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December, 2020 Maximal Density of Sets with Missing Differences and Various Coloring Parameters of Distance Graphs
Ram Krishna Pandey, Neha Rai
Taiwanese J. Math. 24(6): 1383-1397 (December, 2020). DOI: 10.11650/tjm/200403

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 $\mu(M)$, 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 $\mu(M)$ for $|M| \leq 2$. Partial results are known in the case $|M| \geq 3$ 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 $\mu(M)$ 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 $\mu(M)$. 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.

Citation

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Ram Krishna Pandey. Neha Rai. "Maximal Density of Sets with Missing Differences and Various Coloring Parameters of Distance Graphs." Taiwanese J. Math. 24 (6) 1383 - 1397, December, 2020. https://doi.org/10.11650/tjm/200403

Information

Received: 24 December 2019; Revised: 29 March 2020; Accepted: 16 April 2020; Published: December, 2020
First available in Project Euclid: 19 November 2020

MathSciNet: MR4176878
Digital Object Identifier: 10.11650/tjm/200403

Subjects:
Primary: 11B05
Secondary: 05C15

Rights: Copyright © 2020 The Mathematical Society of the Republic of China

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Vol.24 • No. 6 • December, 2020
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