Cliques in \(C_4\) -free graphs of large minimum degree

A graph G is called C4-free if it does not contain the cycle C4 as an induced subgraph. Hubenko, Solymosi and the first author proved (answering a question of Erdős) a peculiar property of C4-free graphs: C4 graphs with n vertices and average degree at least cn contain a complete subgraph (clique) of size at least c′n (with c′ = 0.1c2n). We prove here better bounds (c2n/3 in general and (c− 1/3)n in the range 0.382 ≤ c ≤ 0.733) from the stronger assumption that the C4-free graphs have minimum degree at least cn. Our main result is a theorem for regular graphs, conjectured in the paper mentioned above: 2kregular C4-free graphs on 4k + 1 vertices contain a clique of size k + 1. This is best possible shown by the k-th power of the cycle C4k+1.