Cliques in $$C_4$$-free graphs of large minimum degree

A graph G is called (C_4)-free if it does not contain the cycle (C_4) as an induced subgraph. Hubenko, Solymosi and the first author proved (answering a question of Erd?s) a peculiar property of (C_4)-free graphs: (C_4)-free graphs with n vertices and average degree at least cn contain a complete subgraph (clique) of size at least (c'n) (with (c'= 0.1c^2)). We prove here better bounds (big ({c^2nover 2+c}) in general and ((c-1/3)n) when ( c le 0.733big )) from the stronger assumption that the (C_4)-free graphs have minimum degree at least cn. Our main result is a theorem for regular graphs, conjectured in the paper mentioned above: 2k-regular (C_4)-free graphs on (4k+1) vertices contain a clique of size (k+1). This is the best possible as shown by the kth power of the cycle (C_{4k+1}).