University Of Tehran Press Journal of Algorithms and Computation 2476-2776 47 1 2016 06 10 Zarankiewicz Numbers and Bipartite Ramsey Numbers 63 78 7943 10.22059/jac.2016.7943 EN Alex F. Collins Rochester Institute of Technology, School of Mathematical Sciences, Rochester, NY 14623 Alexander W. N. Riasanovsky University of Pennsylvania, Department of Mathematics, Philadelphia, PA 19104, USA John C. Wallace Trinity College, Department of Mathematics, Hartford, CT 06106, USA Stanis Law P. Radziszowski Rochester Institute of Technology, Department of Computer Science, Rochester, NY 14623 Journal Article 2016 02 08 The Zarankiewicz number <em>z</em>(<em>b;</em> <em>s</em>) is the maximum size of a subgraph of <em>K</em>_{<em>b</em>,<em>b</em>} which does not contain <em>K</em>_{<em>s</em>,<em>s</em>} as a subgraph. The two-color bipartite Ramsey number <em>b</em>(<em>s</em>, <em>t</em>) is the smallest integer <em>b</em> such that any coloring of the edges of <em>K</em>_{<em>b</em>,<em>b</em>} with two colors contains a <em>K</em>_{<em>s</em>,<em>s</em>} in the rst color or a <em>K</em>_{<em>t</em>,<em>t</em>} in the second color.<br />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 <em>z</em>(<em>b;</em> <em>s</em>) for 3≤s≤6. Our approach and new knowledge about <em>z</em>(<em>b;</em> <em>s</em>) permit us to improve some of the results on bipartite Ramsey numbers obtained by https://jac.ut.ac.ir/article_7943_0d5d2a7f40f78dfe0e529df98f3049dd.pdf