The regularity conjecture for $H(I,I+2)$

About 12 years old · traced to 2014

Let $H(I,J)$ be the least positive integer $n$ such that there exist $I$ disjoint Golomb rulers, each a $J$ -subset of $\\{1,2,\ldots,n\\}$ . Regularity conjecture. For every integer $I\geq8$ ,

H(I,I+2)=I(I+2).

Thus the conjecture predicts regular disjoint Golomb rulers in this parameter range. The paper reports that the assertion has been confirmed computationally for $I\in\\{8,\ldots,150\\}$ , but treats the unrestricted statement as conjectural.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Regularity conjecture for $H(I,I+2)$
For positive integers $I$ and $J$ , let $H(I,J)$ be the least $n$ for which there are $I$ disjoint $J$ -mark Golomb rulers in $\{1,\ldots,n\}$ . Regularity conjecture for $H(I,I+2)$ . For every integer $I\geq4$ ,
$H(I,I+2)=I(I+2).$
The source reports that this is known for $I\in\{4,5,6,7\}$ from results of Kløve and Shearer, while the general assertion remains open.
source: Xiaodong Xu, Baoxin Xiu, Changjun Fan and Meilian Liang, “Some constructive results on Disjoint Golomb Rulers”, arXiv:2409.14409 (2024).

References

Primary source

Xiu Baoxin, Changjun Fan and Meilian Liang, “On Disjoint Golomb Rulers”, arXiv:1405.4535 (2014).

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Regularity conjecture for H(I,I+2)H(I,I+2)H(I,I+2)

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Regularity conjecture for $H(I,I+2)$