Some constructive results on Disjoint Golomb Rulers

TL;DR

This paper explores the structure and construction of disjoint Golomb rulers, proposing conjectures on their existence and providing constructive results that extend previous inequalities in the field.

Contribution

It introduces new conjectures on the construction of disjoint Golomb rulers and offers constructive proofs that generalize existing inequalities.

Findings

Proposes a main conjecture on the existence of disjoint Golomb rulers within any set of size H(I, J).

Provides constructive results that improve or extend known inequalities on DGR.

Suggests a generalization of a 1975 conjecture by Komlós, Sulyok, and Szemerédi.

Abstract

A set $\{a_i\:|\: 1\leq i \leq k\}$ of non-negative integers is a Golomb ruler if differences $a_i-a_j$, for any $i \neq j$, are all distinct.All finite Sidon sets are Golomb rulers, and vice versa. A set of $I$ disjoint Golomb rulers (DGR) each being a $J$-subset of $\{1,2,\cdots, n\}$ is called an $(I,J,n)$-DGR. Let $H(I, J)$ be the least positive integer $n$ such that there is an $(I,J,n)$-DGR. In this paper, we propose a series of conjectures on the constructions and structures of DGR. The main conjecture states that if $A$ is any set of positive integers such that $|A| = H(I, J)$, then there are $I$ disjoint Golomb rulers, each being a $J$-subset of $A$, which generalizes the conjecture proposed by Koml{\'o}s, Sulyok and Szemer{\'e}di in 1975 on the special case $I = 1$. This main conjecture implies some interesting conjectures on disjoint Golomb rulers. We also prove some constructive results on DGR, which improve or generalize some basic inequalities on DGR proved by Kl{\o}ve.