On optimal \(M\)-sets related to Motzkin's problem (Q1983376)

Summary: Let \(M\) be a set of positive integers. A set \(S\) of nonnegative integers is called an \(M\)-set if \(a\) and \(b\in S\), then \(a-b\notin M\). If \(S\subseteq\{ 0, 1, \dots, n\}\) is an \(M\)-set with the maximal cardinality, then \(S\) is called a maximal \(M\)-set of \(\{0, 1, \dots, n\}\). If \(S\cap\{0, 1, \dots, n\}\) is a maximal \(M\)-set of \(\{0, 1, \dots, n\}\) for all integers \(n\geq0\), then we call \(S\) an optimal \(M\)-set. In this paper, we study the existence of an optimal \(M\)-set.