{"title":"Subsquares in random Latin squares","authors":"Jack Allsop, Ian M. Wanless","doi":"arxiv-2409.08446","DOIUrl":null,"url":null,"abstract":"We prove that with probability $1-o(1)$ as $n \\to \\infty$, a uniformly random\nLatin square of order $n$ contains no subsquare of order $4$ or more, resolving\na conjecture of McKay and Wanless. We also show that the expected number of\nsubsquares of order 3 is bounded.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"92 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08446","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that with probability $1-o(1)$ as $n \to \infty$, a uniformly random
Latin square of order $n$ contains no subsquare of order $4$ or more, resolving
a conjecture of McKay and Wanless. We also show that the expected number of
subsquares of order 3 is bounded.
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