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)1 / t nm1 - 1 / t + kn + (t - 1) m1 + 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 Ks, t, then the following bounds holdμ ≤ (s - t + 1)1 / t n1 - 1 / t + (t - 1) n1 - 2 / t + t - 2,ande (G) < frac(1, 2) (s - t + 1)1 / t n2 - 1 / t + frac(1, 2) (t - 1) n2 - 2 / t + frac(1, 2) (t - 2) n . © 2009 Elsevier Inc. All rights reserved.
Linear Algebra and Its Applications
Nikiforov, V. (2010). A contribution to the Zarankiewicz problem. Linear Algebra and Its Applications, 432 (6), 1405-1411. https://doi.org/10.1016/j.laa.2009.10.040