Zarankiewicz numbers from pairwise balanced designs

For positive integers s, t, m and n, the Zarankiewicz number Zs,t(m, n) is usually defined to be the maximum number of edges in a bipartite graph with parts of sizes m and n that has no complete bipartite subgraph containing s vertices in the part of size m and t vertices in the part of size n. In this talk, we obtain some new exact Zarankiewicz numbers by using pairwise balanced designs.