The Zarankiewicz number z(43;2) is the largest number of edges in a four-cycle-free bipartite graph with two parts of size 43. Reiman's bound gives z(43;2) <= 301, with equality only for the incidence graph of a projective plane of order six; no such plane exists, so z(43;2) <= 300. We prove z(43;2) <= 299. The argument is elementary and uses no computer search: a counting identity for the leave of the configuration shows that a hypothetical 300-edge graph admits one of exactly twenty-seven degree profiles per side, of which only four combinations are locally compatible. Three force two vertices to share six neighbours; the fourth forces a transversal design TD(6,6), hence four mutually orthogonal Latin squares of order six, contradicting Tarry's theorem. We also give an explicit 284-edge construction inside PG(2,7), so that 284 <= z(43;2) <= 299.

翻译：暂无翻译