For a given set M of positive integers, a well-known problem of Motzkin asked to determine the maximal asymptotic density of M -sets, denoted by µ(M ), where an M -set is a set of non-negative integers in which no two elements differ by an element in M . In 1973, Cantor and Gordon found µ(M ) for |M | ≤ 2. Partial results are known in the case |M | ≥ 3 including results in the case when M is an infinite set. This number theory problem is also related to various types of coloring problems of the distance graphs generated by M . In particular, it is known that the reciprocal of the fractional chromatic number of the distance graph generated by M is equal to the value µ(M ) when M is finite. Motivated by the families M = {a, b, a + b} and M = {a, b, a + b, b − a} discussed by Liu and Zhu, we study two families of sets M , namely, M = {a, b, b − a, n(a + b)} and M = {a, b, a + b, n(b − a)}.For both of these families, we find some exact values and some bounds on µ(M ). We also find bounds on the fractional and circular chromatic numbers of the distance graphs generated by these families. Furthermore, we determine the exact values of chromatic number of the distance graphs generated by these two families.
For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.
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| ≤ 2. Since then, many researchers have worked upon the problem and found several partial results in the case |M| ≥ 3, including some results in the case when 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 and n is sufficiently large. In most of the cases, we find bounds for the parameter 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 problems such as the “lonely runner conjecture” in Diophantine approximation and coloring parameters such as “circular coloring” and “fractional coloring” in graph theory. We also give some partial results for the general family M = {a, a +1, 2a + 1,…, (s – 2)a + 1, n}, where s ≥ 5 and mention related problems in the remaining cases for future work.
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