{"title":"Tiling with monochromatic bipartite graphs of bounded maximum degree","authors":"António Girão, Oliver Janzer","doi":"10.1112/mtk.12280","DOIUrl":null,"url":null,"abstract":"<p>We prove that for any <span></span><math></math>, there exists a constant <span></span><math></math> such that the following is true. Let <span></span><math></math> be an infinite sequence of bipartite graphs such that <span></span><math></math> and <span></span><math></math> hold for all <span></span><math></math>. Then, in any <span></span><math></math>-edge-coloured complete graph <span></span><math></math>, there is a collection of at most <span></span><math></math> monochromatic subgraphs, each of which is isomorphic to an element of <span></span><math></math>, whose vertex sets partition <span></span><math></math>. This proves a conjecture of Corsten and Mendonça in a strong form and generalises results on the multi-colour Ramsey numbers of bounded-degree bipartite graphs. It also settles the bipartite case of a general conjecture of Grinshpun and Sárközy.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12280","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12280","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that for any , there exists a constant such that the following is true. Let be an infinite sequence of bipartite graphs such that and hold for all . Then, in any -edge-coloured complete graph , there is a collection of at most monochromatic subgraphs, each of which is isomorphic to an element of , whose vertex sets partition . This proves a conjecture of Corsten and Mendonça in a strong form and generalises results on the multi-colour Ramsey numbers of bounded-degree bipartite graphs. It also settles the bipartite case of a general conjecture of Grinshpun and Sárközy.
期刊介绍：
Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
