The Zarankiewicz Problem via Chow Forms

The well-known Zarankiewicz problem [Za] is to determine the least positive integer Z(m;n;r;s) such that each m £ n 0-1 matrix containing Z(m;n;r;s) ones has an r £ s submatrix consisting entirely of ones. In graph-theoretic language, this is equivalent to finding the least positive integer Z(m;n;r;s) such that each bipartite graph on m black vertices and n white vertices with Z(m;n;r;s) edges has a complete bipartite subgraph on r black vertices and s white vertices. A complete solution of the Zarankiewicz problem has not been given. While exact values of Z(m;n;r;s) are known for certain infinite subsets of m;n;r and s, only asymptotic bounds are known in the general case; for example, see ˇ