On Some Zarankiewicz Numbers and Bipartite Ramsey Numbers for Quadrilateral

The Zarankiewicz number z(m, n; s, t) is the maximum number of edges in a subgraph of K-m,K-n that does not contain K-s,K-t as a subgraph. The bipartite Ramsey number b(ni, ... , n(k)) is the least positive integer b such that any coloring of the edges of K-b,K-b with k colors will result in a monochromatic copy of K-n,K-ni in the i-th color, for some i, 1 <= i <= k. If n(i) = m for all i, then we denote this number by b(k)(m). In this paper we obtain the exact values of some Zarankiewicz numbers for quadrilateral (s = t = 2), and we derive new bounds for diagonal multicolor bipartite Ramsey numbers avoiding quadrilateral. In particular, we prove that b(4)(2) = 19, and establish new general lower and upper bounds on b(k) (2).