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\H os) a peculiar property of $C_4$-free graphs: $C_4$ graphs with $n$ vertices and average degree at least $cn$ contain a complete subgraph (clique) of size at least $c'n$ (with $c'= 0.1c^2n$). We prove here better bounds (${c^2n\over 2+c}$ in general and $(c-1/3)n$ when $c \le 0.733$) 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 best possible shown by the $k$-th power of the cycle $C_{4k+1}$.

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