Date of Final Oral Examination (Defense)

12-2011

Type of Culminating Activity

Thesis

Degree Title

Master of Science in Mathematics

Department

Mathematics

Major Advisor

Andres E. Caicedo, Ph.D.

Abstract

This thesis considers several coloring problems all of which have a combinatorial flavor. We review some results on the chromatic number of the plane, and improve a bound on the value of regressive Ramsey numbers. The main work of this thesis considers the problem of whether given any n ≥ 1; one can color Z+ in such a way that for all a ϵ Z+ the numbers a, 2a, 3a, ..., na are assigned different colors. Such colorings are referred to as satisfactory. We provide a sufficient condition for guaranteeing the existence of satisfactory colorings and analyze the resulting structure. Explicit constructions are given for n ≤ 54: The thesis concludes with some suggestions towards a general argument.

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