{"title":"注意下限阈值","authors":"Lutz Warnke","doi":"10.1002/rsa.21194","DOIUrl":null,"url":null,"abstract":"Gunby–He–Narayanan showed that the logarithmic gap predictions of Kahn–Kalai and Talagrand (proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about thresholds of up‐sets do not apply to down‐sets. In particular, for the down‐set of triangle‐free graphs, they showed that there is a polynomial gap between the threshold and the factional expectation threshold. In this short note we give a simpler proof of this result, and extend the polynomial threshold gap to down‐sets of ‐free graphs.","PeriodicalId":20948,"journal":{"name":"Random Structures and Algorithms","volume":"2 6","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Note on down-set thresholds\",\"authors\":\"Lutz Warnke\",\"doi\":\"10.1002/rsa.21194\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Gunby–He–Narayanan showed that the logarithmic gap predictions of Kahn–Kalai and Talagrand (proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about thresholds of up‐sets do not apply to down‐sets. In particular, for the down‐set of triangle‐free graphs, they showed that there is a polynomial gap between the threshold and the factional expectation threshold. In this short note we give a simpler proof of this result, and extend the polynomial threshold gap to down‐sets of ‐free graphs.\",\"PeriodicalId\":20948,\"journal\":{\"name\":\"Random Structures and Algorithms\",\"volume\":\"2 6\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-11-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Random Structures and Algorithms\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1002/rsa.21194\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures and Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1002/rsa.21194","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
Gunby-He-Narayanan表明，Kahn-Kalai和Talagrand(由Park-Pham和Frankston-Kahn-Narayanan-Park证明)关于上升阈值的对数间隙预测不适用于下降阈值。特别地，对于无三角图的下集，他们证明了阈值与分式期望阈值之间存在多项式间隙。在这篇简短的笔记中，我们给出了这个结果的一个更简单的证明，并将多项式阈值间隙扩展到F $$ F $$自由图的下集。
Gunby–He–Narayanan showed that the logarithmic gap predictions of Kahn–Kalai and Talagrand (proved by Park–Pham and Frankston–Kahn–Narayanan–Park) about thresholds of up‐sets do not apply to down‐sets. In particular, for the down‐set of triangle‐free graphs, they showed that there is a polynomial gap between the threshold and the factional expectation threshold. In this short note we give a simpler proof of this result, and extend the polynomial threshold gap to down‐sets of ‐free graphs.
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