![Page 1: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/1.jpg)
New developments in hypergraph Ramsey theory
Andrew Suk (UC San Diego)
April 24, 2018
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 2: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/2.jpg)
Origins of Ramsey theory
“A combinatorial problem in geometry,” by Paul Erdos and GeorgeSzekeres (1935)
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 3: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/3.jpg)
Erdos-Szkeres 1935
Theorem (Monotone subsequence)
Any sequence of (n − 1)2 + 1 integers contains a monotone
subsequence of length n.
Theorem (Convex polygon)
For any n > 0, there is a minimal ES(n), such that every set of
ES(n) points in the plane in general position contains n members
in convex position.
Theorem (Ramsey numbers)
New proof of Ramsey’s theorem.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 4: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/4.jpg)
Convex polygon theorem
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Theorem (Erdos-Szekeres 1935, 1960)
2n−2 + 1 ≤ ES(n) ≤(
2n − 4
n − 2
)
+ 1 = O(4n/√n).
Conjecture: ES(n) = 2n−2 + 1, n ≥ 3.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 5: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/5.jpg)
Convex polygon theorem
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Theorem (Erdos-Szekeres 1935, 1960)
2n−2 + 1 ≤ ES(n) ≤(
2n − 4
n − 2
)
+ 1 = O(4n/√n).
Theorem (S. 2016)
ES(n) = 2n+o(n)
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 6: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/6.jpg)
Erdos-Szekeres 1935
Theorem (Monotone subsequence)
Any sequence of (n − 1)2 + 1 integers contains a monotone
subsequence of length n.
Theorem (Convex polygon)
For any n > 0, there is a minimal ES(n), such that every set of
ES(n) points in the plane in general position contains n members
in convex position.
Theorem (Ramsey numbers)
New proof of Ramsey’s theorem.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 7: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/7.jpg)
Ramsey theory
Formal definition: For any integers k ≥ 1, s, n ≥ k , there is aminimum rk(s, n) = N, such that for every red/blue coloring of thek-tuples of {1, 2, . . . ,N},
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 8: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/8.jpg)
Ramsey theory
Formal definition: For any integers k ≥ 1, s, n ≥ k , there is aminimum rk(s, n) = N, such that for every red/blue coloring of thek-tuples of {1, 2, . . . ,N},
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 9: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/9.jpg)
Ramsey theory
Formal definition: For any integers k ≥ 1, s, n ≥ k , there is aminimum rk(s, n) = N, such that for every red/blue coloring of thek-tuples of {1, 2, . . . ,N},
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 10: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/10.jpg)
Ramsey theory
Formal definition: For any integers k ≥ 1, s, n ≥ k , there is aminimum rk(s, n) = N, such that for every red/blue coloring of thek-tuples of {1, 2, . . . ,N},
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 11: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/11.jpg)
Ramsey theory
Formal definition: For any integers k ≥ 1, s, n ≥ k , there is aminimum rk(s, n) = N, such that for every red/blue coloring of thek-tuples of {1, 2, . . . ,N},
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 12: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/12.jpg)
Ramsey theory
Formal definition: For any integers k ≥ 1, s, n ≥ k , there is aminimum rk(s, n) = N, such that for every red/blue coloring of thek-tuples of {1, 2, . . . ,N},
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 13: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/13.jpg)
Ramsey theory
Formal definition: For any integers k ≥ 1, s, n ≥ k , there is aminimum rk(s, n) = N, such that for every red/blue coloring of thek-tuples of {1, 2, . . . ,N},
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rk(s, n) = Ramsey numbers
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 14: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/14.jpg)
Graph Ramsey theorem
Theorem (Erdos-Szekeres 1935)
r2(s, n) ≤(
n + s − 2
s − 1
)
r2(n, n) ≤(
2n − 2
n − 1
)
≈ 4n√n
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 15: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/15.jpg)
Graph Ramsey number r2(n, n) ≤ 4n
Let G be the complete graph with 4n vertices, every edge has colorred or blue.
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 21: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/21.jpg)
Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 25: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/25.jpg)
Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 26: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/26.jpg)
Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 29: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/29.jpg)
Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 30: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/30.jpg)
Graph Ramsey number r2(n, n) ≤ 4n
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 31: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/31.jpg)
Diagonal graph Ramsey numbers
Theorem (Erdos 1947, Erdos-Szekeres 1935)
(1 + o(1))n
e2n/2 < r2(n, n) <
4n√n.
Theorem (Spencer 1977, Conlon 2008)
(1 + o(1))
√2
en2n/2 < r2(n, n) <
4n
nc log n/ log log n
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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...1 2 43 22cn
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 41: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/41.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
�� ���� �� ���� ���� �� ���� ���� �� ������
...1 2 43 22cn
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 42: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/42.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
�� ���� �� ���� ���� �� ���� ���� �� ������
...1 2 43 22cn
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 43: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/43.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
�� ���� �� �� ���� ������
1 2 3 22cn
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 44: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/44.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 45: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/45.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 46: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/46.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 47: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/47.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 48: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/48.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 49: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/49.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 50: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/50.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 51: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/51.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 52: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/52.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 53: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/53.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 54: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/54.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 55: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/55.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 56: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/56.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 57: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/57.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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4n1 2
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 58: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/58.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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n
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 59: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/59.jpg)
3-uniform hypergraphs
Greedy argument: r3(n, n) ≤ 22O(n)
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n
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 60: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/60.jpg)
Hypergraph Ramsey numbers
Theorem (Erdos-Rado 1952)
2cn2< r3(n, n) < 22
c′n
Conjecture (Erdos, $500)
r3(n, n) > 22cn
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Erdos-Rado upper bound argument
������������
������������
H = k−uniform (k−1)−uniform
Greedy argument to reduce the problem to a (k − 1)-uniformhypergraph problem. Argument shows
rk(s, n) ≤ 2(rk−1(s−1,n−1))k−1.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 62: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/62.jpg)
Upper bounds for diagonal hypergraph Ramsey numbers
Applying rk(s, n) ≤ 2(rk−1(s−1,n−1))k−1.
Conlon (2008): r2(n, n) <4n
nc log n/ log log n
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Upper bounds for diagonal hypergraph Ramsey numbers
Applying rk(s, n) ≤ 2(rk−1(s−1,n−1))k−1.
Conlon (2008): r2(n, n) <4n
nc log n/ log log n
Erdos-Rado (1952): r3(n, n) < 22cn
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 64: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/64.jpg)
Upper bounds for diagonal hypergraph Ramsey numbers
Applying rk(s, n) ≤ 2(rk−1(s−1,n−1))k−1.
Conlon (2008): r2(n, n) <4n
nc log n/ log log n
Erdos-Rado (1952): r3(n, n) < 22cn
Erdos-Rado (1952): r4(n, n) < 222cn
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 65: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/65.jpg)
Upper bounds for diagonal hypergraph Ramsey numbers
Applying rk(s, n) ≤ 2(rk−1(s−1,n−1))k−1.
Conlon (2008): r2(n, n) <4n
nc log n/ log log n
Erdos-Rado (1952): r3(n, n) < 22cn
Erdos-Rado (1952): r4(n, n) < 222cn
...
Erdos-Rado (1952): rk(n, n) < twrk(cn)
twr1(x) = x and twri+1(x) = 2twri (x).
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 66: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/66.jpg)
Lower bounds for diagonal hypergraph Ramsey numbers
Random constructions
Spencer (1977):√2en2n/2 < r2(n, n) <
4n
nc log n/ log log n
Erdos (1947): 2c′n2 < r3(n, n) < 22
cn
Erdos-Rado (1952): r4(n, n) < 222cn
...
Erdos-Rado (1952): rk(n, n) < twrk(cn)
twr1(x) = x and twri+1(x) = 2twri (x).
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 67: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/67.jpg)
Lower bounds for diagonal hypergraph Ramsey numbers
Erdos-Hajnal stepping up lemma: k ≥ 3, rk+1(n, n) > 2rk (n/4,n/4).
Spencer (1977):√2en2n/2 < r2(n, n) <
4n
nc log n/ log log n
Erdos (1947): 2c′n2 < r3(n, n) < 22
cn
Erdos-Rado (1952): r4(n, n) < 222cn
...
Erdos-Rado (1952): rk(n, n) < twrk(cn)
twr1(x) = x and twri+1(x) = 2twri (x).
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 68: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/68.jpg)
Lower bounds for diagonal hypergraph Ramsey numbers
Erdos-Hajnal stepping up lemma: k ≥ 3, rk+1(n, n) > 2rk (n/4,n/4).
Spencer (1977):√2en2n/2 < r2(n, n) <
4n
nc log n/ log log n
Erdos (1947): 2c′n2 < r3(n, n) < 22
cn
Erdos-Hajnal: 22c′n2
< r4(n, n) < 222cn
...
Erdos-Rado (1952): rk(n, n) < twrk(cn)
twr1(x) = x and twri+1(x) = 2twri (x).
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 69: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/69.jpg)
Lower bounds for diagonal hypergraph Ramsey numbers
Erdos-Hajnal stepping up lemma: k ≥ 3, rk+1(n, n) > 2rk (n/4,n/4).
Spencer (1977):√2en2n/2 < r2(n, n) <
4n
nc log n/ log log n
Erdos (1947): 2c′n2 < r3(n, n) < 22
cn
Erdos-Hajnal: 22c′n2
< r4(n, n) < 222cn
...
Erdos-Hajnal: twrk−1(c′n2) < rk(n, n) < twrk(cn)
twr1(x) = x and twri+1(x) = 2twri (x).
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 70: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/70.jpg)
Conjecture: r3(n, n) > 22cn
Theorem (Erdos-Hajnal-Rado 1952/1965)
2cn2< r3(n, n) < 22
c′n
Theorem (Erdos-Hajnal)
r3(n, n, n, n) > 22cn
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Off-diagonal Ramsey numbers
rk(s, n) where s is fixed, and n → ∞. rk(s, n) ≪ rk(n, n)
Graphs:
Theorem (Ajtai-Komlos-Szemeredi 1980, Kim 1995)
r2(3, n) = Θ(
n2
log n
)
Theorem
For fixed s > 3
n(s+1)/2+o(1) < r2(s, n) < ns−1+o(1)
2n/2 < r2(n, n) < 4n
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Upper bounds for off-diagonal Ramsey numbers
3-uniform hypergraphs:
Theorem (Erdos-Hajnal-Rado)
For fixed s ≥ 4,2csn < r3(s, n) < 2c
′n2s−4.
Theorem (Conlon-Fox-Sudakov 2010)
For fixed s ≥ 4,
2csn log n < r3(s, n) < 2c′ns−2 log n.
2cn2< r3(n, n) < 22
c′n
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Upper bounds for off-diagonal hypergraph Ramsey numbers
Fixed s ≥ k + 1.
Erdos-Szekeres (1935): r2(s, n) < ns−1+o(1)
Conlon-Fox-Sudakov (2010): r3(s, n) < 2cns−2 log n
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Upper bounds for off-diagonal hypergraph Ramsey numbers
Fixed s ≥ k + 1.
Erdos-Szekeres (1935): r2(s, n) < ns−1+o(1)
Conlon-Fox-Sudakov (2010): r3(s, n) < 2cns−2 log n
Erdos-Rado (1952): r4(s, n) < 22nc
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 75: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/75.jpg)
Upper bounds for off-diagonal hypergraph Ramsey numbers
Fixed s ≥ k + 1.
Erdos-Szekeres (1935): r2(s, n) < ns−1+o(1)
Conlon-Fox-Sudakov (2010): r3(s, n) < 2cns−2 log n
Erdos-Rado (1952): r4(s, n) < 22nc
Erdos-Rado (1952): r5(s, n) < 222n
c
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 76: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/76.jpg)
Upper bounds for off-diagonal hypergraph Ramsey numbers
Fixed s ≥ k + 1.
Erdos-Szekeres (1935): r2(s, n) < ns−1+o(1)
Conlon-Fox-Sudakov (2010): r3(s, n) < 2cns−2 log n
Erdos-Rado (1952): r4(s, n) < 22nc
Erdos-Rado (1952): r5(s, n) < 222n
c
...
Erdos-Rado: rk(s, n) < twrk−1(nc)
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 77: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/77.jpg)
Lower bounds for off-diagonal hypergraph Ramsey numbers
Fixed s ≥ k + 1.
Bohman-Keevash (2010): n(s+1)/2+o(1) < r2(s, n) < ns−1+o(1)
Conlon-Fox-Sudakov (2010): 2c′n log n < r3(s, n) < 2cn
s−2 log n
Erdos-Rado (1952): r4(s, n) < 22nc
Erdos-Rado (1952): r5(s, n) < 222n
c
...
Erdos-Rado: rk(s, n) < twrk−1(nc)
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Lower bounds for off-diagonal hypergraph Ramsey numbers
Fixed s ≥ k + 1.
Bohman-Keevash (2010): n(s+1)/2+o(1) < r2(s, n) < ns−1+o(1)
Conlon-Fox-Sudakov (2010): 2c′n log n < r3(s, n) < 2cn
s−2 log n
Erdos-Rado (1952): r4(s, n) < 22nc
Erdos-Rado (1952): r5(s, n) < 222n
c
...
Erdos-Rado: rk(s, n) < twrk−1(nc)
Tower growth rate for r4(5, n) is unknown.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 79: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/79.jpg)
Lower bounds for off-diagonal hypergraph Ramsey numbers
Fixed s ≥ k + 1.
Bohman-Keevash (2010): n(s+1)/2+o(1) < r2(s, n) < ns−1+o(1)
Conlon-Fox-Sudakov (2010): 2c′n log n < r3(s, n) < 2cn
s−2 log n
Erdos-Hajnal: 22c′n
< r4(7, n) < 22nc
Erdos-Rado (1952): r5(s, n) < 222n
c
...
Erdos-Rado: rk(s, n) < twrk−1(nc)
Tower growth rate for r4(5, n) is unknown.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 80: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/80.jpg)
Lower bounds for off-diagonal hypergraph Ramsey numbers
Fixed s ≥ k + 1.
Bohman-Keevash (2010): n(s+1)/2+o(1) < r2(s, n) < ns−1+o(1)
Conlon-Fox-Sudakov (2010): 2c′n log n < r3(s, n) < 2cn
s−2 log n
Erdos-Hajnal: 22c′n
< r4(7, n) < 22nc
MS and CFS (2015): 222c
′n
< r5(8, n) < 222n
c
...
Erdos-Rado: rk(s, n) < twrk−1(nc)
Tower growth rate for r4(5, n) is unknown.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 81: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/81.jpg)
Lower bounds for off-diagonal hypergraph Ramsey numbers
Fixed s ≥ k + 1.
Bohman-Keevash (2010): n(s+1)/2+o(1) < r2(s, n) < ns−1+o(1)
Conlon-Fox-Sudakov (2010): 2c′n log n < r3(s, n) < 2cn
s−2 log n
Erdos-Hajnal: 22c′n
< r4(7, n) < 22nc
MS and CFS (2015): 222c
′n
< r5(8, n) < 222n
c
...
MS and CFS (2015): twrk−1(c′n) < rk(k + 3, n) < twrk−1(n
c)
Tower growth rate for r4(5, n) is unknown.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 82: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/82.jpg)
Lower bounds for off-diagonal hypergraph Ramsey numbers
Fixed s ≥ k + 1.
Bohman-Keevash (2010): n(s+1)/2+o(1) < r2(s, n) < ns−1+o(1)
Conlon-Fox-Sudakov (2010): 2c′n log n < r3(s, n) < 2cn
s−2 log n
Erdos-Hajnal: 22c′n
< r4(7, n) < 22nc
MS and CFS (2015): 222c
′n
< r5(8, n) < 222n
c
...
MS and CFS (2015): twrk−1(c′n) < rk(k + 3, n) < twrk−1(n
c)
What is the tower growth rate of rk(k + 1, n) and rk(k + 2, n)?
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 83: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/83.jpg)
Lower bounds for off-diagonal hypergraph Ramsey numbers
Fixed s ≥ k + 1.
Bohman-Keevash (2010): n(s+1)/2+o(1) < r2(s, n) < ns−1+o(1)
Conlon-Fox-Sudakov (2010): 2c′n log n < r3(s, n) < 2cn
s−2 log n
Erdos-Hajnal: 22c′n
< r4(7, n) < 22nc
MS and CFS (2015): 222c
′n
< r5(8, n) < 222n
c
...
MS and CFS (2015): twrk−1(c′n) < rk(k + 3, n) < twrk−1(n
c)
What is the tower growth rate of r4(5, n) and r4(6, n)?
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 84: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/84.jpg)
Towards the Erdos-Hajnal conjecture
Conjecture (Erdos-Hajnal)
r4(5, n), r4(6, n) > 22cn
Erdos-Hajnal (1972): r4(5, n), r4(6, n) > 2cn
Mubayi-S. (2017): r4(5, n) > 2n2
r4(6, n) > 2nc log n
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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New lower bounds for off-diagonal hypergraph Ramseynumbers
Theorem (Mubayi-S., 2018)
r4(5, n) > 2nc log n
r4(6, n) > 22cn
1/5
.
for fixed k > 4
rk(k + 1, n) > twrk−2(nc log n) rk(k + 2, n) > twrk−1(cn
1/5).
rk(k + 2, n) = twrk−1(nΘ(1))
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Open problems
Diagonal Ramsey problem ($500 Erdos):
2cn2< r3(n, n) < 22
cn
.
Off-diagonal Ramsey problem:
2nc log n
< r4(5, n) < 22cn
.
Theorem (Mubayi-S. 2017)
Showing r3(n, n) > 22cn
implies that r4(5, n) > 22c′n
.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 87: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/87.jpg)
More off-diagonal?
Off diagonal hypergraph Ramsey numbers: rk(k + 1, n)
Red clique size k + 1 or Blue clique of size n.
rk(k , n) = n (trivial)
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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More off-diagonal?
Off diagonal hypergraph Ramsey numbers: rk(k + 1, n)
Red clique size k + 1 or Blue clique of size n.
rk(k , n) = n (trivial)
A more off diagonal Ramsey number:
Almost Red clique size k + 1 or Blue clique of size n.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 89: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/89.jpg)
Very off-diagonal
Another Ramsey function: Let rk(k + 1, t ; n) be the minimumN, such that for every red/blue coloring of the k-tuples of{1, 2, . . . ,N},
����
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...21 3 4 N
1 k + 1 integers which induces at least t red k-tuples, or
2 n integers for which every k-tuple is blue.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 90: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/90.jpg)
Very off-diagonal
Another Ramsey function: Let rk(k + 1, t ; n) be the minimumN, such that for every red/blue coloring of the k-tuples of{1, 2, . . . ,N},
����
����
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����
����
����
����
����
����
����
...21 3 4 N
1 k + 1 integers which induces at least t red k-tuples, or
2 n integers for which every k-tuple is blue.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 91: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/91.jpg)
Very off-diagonal
Another Ramsey function: Let rk(k + 1, t ; n) be the minimumN, such that for every red/blue coloring of the k-tuples of{1, 2, . . . ,N},
����
����
����
����
����
����
����
����
����
����
...21 3 4 N
1 k + 1 integers which induces at least t red k-tuples, or
2 n integers for which every k-tuple is blue.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 92: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/92.jpg)
Very off-diagonal
Another Ramsey function: Let rk(k + 1, t ; n) be the minimumN, such that for every red/blue coloring of the k-tuples of{1, 2, . . . ,N},
����
����
����
����
����
����
����
����
����
����
...21 4 N3
1 k + 1 integers which induces at least t red k-tuples, or
2 n integers for which every k-tuple is blue.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 93: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/93.jpg)
Very off-diagonal
Another Ramsey function: Let rk(k + 1, t ; n) be the minimumN, such that for every red/blue coloring of the k-tuples of{1, 2, . . . ,N},
����
����
����
����
����
����
����
����
����
����
...21 4 N3
1 k + 1 integers which induces at least t red k-tuples, or
2 n integers for which every k-tuple is blue.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 94: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/94.jpg)
Very off-diagonal
Another Ramsey function: Let rk(k + 1, t ; n) be the minimumN, such that for every red/blue coloring of the k-tuples of{1, 2, . . . ,N},
����
����
����
����
����
����
����
����
����
����
...2 31 4 N
1 k + 1 integers which induces at least t red k-tuples, or
2 n integers for which every k-tuple is blue.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 95: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/95.jpg)
Very off-diagonal
Another Ramsey function: Let rk(k + 1, t ; n) be the minimumN, such that for every red/blue coloring of the k-tuples of{1, 2, . . . ,N},
����
����
����
����
����
����
����
����
����
����
...2 31 4 N
1 k + 1 integers which induces at least t red k-tuples, or
2 n integers for which every k-tuple is blue.
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 96: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/96.jpg)
An old problem of Erdos and Hajnal 1972
Problem (Erdos-Hajnal 1972)
For k ≥ 3 and t ∈ [k + 1], estimate rk(k + 1, t; n).
rk(k + 1, 1; n) = n
...
twrk−2(nc′ log n) ≤ rk(k + 1, k + 1; n) = rk(k + 1, n) ≤ twrk−1(n
c)
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 97: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/97.jpg)
Known results
rk(k + 1, 2; n) < O(nk−1) = N.
Case 1: (k − 1)-tuple with 2 red edges.
���� �� ���� �� ������ �� �� ���� ����
...1 2 43 N
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 98: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/98.jpg)
Known results
rk(k + 1, 2; n) < O(nk−1) = N.
Case 1: (k − 1)-tuple with 2 red edges.
���� �� ���� �� ������ �� �� ���� ����
...1 2 43 N
k−1
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 99: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/99.jpg)
Known results
rk(k + 1, 2; n) < O(nk−1) = N.
Case 1: (k − 1)-tuple with 2 red edges.
���� �� ���� �� ������ �� �� ���� ����
...1 2 43 N
k−1
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 100: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/100.jpg)
Known results
rk(k + 1, 2; n) < O(nk−1) = N.
Case 1: (k − 1)-tuple with 2 red edges.
���� �� ���� �� ������ �� �� ���� ����
...1 2 43 N
k−1
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 101: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/101.jpg)
Known results
rk(k + 1, 2; n) < O(nk−1) = N.
Case 1: (k − 1)-tuple with 2 red edges.
���� �� �� ������ �� ������ ��������
...1 2 43 N
k−1
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 102: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/102.jpg)
Known results
rk(k + 1, 2; n) < O(nk−1) = N.
Case 1: (k − 1)-tuple with 2 red edges.
Case 2: Less than(
N
k−1
)
k-tuples are red implies blue copy of K(k)n .
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
![Page 103: New developments in hypergraph Ramsey theoryasuk/CSUN042418.pdf · 2018-04-26 · Origins of Ramsey theory “A combinatorial problem in geometry,” by Paul Erdo˝s and George Szekeres](https://reader035.vdocuments.us/reader035/viewer/2022081613/5fba3f70cccc5312c77936c2/html5/thumbnails/103.jpg)
Slightly better bounds
Theorem (Rodl-Sinajova and Kostochka-Mubayi-Verstraete)
cnk−1
log n≤ rk(k + 1, 2; n) ≤ c ′
nk−1
log n
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Upper bounds
Erdos-Rado argument: rk(k + 1, t; n) ≤ 2(rk−1(k,t−1;n))k−1.
Theorem (Erdos-Hajnal 1972)
rk(k + 1, 2; n) < cnk−1
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Upper bounds
Erdos-Rado argument: rk(k + 1, t; n) ≤ 2(rk−1(k,t−1;n))k−1.
Theorem (Erdos-Hajnal 1972)
rk(k + 1, 2; n) < cnk−1
rk(k + 1, 3; n) < 2nc
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Upper bounds
Erdos-Rado argument: rk(k + 1, t; n) ≤ 2(rk−1(k,t−1;n))k−1.
Theorem (Erdos-Hajnal 1972)
rk(k + 1, 2; n) < cnk−1
rk(k + 1, 3; n) < 2nc
rk(k + 1, 4; n) < 22nc
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Upper bounds
Erdos-Rado argument: rk(k + 1, t; n) ≤ 2(rk−1(k,t−1;n))k−1.
Theorem (Erdos-Hajnal 1972)
rk(k + 1, 2; n) < cnk−1
rk(k + 1, 3; n) < 2nc
rk(k + 1, 4; n) < 22nc
...
rk(k + 1, t; n) < twrt−1(nc)
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Upper bounds
Erdos-Rado argument: rk(k + 1, t; n) ≤ 2(rk−1(k,t−1;n))k−1.
Theorem (Erdos-Hajnal 1972)
rk(k + 1, 2; n) < cnk−1
rk(k + 1, 3; n) < 2nc
rk(k + 1, 4; n) < 22nc
...
rk(k + 1, k ; n) < twrk−1(nc)
rk(k + 1, k + 1; n) < twrk−1(nc)
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Upper bounds
Erdos-Rado argument: rk(k + 1, t; n) ≤ 2(rk−1(k,t−1;n))k−1.
Theorem (Erdos-Hajnal 1972)
rk(k + 1, 2; n) < cnk−1
rk(k + 1, 3; n) < 2nc
rk(k + 1, 4; n) < 22nc
...
rk(k + 1, t; n) < twrt−1(nc)
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Lower bounds
Erdos-Rado argument: rk(k + 1, t; n) ≤ 2(rk−1(k,t−1;n))k−1
.
Theorem
c ′nk−1
log n< rk(k + 1, 2; n) < c
nk−1
log n
rk(k + 1, 3; n) < 2nc
rk(k + 1, 4; n) < 22nc
...
rk(k + 1, t; n) < twrt−1(nc)
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Lower bounds
Erdos-Rado argument: rk(k + 1, t; n) ≤ 2(rk−1(k,t−1;n))k−1
.
Theorem
c ′nk−1
log n< rk(k + 1, 2; n) < c
nk−1
log n
2c′n < rk(k + 1, 3; n) < 2n
c
rk(k + 1, 4; n) < 22nc
...
rk(k + 1, t; n) < twrt−1(nc)
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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Lower bounds
Erdos-Rado argument: rk(k + 1, t; n) ≤ 2(rk−1(k,t−1;n))k−1
.
Theorem
c ′nk−1
log n< rk(k + 1, 2; n) < c
nk−1
log n
2c′n < rk(k + 1, 3; n) < 2n
c
2c′n < rk(k + 1, 4; n) < 22
nc
...
rk(k + 1, t; n) < twrt−1(nc)
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Lower bounds
Erdos-Rado argument: rk(k + 1, t; n) ≤ 2(rk−1(k,t−1;n))k−1
.
Theorem
c ′nk−1
log n< rk(k + 1, 2; n) < c
nk−1
log n
2c′n < rk(k + 1, 3; n) < 2n
c
2c′n < rk(k + 1, 4; n) < 22
nc
...
2c′n < rk(k + 1, t; n) < twrt−1(n
c)
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New bounds for rk(k + 1, t; n)
Improvement on the Erdos-Rado upper bound argument.
Theorem (Mubayi-S. 2018)
For k ≥ 3, and 2 ≤ t ≤ k, we have
rk(k + 1, t; n) < twrt−1(cnk−t+1 log n)
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New bounds for rk(k + 1, t; n)
For 3 ≤ t ≤ k − 2
Theorem (Mubayi-S. 2018)
For k ≥ 6, and 3 ≤ t ≤ k − 2, we have
twrt−1(c′nk−t+1) < rk(k + 1, t; n) < twrt−1(cn
k−t+1 log n)
when k − t is even, and
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory
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New bounds for rk(k + 1, t; n)
For 3 ≤ t ≤ k − 2
Theorem (Mubayi-S. 2018)
For k ≥ 6, and 3 ≤ t ≤ k − 2, we have
twrt−1(c′nk−t+1) < rk(k + 1, t; n) < twrt−1(cn
k−t+1 log n)
when k − t is even, and
twrt−1(c′n(k−t+1)/2) < rk(k + 1, t; n) < twrt−1(cn
k−t+1 log n)
when k − t is odd.
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New bounds for rk(k + 1, t; n)
For t = k − 1, k , k + 1
Theorem (Mubayi-S. 2018)
For k ≥ 6,
twrk−3(cn3) < rk(k + 1, k − 1; n) < twrk−2(c
′n2 log n)
twrk−3(cn3) < rk(k + 1, k ; n) < twrk−1(c
′n log n)
twrk−2(nc log n) < rk(k + 1, k + 1; n) < twrk−1(c
′n2 log n).
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Open problem
2c′n3 < r5(6, 5; n) < 22
2nc
Improve the upper and lower bounds for r5(6, 5; n).
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Thank you!
Andrew Suk (UC San Diego) New developments in hypergraph Ramsey theory