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

Abstract

Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n 1/2−ɛ ) ≤ f s,s+k (n) ≤ O(n 1/2+ɛ . In addition, we also discuss some connections between the function f s,t and vertex Folkman numbers.