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

Publication Date


Document Type


Student Type


Degree Name

Applied and Computational Mathematics (MS)

Department, Program, or Center

School of Mathematical Sciences (COS)


Stanisław Radziszowski

Advisor/Committee Member

Darren Narayan

Advisor/Committee Member

Hossein Shahmohamad


Physical copy available from RIT's Wallace Library at QA166 .C654 2015


RIT – Main Campus

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