Triangle xyz in GA ⇐⇒2x + y , x − z , −y − 2z ∈ A
It’s a 3-AP with diff −x − y − z
GA
X
Y Z
x
y z
x ∼ y iff
2x + y ∈ A
x ∼ z iff
x − z ∈ A
y ∼ z iff
−y − 2z ∈ A
No triangles? Only triangles ←→ trivial 3-APs with diff 0.Every edge of the graph is contained in exactly one triangle(the one with x + y + z = 0).
![Page 110: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/110.jpg)
Proof of Roth’s theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Given A, construct tripartitegraph GA with vertex setsX = Y = Z = ZN .
Triangle xyz in GA ⇐⇒2x + y , x − z , −y − 2z ∈ A
It’s a 3-AP with diff −x − y − z
GA
X
Y Z
x
y z
x ∼ y iff
2x + y ∈ A
x ∼ z iff
x − z ∈ A
y ∼ z iff
−y − 2z ∈ ANo triangles?
No triangles? Only triangles ←→ trivial 3-APs with
diff 0.Every edge of the graph is contained in exactly one triangle(the one with x + y + z = 0).
![Page 111: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/111.jpg)
Proof of Roth’s theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Given A, construct tripartitegraph GA with vertex setsX = Y = Z = ZN .
Triangle xyz in GA ⇐⇒2x + y , x − z , −y − 2z ∈ A
It’s a 3-AP with diff −x − y − z
GA
X
Y Z
x
y z
x ∼ y iff
2x + y ∈ A
x ∼ z iff
x − z ∈ A
y ∼ z iff
−y − 2z ∈ A
No triangles? Only triangles ←→ trivial 3-APs with diff 0.
Every edge of the graph is contained in exactly one triangle(the one with x + y + z = 0).
![Page 112: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/112.jpg)
Proof of Roth’s theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Given A, construct tripartitegraph GA with vertex setsX = Y = Z = ZN .
Triangle xyz in GA ⇐⇒2x + y , x − z , −y − 2z ∈ A
It’s a 3-AP with diff −x − y − z
GA
X
Y Z
x
y z
x ∼ y iff
2x + y ∈ A
x ∼ z iff
x − z ∈ A
y ∼ z iff
−y − 2z ∈ A
No triangles? Only triangles ←→ trivial 3-APs with diff 0.Every edge of the graph is contained in exactly one triangle(the one with x + y + z = 0).
![Page 113: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/113.jpg)
Proof of Roth’s theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Constructed a graph with
3N vertices
3N|A| edges
every edge in exactly one triangle
Theorem (Ruzsa & Szemeredi ’76)
If every edge in a graph G = (V ,E ) is contained in exactly one
triangle, then |E | = o(|V |2).
(a consequence of the triangle removal lemma)
So 3N|A| = o(N2). Thus |A| = o(N).
![Page 114: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/114.jpg)
Proof of Roth’s theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Constructed a graph with
3N vertices
3N|A| edges
every edge in exactly one triangle
Theorem (Ruzsa & Szemeredi ’76)
If every edge in a graph G = (V ,E ) is contained in exactly one
triangle, then |E | = o(|V |2).
(a consequence of the triangle removal lemma)
So 3N|A| = o(N2). Thus |A| = o(N).
![Page 115: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/115.jpg)
Relative Roth theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Relative Roth theorem (Conlon, F., Zhao)
If S ⊆ ZN satisfies the 3-linear forms condition,
and A ⊆ S is 3-AP-free, then |A| = o(|S |).
ZN
ZNZN
GS x
y z
x ∼ y iff
2x + y ∈ S
x ∼ z iff
x − z ∈ S
y ∼ z iff
−y − 2z ∈ S
3-linear forms condition:GS has asymp. the expected number ofembeddings of K2,2,2 & its subgraphs(compared to random graph of same density)
K2,2,2 & subgraphs,
e.g.,
![Page 116: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/116.jpg)
Relative Roth theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Relative Roth theorem (Conlon, F., Zhao)
If S ⊆ ZN satisfies the 3-linear forms condition,
and A ⊆ S is 3-AP-free, then |A| = o(|S |).
ZN
ZNZN
GS x
y z
x ∼ y iff
2x + y ∈ S
x ∼ z iff
x − z ∈ S
y ∼ z iff
−y − 2z ∈ S
3-linear forms condition:GS has asymp. the expected number ofembeddings of K2,2,2 & its subgraphs(compared to random graph of same density)
K2,2,2 & subgraphs,
e.g.,
![Page 117: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/117.jpg)
Relative Roth theorem
Roth’s theorem (1952)
If A ⊆ ZN is 3-AP-free, then |A| = o(N).
Relative Roth theorem (Conlon, F., Zhao)
If S ⊆ ZN satisfies the 3-linear forms condition,
and A ⊆ S is 3-AP-free, then |A| = o(|S |).
ZN
ZNZN
GS x
y z
x ∼ y iff
2x + y ∈ S
x ∼ z iff
x − z ∈ S
y ∼ z iff
−y − 2z ∈ S
3-linear forms condition:GS has asymp. the expected number ofembeddings of K2,2,2 & its subgraphs(compared to random graph of same density)
K2,2,2 & subgraphs,
e.g.,
![Page 118: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/118.jpg)
Relative Szemeredi theorem
Relative Szemeredi theorem (Conlon, F., Zhao)
Fix k ≥ 3. If S ⊆ ZN satisfies the k-linear forms condition,
and A ⊆ S is k-AP-free, then |A| = o(|S |).
k = 4: build a weighted 4-partite 3-uniform hypergraph
Vertex sets W = X = Y = Z = ZN
x y
zw
X Y
ZWxyz ∈ E ⇐⇒ 3w + 2x + y ∈ S
wxz ∈ E ⇐⇒ 2w + x − z ∈ S
wyz ∈ E ⇐⇒ w − y − 2z ∈ S
xyz ∈ E ⇐⇒ −x − 2y − 3z ∈ S
common diff: −w − x − y − z
4-linear forms condition: correct count of the 2-blow-up of the
simplex K(3)4 (as well as its subgraphs)
![Page 119: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/119.jpg)
Relative Szemeredi theorem
Relative Szemeredi theorem (Conlon, F., Zhao)
Fix k ≥ 3. If S ⊆ ZN satisfies the k-linear forms condition,
and A ⊆ S is k-AP-free, then |A| = o(|S |).
k = 4: build a weighted 4-partite 3-uniform hypergraph
Vertex sets W = X = Y = Z = ZN
x y
zw
X Y
ZWxyz ∈ E ⇐⇒ 3w + 2x + y ∈ S
wxz ∈ E ⇐⇒ 2w + x − z ∈ S
wyz ∈ E ⇐⇒ w − y − 2z ∈ S
xyz ∈ E ⇐⇒ −x − 2y − 3z ∈ S
common diff: −w − x − y − z
4-linear forms condition: correct count of the 2-blow-up of the
simplex K(3)4 (as well as its subgraphs)
![Page 120: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/120.jpg)
Avoiding regularity
A major drawback of the regularity lemma is the tower dependence.
Problem
Find alternative proofs of the applications which avoid using the
regularity lemma and give improved bounds.
![Page 121: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/121.jpg)
Preview of the next lecture
Green developed an arithmetic regularity lemma and used it toprove the following two extensions of Roth’s theorem:
Green’s Arithmetic Removal Lemma
For each ε > 0 there is δ > 0 such that if G is an abelian group
and A,B,C ⊂ G with at most δ|G |2 triples (a, b, c) ∈ A× B × C
with a + b + c = 0, then we can delete ε|G | elements from A,B,C
and get rid of all solutions.
Green’s Roth theorem with popular differences
For each ε > 0 there is N(ε) such that if G is an abelian group
with |G | ≥ N(ε) and A ⊂ G with |A| = α|G |, then there is a
nonzero d ∈ G such that the density of three-term arithmetic
progressions with common difference d in A is at least α3 − ε.
Next lecture: tight quantitative bounds in vector spaces.
![Page 122: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/122.jpg)
Preview of the next lecture
Green developed an arithmetic regularity lemma and used it toprove the following two extensions of Roth’s theorem:
Green’s Arithmetic Removal Lemma
For each ε > 0 there is δ > 0 such that if G is an abelian group
and A,B,C ⊂ G with at most δ|G |2 triples (a, b, c) ∈ A× B × C
with a + b + c = 0, then we can delete ε|G | elements from A,B,C
and get rid of all solutions.
Green’s Roth theorem with popular differences
For each ε > 0 there is N(ε) such that if G is an abelian group
with |G | ≥ N(ε) and A ⊂ G with |A| = α|G |, then there is a
nonzero d ∈ G such that the density of three-term arithmetic
progressions with common difference d in A is at least α3 − ε.
Next lecture: tight quantitative bounds in vector spaces.
![Page 123: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/123.jpg)
Preview of the next lecture
Green developed an arithmetic regularity lemma and used it toprove the following two extensions of Roth’s theorem:
Green’s Arithmetic Removal Lemma
For each ε > 0 there is δ > 0 such that if G is an abelian group
and A,B,C ⊂ G with at most δ|G |2 triples (a, b, c) ∈ A× B × C
with a + b + c = 0, then we can delete ε|G | elements from A,B,C
and get rid of all solutions.
Green’s Roth theorem with popular differences
For each ε > 0 there is N(ε) such that if G is an abelian group
with |G | ≥ N(ε) and A ⊂ G with |A| = α|G |, then there is a
nonzero d ∈ G such that the density of three-term arithmetic
progressions with common difference d in A is at least α3 − ε.
Next lecture: tight quantitative bounds in vector spaces.
![Page 124: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/124.jpg)
Preview of the next lecture
Green developed an arithmetic regularity lemma and used it toprove the following two extensions of Roth’s theorem:
Green’s Arithmetic Removal Lemma
For each ε > 0 there is δ > 0 such that if G is an abelian group
and A,B,C ⊂ G with at most δ|G |2 triples (a, b, c) ∈ A× B × C
with a + b + c = 0, then we can delete ε|G | elements from A,B,C
and get rid of all solutions.
Green’s Roth theorem with popular differences
For each ε > 0 there is N(ε) such that if G is an abelian group
with |G | ≥ N(ε) and A ⊂ G with |A| = α|G |, then there is a
nonzero d ∈ G such that the density of three-term arithmetic
progressions with common difference d in A is at least α3 − ε.
Next lecture: tight quantitative bounds in vector spaces.
![Page 125: The Graph Regularity Method - Video Lectures lecture 1.pdf · The Graph Regularity Method Jacob Fox Stanford University Marston Morse Lecture Series Institute for Advanced Studies](https://reader033.vdocuments.us/reader033/viewer/2022060407/5f0fbcf87e708231d445a3fc/html5/thumbnails/125.jpg)
Preview of the next lecture
Green developed an arithmetic regularity lemma and used it toprove the following two extensions of Roth’s theorem:
Green’s Arithmetic Removal Lemma
For each ε > 0 there is δ > 0 such that if G is an abelian group
and A,B,C ⊂ G with at most δ|G |2 triples (a, b, c) ∈ A× B × C
with a + b + c = 0, then we can delete ε|G | elements from A,B,C
and get rid of all solutions.
Green’s Roth theorem with popular differences
For each ε > 0 there is N(ε) such that if G is an abelian group
with |G | ≥ N(ε) and A ⊂ G with |A| = α|G |, then there is a
nonzero d ∈ G such that the density of three-term arithmetic
progressions with common difference d in A is at least α3 − ε.
Next lecture: tight quantitative bounds in vector spaces.