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

Abstract

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