  The Zarankiewicz number $z(m,n;s,t)$ is the maximum number of edges in a
subgraph of $K_{m,n}$ that does not contain $K_{s,t}$ as a subgraph. The
bipartite Ramsey number $b(n_1, \cdots, n_k)$ is the least positive integer $b$
such that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in
a monochromatic copy of $K_{n_i,n_i}$ in the $i$-th color, for some $i$, $1 \le
i \le k$. 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 $b_k(2)$.

Combinatorics;Discrete Mathematics