Large Cliques in C4-Free Graphs

A graph is called C4-free if it contains no cycle of length four as an induced subgraph. We prove that if a C4-free graph has n vertices and at least c1n 2 edges then it has a complete subgraph of c2n vertices, where c2 depends only on c1. We also give estimates on c2 and show that a similar result does not hold for H-free graphs—unless H is an induced subgraph of C4. The best value of c2 is determined for chordal graphs. Graphs are understood to be simple, i.e. without loops or multiple edges and this is essential. The order of the largest complete subgraph of G is denoted by ω(G) and the order of the largest independent set of G is denoted by α(G). A graph is called C4-free if it contains no cycle of length four as an induced subgraph. The following question has been asked by Paul Erdős: is it true that C4-free graphs with n vertices and at least c1n edges must contain complete subgraphs of c2n vertices, where c2 depends only on c1? We give the affirmative answer (Corollary 1) with c2=0.4c1. In fact, we shall prove a more general result, Theorem 1: A C4-free graph with n vertices and average degree at least a must contain a complete subgraph of order at least 0.1a2n−1. The role of C4 is very important, similar results are not true for H-free graphs as the next proposition shows.