Topics in Ramsey Theory

Abstract

Ramsey theory is the study of conditions under which mathematical objects show order when partitioned. Ramsey theory on the integers concerns itself with partitions of $[1,n]$ into $r$ subsets and asks the question whether one (or more) of these $r$ subsets contains a $k$-term member of $\mathcal{F}$, where $[1,n]=\{1,2,3,\ldots,n\}$ and $\mathcal{F}$ is a certain family of subsets of $\mathbb{Z}^+$. When $\mathcal{F}$ is fixed to be the set of arithmetic progressions, the corresponding Ramsey-type numbers are called the van der Waerden numbers. I started the project choosing $\mathcal{F}$ to be the set of semi-progressions of scope $m$. A semi-progression of scope $m\in \mathbb{Z}^+$ is a set of integers $\{x_1,x_2,\ldots,x_k\}$ such that for some $d\in\mathbb{Z}^+$, $x_{i}-x_{i-1}\in\{d,2d,\ldots,md\}$ for all $i\in\{2,3,\ldots,k\}$. The exact values of Ramsey-type functions corresponding to semi-progressions are not known. We use $SP_m(k)$ to denote these numbers as a Ramsey-type function of $k$ for a fixed scope $m$. During this project, I used the probabilistic method to get an exponential lower bound for any fixed $m$. The first chapter starts with a brief introduction to Ramsey theory and then explains the problem considered. In the second chapter, I give the results obtained on semi-progressions. In the third chapter, I will discuss the lower bound obtained on $Q_1(k)$. When $\mathcal{F}$ is chosen to be quasi-progressions of diameter $n$, the corresponding Ramsey-type numbers obtained are denoted as $Q_n(k)$. The last chapter gives an exposition of advanced probabilistic techniques, in particular concentration inequalities and how to apply them.