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

Abstract

AbstractIn this paper we present three Ramsey‐type results, which we derive from a simple and yet powerful lemma, proved using probabilistic arguments. Let 3 ≤ r < s be fixed integers and let G be a graph on n vertices not containing a complete graph Ks on s vertices. More than 40 years ago Erdős and Rogers posed the problem of estimating the maximum size of a subset of G without a copy of the complete graph Kr. Our first result provides a new lower bound for this problem, which improves previous results of various researchers. It also allows us to solve some special cases of a closely related question posed by Erdős. For two graphs G and H, the Ramsey number R(G, H) is the minimum integer N such that any red‐blue coloring of the edges of the complete graph KN, contains either a red copy of G or a blue copy of H. The book with n pages is the graph Bn consisting of n triangles sharing one edge. Here we study the book‐complete graph Ramsey numbers and show that R(Bn, Kn) ≤ O(n3/log3/2n), improving the bound of Li and Rousseau. Finally, motivated by a question of Erdős, Hajnal, Simonovits, Sós, and Szemerédi, we obtain for all 0 < δ < 2/3 an estimate on the number of edges in a K4‐free graph of order n which has no independent set of size n1‐δ. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005