On Disjoint Golomb Rulers

A set {ai | 1 ≤ i ≤ k} of non-negative integers is a Golomb ruler if differences ai − aj , for any i 6= j, are all distinct. A set of I disjoint Golomb rulers (DGR) each being a J-subset of {1, 2, · · · , n} is called an (I, J, n) − DGR. Let H(I, J) be the least positive 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 Komlós, Sulyok and Szemerédi in 1975 on the special case I = 1. These conjectures are computationally verified for some values of I and J through modest computation. Eighteen exact values of H(I, J) and ten upper bounds on H(I, J) are obtained by computer search for 7 ≤ I ≤ 13 and 10 ≤ J ≤ 13. Moveover for I > 13 and 10 ≤ J ≤ 13, H(I, J) = IJ are determined without difficulty.