Avoidable Sets in Groups

Abstract

In a set equipped with a binary operation, \((S, \cdot)\), a subset \(U\) is defined to be avoidable if there exists a partition \(\{A, B\}\) of \(S\) such that no element of \(U\) is the product of two distinct elements of \(A\) or of two distinct elements of \(B\). For more than two decades, avoidable sets in the natural numbers (under addition) have been studied by renowned mathematicians such as Erdős, and a few families of sets have been shown to be avoidable in that setting. In this paper, we investigate the generalized notion of an avoidable set and determine the avoidable sets in several families of groups; previous work in this field considered only the case \((S, \cdot) = (\mathbb{N}, +)\).