The bipartite Ramsey number b(m, n) is the minimum b such that any 2-coloring of Kb,b results in a monochromatic Km,m subgraph in the first color or a monochromatic Kn,n subgraph in the second color. The Zarankiewicz number z(m, n; s, t) is the maximum size among Ks,t-free subgraphs of Km,n. In this work, we discuss the intimate relationship between the two numbers as well as propose a new method for bounding the Zarankiewicz numbers. We use the better bounds to improve the upper bound on b(2, 5), in addition we improve the lower bound of b(2, 5) by construction. The new bounds are shown to be 17 ≤ b(2, 5) ≤ 18. Additionally, we apply the same methods to the multicolor case b(2, 2, 3) which has previously not been studied and determine bounds to be 16 ≤ b(2, 2, 3) ≤ 23.
Library of Congress Subject Headings
Ramsey numbers; Graph coloring
Applied and Computational Mathematics (MS)
Department, Program, or Center
School of Mathematical Sciences (COS)
Collins, Alex F., "Bipartite Ramsey Numbers and Zarankiewicz Numbers" (2015). Thesis. Rochester Institute of Technology. Accessed from
RIT – Main Campus