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 Goddard, Henning and Oellermann in 2000. In particular, we compute the smallest previously unknown bipartite Ramsey number, b(2; 5) = 17. Moreover, we prove that up to isomorphism there exists a unique 2-coloring which witnesses the lower bound 16 < b(2; 5). We also nd tight bounds on b(2; 2; 3), 17 b(2; 2; 3) 18, which currently is the smallest open case for multicolor bipartite Ramsey numbers.