A contribution to the Zarankiewicz problem

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× n that does not contain an all ones submatrix of size s× t. We show that if s ≥ 2 and t ≥ 2, then for every k = 0, . . . , s− 2, z (m,n, s, t) ≤ (s− k − 1) nm + kn+ (t− 1)m. This generic bound implies the known bounds of Kövari, Sós and Turán, and of Füredi. As a consequence, we also obtain the following results: Let G be a graph of n vertices and e (G) edges, and let μ be the spectral radius of its adjacency matrix. If G does not contain a complete bipartite subgraph Ks,t, then the following bounds hold μ ≤ (s− t+ 1) n + (t− 1)n + t− 2, and e (G) < 1 2 (s− t+ 1) n + 1 2 (t− 1)n + 1 2 (t− 2)n.