A polychromatic approach to a Turán-type problem in finite groups

Abstract

Let $G$ be a finite abelian group. Given $S \subseteq G$, $a \in G$, any set of the form $a + S = \{a \} + S$ is called a {\em translate} of $S$. A coloring of the elements of $G$ is {\em $S-$polychromatic} if every translate of $S$ contains an element of each color. The largest number of colors allowing an $S-$polychromatic coloring of the translates of $S$ is known as the {\em polychromatic number} of $S$, denoted $p_G(S)$. Determining the polychromatic number of finite abelian groups is a relatively new and unexplored method that can be used to solve the following problem: What is the maximum number of elements in a subset of $G$ which does not contain a translate of $S$? This type of problem called a Tur\'{a}n-type problem is common in extremal graph theory, but is new to the realm of algebra. This dissertation aims to determine bounds on the desired maximum number of elements, referred to as the {\em Tur\'{a}n number}, using polychromatic colorings on the desired maximum number of elements within the context of the well known abelian group the integers modulo $n$, denoted $\mathbb{Z}_n$ for all $n \geq 3$. The problem is also redefined and explored within the context of nonabelian groups such as the dihedral group and the dicyclic group.\\ \indent Trivial bounds are first presented on the Tur\'{a}n number for any group. Results on the improvement of the trivial lower bound are then presented. The results involve determining the polychromatic number for various subsets. The polychromatic number of any subset of cardinality two of any group is determined. Results related to the polychromatic number of any subset of odd prime cardinality of $\mathbb{Z}_n$ are presented. Results related to the polychromatic number of subsets of cardinality three and $n$ of $D_{2n}$ and subsets of cardinality three of $Dic_n$ are also presented.</p