zarankiewicz numbers and bipartite ramsey numbers

the zarankiewicz number z(b; s) is the maximum size of a subgraph of kb,b which does not contain ks,s as a subgraph. the two-color bipartite ramsey number b(s, t) is the smallest integer b such that any coloring of the edges of kb,b with two colors contains a ks,s in the rst color or a kt,t in the second color.in this work, we design and exploit a computational method for bounding and computing zarankiewicz numbers. using it, we obtain several new values and bounds on z(b; s) for 3≤s≤6. our approach and new knowledge about z(b; s) permit us to improve some of the results on bipartite ramsey numbers obtained by