An Upper Bound on Zarankiewicz' Problem

Let ex( n, K 3,3 ) denote the maximum number of edges of a K 3,3 -free graph on n vertices. Improving earlier results of Kővári, T. Sós and Turán on Zarankiewicz' problem, we obtain that Brown's example for a maximal K 3,3 -free graph is asymptotically optimal. Hence .