Maximal Density of M-sets and its applications

Abstract

Suppose 𝑀 is any given set of positive integers and we have to find the maximal density of 𝑀-sets which is denoted by 𝜇(𝑀), where 𝑀-sets are defined as the set of non–negative integers in which the difference of any two elements of 𝑀, does not lie in 𝑀. Maximal density for 𝜇(𝑀) will be given for |𝑀| ≤ 2 and some partial results are known for the case |𝑀| ≥ 3 including when 𝑀 is infinite. This number theory problem of getting maximal density will be further associated in various coloring problem of distance graphs generated by 𝑀 in Graph Theory. It is known that the reciprocal of fractional chromatic number of distance graph generated by 𝑀 is equal to the maximal density of 𝑀-sets given 𝑀 is finite. Some special family of finite sets known as almost difference Closed Sets and some infinite sets can be discussed further for which value of 𝜇(𝑀) will be find. For some of these, exact value of 𝜇(𝑀) will be known and for some we can get the bounds on 𝜇(𝑀) which will further help in finding the fractional chromatic number of the distance graph generated by that 𝑀-set. Using all the methods elucidated here to find 𝜇(𝑀) can be used further to get fractional chromatic number of distance graphs generated by these special family of sets as both are indeed the same problem.