On the Turán Number for the Hexagon

A long-standing conjecture in combinatorics, made by Erdős and Simonovits, is that the maximum number of edges in an n-vertex graph without a hexagon is asymptotically 1 2n 4/3 as n→∞. This conjecture corresponds to the asymptotic optimality of constructions of generalized quadrangles as a source of dense hexagon-free graphs. In this paper, we construct a counterexample to this conjecture. For infinitely many n, we construct an n-vertex hexagon-free graph of size 3( √ 5−2) ( √ 5−1)4/3 n 4/3 + O(n) ≈ 0.534n. On the positive side, we obtain the best known upper bound for the maximum number of edges in an n-vertex hexagon free graph: such a graph has size at most λn + O(n) ≈ 0.627n where λ is the real root of 16λ − 4λ + λ − 3 = 0. The same methods are applied to give an upper bound for the maximum number of edges in a hexagon-free m by n bipartite graph, and the bound is asymptotically tight when 2m = n or 2n = m.