ramsey properties of random graphs; a sharp threshold proven via a hypergraph regularity lemma. ehud...
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Ramsey Properties
ofRandom Graphs;
A Sharp Threshold
Proven via
A Hypergraph Regularity Lemma.
Ehud Friedgut, Vojtech Rödl,Andrzej Rucinski, Prasad Tetali
Special thanks to the
Tetali family
for
costume design.
Chapter I
We will say a graph is a Ramsey graph
if every bi-coloring of its edges contains a monochromatic triangle.
e.g. Why?!
Ramseynot is 5K Ramsey 6 isK
Is there a sharp threshold?
1
4
if 0
10 if 1]),(Pr[
:)(Thm
ec
cRamsey
ncnGLim
LRRV
??
if 0
if 1]),(Pr[
such thatconstant aexist thereDoes :Question
*
*
*
cc
ccRamsey
ncnGLim
c
.10 and e1 between oscillatemay
principle in and ),( :Correction4
** ncc
1]))1(
,(Pr[
0]))1(
,(Pr[
*
*
Ramseyn
cnGLim
Ramseyn
cnGLim
Theorem: Yes, there does.
such that *c
n
cp Why is the critical edge probability?
22)2( cpn
The expected number of triangles per edge is
Chapter II
U V
||||
),(),(
is ),,(graph bipartite a ofdensity The
VU
VUEVUd
EUVG
)','(),(
|||'| ,|||'| with ' ,'
everyfor ifregular - isit say We
VUdVUd
VVUUVVUU
'U'V
A multi-partite graph on vertex sets kVV ,,1
is -regular if all but of the pairs are -regular
2
k
Easy if k is very small or very large…
Szemerédi’s Regularity Lemma:
regular.- isgraph temultiparti induced
resulting thesuch that with
parts equalalmost into dpartitione becan
with ),(graph
everyfor such that K, ,N ,
0
0
Ktk
tV
NVEVG
k
Weighted variations? Sparse graphs? Hypergraphs?
A hitting set of a graph G is a set of verticesthat intersects every edge. In a dense graph
on vertices there may be hitting sets.n )(2 n
We would like to capture all hitting sets bya family of cores so that:
1. Every hitting set contains a core.
2. The number of cores is . )(2 no
3. Every core is of size linear in .n
If G is a complete bipartite graph on
vertex sets U, V take the cores to be
U and V.
If G is -regular bipartite
take all sets U’ or V’
such that
or
UU )1('
VV )1('
U V
1. Every hitting set contains a core.
3. Every core is of size linear in .n
2. The number of cores is . )(2 no
. 2 )(no
n
n
In a general graph – fix a Szemerédi partition.Draw the super-graph of regular pairs.
A core will be any set obtained by takinga hitting set in the super-graph and taking at least of the vertices in all the super-vertices involved.
1
Chapter III
“Theorem”:
Sharp threshold Global property
Coarse threshold Local property
e.g. connectivity has a sharp threshold - whereas containing a triangle has a coarse threshold.
…which means exactly that Ramsinesshas a sharp threshold!
Any such would be sensitive to smallglobal enhancement …
G
If Ramsiness had a coarse threshold it wouldbe local – a typical non-Ramsey would be sensitive to local perturbations…
G )1
,(n
nG
•Let be typical in .
G
)1,(n
nG
•Assume is non-Ramsey.
G
•Assume there exists a small magical graph , say , such thatM 5K
2/1]Ramsey is )Pr[( MG•Show that this implies
999.]Ramsey is ),(Pr[(
nnGG
6K
)1
,(n
nG
is not
seen in !
)1()1( )2/3(156 onn
n
What about
?6K
Many copies of will pose restrictions if they appear – e.g. a problematic copy:
M
We can color M
But in every proper coloring of one of the following will happen:
G
Using probabilistic techniques we can arrange alarge subset of these restrictions as follows:
Every restriction consists of five elementssuch that every proper coloring must agreewith on at least one of them.
B B B R R
For every proper coloring , the set of (graph)edges of on which it agrees with is a hitting set of .
G H
This defines a hypergraph with (hypergraph)edges
of size 5.
H
e
: B B B R R
H
Ge
e
Given a proper coloring of , and an edge of then there exists an element in for which agrees with .
How does one show sensitivity to global enhancement?
•Every large partial coloring survives the addition of a random copy of with probability .
),(n
nG
2/3
2 cn
•There are approximately colorings.2/3
2n
•Union bound: ?? o(1) 2 2?2/32/3
cnn
Depends on the value of !
There may be too many colorings.
c
Last chapter:
We have a hypergraph of restrictions such thatevery proper coloring defines a hitting set of .But, there are too many colorings.We would like to capture them by a family of cores such that :
HH
2. The number of cores is .)( 2/3
2 no
1. Every hitting set contains a core.
3. Every core is of size .)( 2/3n
We then can improve the union boundby clumping:
There are many colorings :
Survivalprobabilityof each.
2/3
2 cn
22/3n Colorings (hitting sets))( 2/3
2 no CoresAll these colorings share a core.
)1(222/32/3 )( ocnno
A Frankl – Rödl partition
2. Partition every one of the bipartite graphs formed into (non-induced) subgraphs.
1. Partition the vertices of (auxiliary partition)G
Choosing five of these bipartite graphsand a subgraph of each gives a polyad, a set of 5 subsets (anologous to a pair of sets in a Szemerédi partition.)
The densityof a polyad
= The number of copies of belonging to H
(The total number of copies of )5
1
n
A regular Polyad – every sufficiently “large”subgraph has density close to that of the polyad.
“Theorem”: If is a typical graph in and is the corresponding restrictionhypergraph then there exists a Frankl-Rödl partition of such that “most”of the polyads formed are -regular.
G )1,(n
nG
H
H
This enables us to define cores, captureall colorings efficiently and finish the proof.
So, what is the definitionof a core?
Believe me, you don’t want toknow.
And In conclusion I wouldlike to say:
•Ramsiness has a sharp threshold because it is a global property.
•Union bounds can be improved by clumping
• Clumping can be done if the underlying structure has an inherent regularity.
•Frankl –Rödl type partitions can extract regularity from various hypergraphs.
Thank you for your attention!!