On sparse Ramsey graphs

Torsten Mütze, ETH ZürichJoint work with Ueli Peter (ETH Zürich)

Introduction• Ramsey theory:

Branch of combinatorics that deals with order arising in large disordered structures

• Ramsey‘s theorem (party version):In any group of 6 people, there are always3 that are mutual friends,or 3 that are mutual strangers.

•Graph theoretic formulation:Any 2-coloring of the edges of K6

contains a monochromatic K3.

• Ramsey‘s theorem (full version, [Ramsey ’30]):For any integer there is a (large) integer suchthat any 2-coloring of the edges of contains a monochromatic .

The smallest such N is calledthe Ramsey number

Introduction• Def: A graph G is called (F, r)-Ramsey,

if any 2-coloring of the edges of G contains a monochromatic copy of F

• [Erdös, Hajnal ’67]:Is there a K3-Ramsey graph with …… no K6 as a subgraph?

[Graham ’68]: Yes!

K8-C5

This is thesmallestsuch graph

… no K5? [Pósa]: Yes!Smallest such graph has v(G)=15 vertices

[Piwakowski et al. ’99]… no K4? [Folkman ‘70]: Yes!Smallest such graph has v(G)%1012 vertices [Frankl, Rödl ’86] %1010 vertices [Spencer ’88]

%104 vertices [Lu ’08]%103 vertices [Dudek, Rödl ’07]… no K3? No!

Ex: K6 is (K3,2)-Ramsey,or simply K3-Ramsey

r

F-Ramsey = (F, 2)-Ramsey

Locally sparse Ramsey graphs

• Folkman’s result:

• Informally: There are Ramsey graphs that are nowhere locally dense (no larger cliques than absolutely necessary)!

•Formally: For any integer there is a (huge!) -Ramsey graph with no cliques of size .

• Later generalized by [Nesetril, Rödl ’76] to the case of more than 2 colors.

• Still, these graphs are very dense globally:they contain many edges!

Is there a K3-Ramsey graph with …

… no K4? [Folkman ‘70]: Yes!

Globally sparse Ramsey graphs• [Rödl, Rucinski ’93]:

How globally sparse can Ramsey graphs possibly be?

•Measure of global sparseness of a graph G:

•m(G) arises naturally in the theory of random graphs

•Ex:

= half the average degree of H

Globally sparse Ramsey graphs• [Rödl, Rucinski ’93]:

How globally sparse can Ramsey graphs possibly be?

•Measure of global sparseness of a graph G:

•Define the Ramsey density of F and r as

Any 2-coloring of the edges of Gcontains a monochromatic copy of F

= half the average degree of H

r

The Ramsey density

•Define the Ramsey density of F and r as

• What do we have to prove?LB: UB:Show that any graph G with Show one (F,r)-Ramsey graph can be properly colored G with

• Ex: F = , r=2, Claim:

UB:

G

LB:

=> G is a forest

The Ramsey density of cliques• Ex: , rR2

• A trivial upper bound: Take as a Ramsey graph a complete graph on as many vertices as the Ramsey number tells us:

• Surprise: Theorem ([Kurek, Rucinski ‘05]):This upper bound is tight, i.e., we have

• Informally:The sparsest -Ramsey graph is a huge complete graph!

• Apart from cliques and some trivial cases (stars, F=P3 and r=2),the Ramsey density is not known for any other graph

• Theorem [M., Peter ’11+]:

•For complete bipartite graphs with we have

•For cycles we have

for even :

for odd :

•For paths we have

Our results

independent of b

independent of

a=2

a=3

( stars (trivial))

Upper bound for even cycles

For cycles we have

for even :

for odd :

• Theorem [M., Peter ’11+]:

• Proof of :

A

A is huge!!!

G

A‘B‘

H

=>

Complete graph on A- edges (r+1)-colored- no grey Kr+1

- by Ramsey‘s theorem: in one of the colors 1,…,r

in this color in G

Lower bound for odd cycles• Theorem [M., Peter ’11+]:

• Proof of :

For cycles we have

for even :

for odd :

Graph G with =>color G with r bipartite graphs ( -free!)

2

1 color

4

2 colors

8

3 colors

2r

r colors

Thank you! Questions?