Maximal Density of Sets with Missing Differences and Various Coloring Parameters of Distance Graphs

Abstract

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.

On some summability methods for a q-analogue of an integral type operator based on multivariate q-Lagrange polynomials

“…There are many ways to define the density of the subsets of natural numbers and these definitions are playing a pivotal role in the areas of Number Theory and Graph Theory (see [16,17]). Any sequence x n is called statistically convergent to l if, for each ǫ > 0, we have the following…”