On Optimal M-Sets Related to Motzkin’s Problem

Abstract

Let $M$ be a set of positive integers. A set $S$ of nonnegative integers is called an $M‐$ set if $a\text{\hspace{0.17em}and\hspace{0.17em}}b\in S$ , then $a-b\notin M$ . If $S\subseteq \left\{\mathrm{0,1},\dots ,n\right\}$ is an $M-$ set with the maximal cardinality, then $S$ is called a maximal $M-$ set of $\left\{\mathrm{0,1},\dots ,n\right\}$ . If $S\cap \left\{\mathrm{0,1},\dots ,n\right\}$ is a maximal $M-$ set of $\left\{\mathrm{0,1},\dots ,n\right\}$ for all integers $n\ge 0$ , then we call $S$ an optimal $M-$ set. In this paper, we study the existence of an optimal $M-$ set.