On the Bounds of Van der Waerden Numbers

Van der Waerden’s Theorem asserts that for any two positive integers k and r, one may find an integer w=w(k,r) known as the Van der Waerden Number such that for every r-coloring of the integers from 1 to w there exists a monochromatic arithmetic progression of length k. This groundbreaking theorem in combinatorics has greatly impacted the field of discrete math for decades. However, it is quite difficult to find the exact values of w. As such, it would be worth more of our time to try and bound such a value, both from below and above, in order to restrict the possible values of the Van der Waerden Numbers. In this thesis we will endeavor to bound such a number; in addition to proving Van der Waerden’s Theorem, we will discuss the unique functions that bound the Van der Waerden Numbers.