Hilton猜想及其推广的组合证明

中文摘要 AI 辅助

利用精细吸收方法，我们证明了对每个整数$g\ge 1$和实数$\gamma > 0$，以及足够大的$n$，存在一个$n^{-1+\gamma}$-扩散分布，作用于阶为$n$、围长至少为$g$且没有真子方阵的拉丁方上。这蕴含了20世纪70年代Hilton猜想（最近由Allsop和Wanless用代数方法证明）的一个组合证明，即对所有足够大的$n$，存在一个阶为$n$的无子方阵拉丁方；实际上，它蕴含至少存在$n^{(1-o(1))n^2}$个无子方阵拉丁方。同时，它也蕴含了高阶围长拉丁方的存在性（最近由Kwan、Sah、Sawhney和Simkin证明），甚至在高围长拉丁方上存在一个$n^{-1+\gamma}$-扩散分布。

英文摘要

Using refined absorption, we prove that for every integer $g\ge 1$ and real $γ> 0$, and for sufficiently large $n$, there exists an $n^{-1+γ}$-spread distribution on Latin squares of order $n$ and girth at least $g$ that have no proper subsquares. This implies a combinatorial proof of Hilton's conjecture from the 1970s (recently proved algebraically by Allsop and Wanless) that for all sufficiently large $n$, there exists a subsquare-free Latin square of order $n$; indeed, it implies there exist at least $n^{(1-o(1))n^2}$ subsquare-free squares. Simultaneously it also implies the existence of high girth Latin squares (recently proved by Kwan, Sah, Sawhney and Simkin) and even an $n^{-1+γ}$-spread distribution on high girth Latin squares.

发表机构

• School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）
• University of Waterloo（滑铁卢大学）

机构由 AI 辅助整理，请以论文原文为准。