Large Kr-free subgraphs in Ks-free graphs and some other Ramsey-type problems

Ks-Free Graphs Without Large Kr-Free Subgraphs

A New Lower Bound For A Ramsey-Type Problem

The Ramsey number R(s,t) is the smallest integer n such that every graph on n vertices contains either a clique of size s or an independent set of size t. The problem of determining or estimating Ramsey numbers is one of the central problems in Combinatorics and it received a considerable amount of attention, see, e.g., [6]. A more general function was first considered (for a special case) by E...

On Ks-free subgraphs in Ks+k-free graphs and vertex Folkman numbers

Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s < t let fs,t(n) = min {max{|S| : S ⊆ V (H) and H[S] contains no Ks}} , where the minimum is taken over all Kt-free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds is still fai...

Subgraphs in vertex neighborhoods of Kr-free graphs

The neighbourhood of every vertex of a triangle-free graph forms an independent set. In 1981, Janos Pach characterized those triangle-free graphs where every independent set belongs to the neighbourhood of a vertex. I will present alternative characterizations and indicate some applications of this result. There are two natural ways to generalize this problem to Kr-free graphs: Characterize tho...

The early evolution of the H-free process

The H-free process, for some fixed graph H , is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c > 0, with high pro...