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Let $M$ be a set of positive integers. We study the maximal density $\mu(M)$of the sets of nonnegative integers $S$ whose elements do not differ by anelement in $M$. In 1973, Cantor and Gordon established a formula for $\mu(M)$for $|M|\leq 2$. Since then, many researchers have worked upon the problem andfound several partial results in the case $|M|\geq 3$, including some resultsin the case, $M$ is an infinite set. In this paper, we study the maximaldensity problem for the families $M=\{a,a+1,2a+1,n\}$ and$M=\{a,a+1,2a+1,3a+1,n\}$, where $a$ and $n$ are positive integers. In most ofthe cases, we find bounds for the parameter \textit{kappa}, denoted by$\kappa(M)$, which actually serves as a lower bound for $\mu(M)$. The parameter$\kappa(M)$ has already got its importance due to its rich connection with theproblems such as the "lonely runner conjecture" in Diophantine approximationsand coloring parameters such as "circular coloring" and "fractional coloring"in graph theory.

Maximal density and the kappa values for the families $\{a,a+1,2a+1,n\}$ and $\{a,a+1,2a+1,3a+1,n\}$