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 Km,n that does not contain Ks,t as a subgraph. The bipartite Ramsey number b(n1, · · · , nk) is the least positive integer b such that any coloring of the edges of Kb,b with k colors will result in a monochromatic copy of Kni,ni in the i-th color, for some i, 1 ≤ i ≤ k. If ni = m for all i, then we denote this number by bk(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 b4(2) = 19, and establish new general lower and upper bounds on bk(2). ∗This research was partially supported by the Polish National Science Centre grant 2011/02/A/ST6/00201.