On the number of edges of quadrilateral-free graphs (Q1924153)

Let \(f(n)\) be the maximum number of edges in a graph of order \(n\) without four-cycles. It has been known that asymptotically \(f(n)\sim n^{3/2}\). In the paper it is proved that \(f(q^2+q+1)\leq q(q+1)^2/2\) for all \(q>13\). Moreover, if \(q\) is a prime power greater than 13, we get equality in the formula.