A contribution to the Zarankiewicz problem

Abstract

Given positive integers m,n,s,t, let z(m,n,s,t) be the maximum number of ones in a (0,1) matrix of size m-by-n that does not contain an all ones submatrix of size s-by-t. We find a flexible upper bound on z(m,n,s,t) that implies the known bounds of Kovari, Sos and Turan, and of Furedi. As a consequence, we find an upper bound on the spectral radius of a graph of order n without a complete bipartite subgraph K_{s,t}.