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 × 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 ) 1 / t nm 1 - 1 / t + kn + ( t - 1 ) m 1 + k / t . 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 K s , t , then the following bounds hold μ ⩽ ( s - t + 1 ) 1 / t n 1 - 1 / t + ( t - 1 ) n 1 - 2 / t + t - 2 , and e ( G ) < 1 2 ( s - t + 1 ) 1 / t n 2 - 1 / t + 1 2 ( t - 1 ) n 2 - 2 / t + 1 2 ( t - 2 ) n .