1. On sparse Ramsey graphs Torsten Mütze, ETH Zürich Joint work with Ueli Peter (ETH Zürich) TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A AA A A

2. 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 always 3 that are mutual friends, or 3 that are mutual strangers. Graph theoretic formulation: Any 2-coloring of the edges of K 6 contains a monochromatic K 3. Ramsey‘s theorem (full version, [Ramsey ’30]): For any integer there is a (large) integer such that any 2-coloring of the edges of contains a monochromatic. The smallest such N is called the Ramsey number

3. 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 K 3 -Ramsey graph with … … no K 6 as a subgraph? [Graham ’68]: Yes! K8-C5K8-C5 This is the smallest such graph … no K 5 ? [Pósa]: Yes! Smallest such graph has v ( G )=15 vertices [Piwakowski et al. ’99] … no K 4 ? [Folkman ‘70]: Yes! Smallest such graph has v ( G ) % 10 12 vertices [Frankl, Rödl ’86] % 10 10 vertices [Spencer ’88] % 10 4 vertices [Lu ’08] % 10 3 vertices [Dudek, Rödl ’07] … no K 3 ? No! Ex: K 6 is ( K 3,2)-Ramsey, or simply K 3 -Ramsey r F -Ramsey = ( F, 2)-Ramsey

4. 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 K 3 -Ramsey graph with … … no K 4 ? [Folkman ‘70]: Yes!

5. 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

6. 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 G contains a monochromatic copy of F = half the average degree of H r

7. 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

8. The Ramsey density of cliques Ex:, r R 2 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 = P 3 and r =2), the Ramsey density is not known for any other graph

9. 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))

10. 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‘A‘ B‘B‘ H => Complete graph on A - edges ( r +1)-colored - no grey K r +1 - by Ramsey‘s theorem: in one of the colors 1,…, r in this color in G

11. 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 2r2r r colors

12. Thank you! Questions?