## Master’s thesis

My student

Thomas Chartier

will be will defending his Master’s thesis for a Mathematics degree, titled

Coloring Problems

Abstract: We consider two coloring problems which have a combinatorial flavor. The chromatic number of the plane, $\chi$, is the least number of colors necessary to color ${\mathbb R}^2$ in such a way that no two points at a unit distance apart receive the same color. It is well known that $4\le\chi\le 7$. We begin by discussing the arguments that give these bounds.

The main point the talk considers the problem of whether given any $n\in{\mathbb Z}^+$, one can color the positive integers in such a way that for all $a\in{\mathbb Z}^+$, the numbers $a,2a,3a,\dots,na$ are assigned different colors. Such colorings are referred to as satisfactory. We begin with an example which provides insight into the underlying structure inherent in all satisfactory colorings, present a sufficient condition for guaranteeing the existence of satisfactory colorings, and analyze the resulting structure.

The defense will be held Thursday, October 13, 2011, 2:40-3:30 pm, in Room MP-201. Refreshments will be served in the math lounge at 2:15 pm.