An extremal characterization of the incidence graphs of projective planes (Q1270652)

Point-line incidence graphs of finite projective planes are known to be \(C_4\)-free graphs with interesting extremal properties; so Füredi proved that, reduced modulo a polarity, they give the \(C_4\)-free graphs of maximum edge number [see \textit{Z. Füredi}, J. Comb. Theory, Ser. B 68, No. 1, 1-6, Art. No. 0052 (1996; Zbl 0858.05063)]. In this paper the authors prove that among the \(C_4\)-free bipartite graphs with maximum edge number and equal partition classes, the point-line incidence graphs of finite projective planes have maximum number of \(C_6\)-subgraphs. This result gives an upper bound for the number of triangles in near-linear spaces with \(n\) points and \(n\) lines, which is reached only by projective planes.