Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Thursday, October 11, 2018, 12:15 pm

**Duration**: 30 minutes

**Location**: CAB G51

**Speaker**: Rajko Nenadov

We say that a graph G is Ramsey for graphs F_{1}, F_{2}, ..., F_{r} if in every colouring of the edges of G with colours {1,...,r} there exists a copy of F_{i} in G which is completely in colour i, for some i. It is well-known that if G is a complete graph on n vertices, for a sufficiently large n depending on F_{i}'s, then G is Ramsey for given graphs. In this talk we consider the threshold probability p for which G = G(n,p), a binomial random graph, has such a Ramsey property.

The case where F_{1} = F_{2} = ... = F_{r} =: F, the so-called symmetric case, was completely resolved in a series of papers by Rödl and Ruciński in the '90s. It turns out that here the threshold coincides with the property that every edge of G(n,p) belongs to many copies of F, and moreover that it can take a role of any edge of F. Soon after, a conjecture of what the threshold should be in the general case was formulated by Kohayakawa and Kreuter. Apart from some special cases such as when F_{i}'s are cycles or complete graphs, it is still wide open. We prove that the 1-statement holds, that is, the threshold is upper bounded by the conjectured value of Kohayakawa and Kreuter.

This is joint work with Frank Mousset and Wojciech Samotij.

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