{"title":"对角线拉姆齐通过有效的准随机","authors":"A. Sah","doi":"10.1215/00127094-2022-0048","DOIUrl":null,"url":null,"abstract":"We improve the upper bound for diagonal Ramsey numbers to \\[R(k+1,k+1)\\le\\exp(-c(\\log k)^2)\\binom{2k}{k}\\] for $k\\ge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal \"effective quasirandomness\" results about convergence of graphs. This optimality represents a natural barrier to improvement.","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"36","resultStr":"{\"title\":\"Diagonal Ramsey via effective quasirandomness\",\"authors\":\"A. Sah\",\"doi\":\"10.1215/00127094-2022-0048\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We improve the upper bound for diagonal Ramsey numbers to \\\\[R(k+1,k+1)\\\\le\\\\exp(-c(\\\\log k)^2)\\\\binom{2k}{k}\\\\] for $k\\\\ge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal \\\"effective quasirandomness\\\" results about convergence of graphs. This optimality represents a natural barrier to improvement.\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2020-05-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"36\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1215/00127094-2022-0048\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1215/00127094-2022-0048","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
We improve the upper bound for diagonal Ramsey numbers to \[R(k+1,k+1)\le\exp(-c(\log k)^2)\binom{2k}{k}\] for $k\ge 3$. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal "effective quasirandomness" results about convergence of graphs. This optimality represents a natural barrier to improvement.
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