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Let $M$ be a given set of positive integers. A set $S$ of nonnegative integers is said to be an $M$-set if $a,b \in S$ implies $a-b \notin M$. In an unpublished problem collection, Motzkin asked to find maximal upper asymptotic density, denoted by $\mu(M)$, of $M$-sets. The first published work on $\mu(M)$ is due to Cantor and Gordon in 1973, in which, they found the exact value of $\mu(M)$ when $|M| \leq 2$. In fact, this is the only general case, in which, we have a closed formulae for $\mu(M)$. If $|M| \geq 3$, then the exact value of $\mu(M)$ is not known for the general set $M$. In the past six decades or so, several attempts have been given to study $\mu(M)$ but $\mu(M)$ has been found exactly or estimated only in very few cases. In this paper, we study $\mu(M)$ for the families $M = \{ a,a+1,x \}$ and $M = \{ a,a+1,x,y \}$, where $y-x \leq 2$ and $y \gt x \gt a+1$. Our results in the case of $M = \{ a,a+1,x \}$ also give counterexamples to a conjecture of Carraher. Although, different counterexamples to this conjecture, were already given by Liu and Robinson in 2020. We also relate our results with the already know results for the families $M = \{ 1,2,x,x+2 \}$ and $M = \{ 2,3,x,x+2 \}$. |
