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!!