关于不相交Golomb规则
On Disjoint Golomb Rulers
摘要：
一个非负整数集合$\{a_i\:|\: 1\leq i \leq k\}$是一个Golomb标尺，如果对于任何$i \neq j$，其差值$a_i-a_j$都是不同的。一组$I$不相交Golomb规则（DGR），每个都是$\{1,2,\cdots, n\}$的$J$ -子集，称为$(I,J,n)-DGR$。令$H(I, J)$为最小正$n$，使得存在$(I,J,n)-DGR$。本文对DGR的构造和结构作了一系列的阐述。主要猜想是：如果$A$是正整数的任意集合，使得$|A| = H(I, J)$，则存在$I$不相交的Golomb规则，每个规则都是$A$的$J$ -子集，这推广了Koml{ó}s，Sulyok和Szemer{é}di在1975年提出的关于特殊情况$I = 1$的猜想。通过适当的计算，对$I$和$J$的某些值，这些结构进行了计算验证。通过对$7 \leq I \leq 13$和$10 \leq J \leq 13$的计算机搜索，得到了$H(I,J)$的18个精确值和10个上界. $I > 13$和$10 \leq J \leq 13$、$H(I,J)=IJ$的迁移没有困难。
英文摘要：
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. 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 $n$ 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{ó}s, Sulyok and Szemer{é}di 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 $7 \leq I \leq 13$ and $10 \leq J \leq 13$. Moveover for $I > 13$ and $10 \leq J \leq 13$, $H(I,J)=IJ$ are determined without difficulty.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
