New results on the Zarankiewicz problem

Let (X,Y) denote a bipartite graph with classes X and Y such that |X|=m and |Y|=n. A complete bipartite subgraph with s vertices in X and t vertices in Y is denoted by K"("s","t"). The Zarankiewicz problem consists in finding the maximum number of edges, denoted by z(m,n;s,t), of a bipartite graph (X,Y) with |X|=m and |Y|=n without a complete bipartite K"("s","t") as a subgraph. First, we prove that z(m,n;s,t)=mn-(m+n-s-t+1) if max{m,n}=