12 problems

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 $H(I,I+2)$.…

Let $G(k)$ be the length of an optimal Golomb ruler with $k$ marks. Singer-type upper-bound conjecture. For every integer $k\geq6$, … The source derives this as a consequence of th…

Let $I$ and $J$ be positive integers. A regular $(I+1,J,(I+1)J)$-DGR is a collection of $I+1$ disjoint $J$-mark Golomb rulers partitioning $\{1,\ldots,(I+1)J\}$. Regular extension…

For positive integers $I$ and $J$, an $(I,J,n)$-DGR is a collection of $I$ disjoint Golomb rulers, each a $J$-subset of $\{1,2,\ldots,n\}$. Let $H(I,J)$ be the least positive integ…

For positive integers $I$ and $J$, let $H(I,J)$ denote the least $n$ admitting $I$ disjoint $J$-mark Golomb rulers in $\{1,\ldots,n\}$. A DGR is regular when $n=IJ$. Regularity pro…

For positive integers $I$ and $J$, let $H(I,J)$ be the least $n$ such that there are $I$ disjoint Golomb rulers, each a $J$-subset of $\{1,2,\ldots,n\}$. Incremental bound conjectu…

Let $A = \{a_1,\ldots,a_n\}$ be a finite set of positive integers, and let an $m$-mark Golomb ruler be a subset of $A$ with $m$ elements and all pairwise differences distinct. Koml…

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…

Let $H(I,J)$ be the least positive integer $n$ such that there exist $I$ disjoint Golomb rulers, each a $J$-subset of $\\{1,2,\ldots,n\\}$. A DGR is regular when $H(I,J)=IJ$. Persi…

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

A Golomb ruler is a set of marks on a line whose pairwise distances are all distinct. Let the length of the ruler be fixed, and consider the maximum number of marks it can contain.…

Let $S$ be an optimal Golomb ruler with $m$ markings whose last marking is at $n$, so that $m$ is maximal for the given $n$ (equivalently, $n$ is minimal for the given $m$). Golomb…