{"title":"图和超图中单调路径的拉姆齐问题","authors":"Lior Gishboliner, Zhihan Jin, Benny Sudakov","doi":"10.1007/s00493-024-00082-7","DOIUrl":null,"url":null,"abstract":"<p>The study of ordered Ramsey numbers of monotone paths for graphs and hypergraphs has a long history, going back to the celebrated work by Erdős and Szekeres in the early days of Ramsey theory. In this paper we obtain several results in this area, establishing two conjectures of Mubayi and Suk and improving bounds due to Balko, Cibulka, Král and Kynčl. For example, in the graph case, we show that the ordered Ramsey number for a fixed clique versus a fixed power of a monotone path of length <i>n</i> is always linear in <i>n</i>. Also, in the 3-graph case, we show that the ordered Ramsey number for a fixed clique versus a tight monotone path of length <i>n</i> is always polynomial in <i>n</i>. As a by-product, we also obtain a color-monotone version of the well-known Canonical Ramsey Theorem of Erdős and Rado, which could be of independent interest.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Ramsey Problems for Monotone Paths in Graphs and Hypergraphs\",\"authors\":\"Lior Gishboliner, Zhihan Jin, Benny Sudakov\",\"doi\":\"10.1007/s00493-024-00082-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The study of ordered Ramsey numbers of monotone paths for graphs and hypergraphs has a long history, going back to the celebrated work by Erdős and Szekeres in the early days of Ramsey theory. In this paper we obtain several results in this area, establishing two conjectures of Mubayi and Suk and improving bounds due to Balko, Cibulka, Král and Kynčl. For example, in the graph case, we show that the ordered Ramsey number for a fixed clique versus a fixed power of a monotone path of length <i>n</i> is always linear in <i>n</i>. Also, in the 3-graph case, we show that the ordered Ramsey number for a fixed clique versus a tight monotone path of length <i>n</i> is always polynomial in <i>n</i>. As a by-product, we also obtain a color-monotone version of the well-known Canonical Ramsey Theorem of Erdős and Rado, which could be of independent interest.</p>\",\"PeriodicalId\":50666,\"journal\":{\"name\":\"Combinatorica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-02-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Combinatorica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00493-024-00082-7\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00493-024-00082-7","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
对图和超图的单调路径的有序拉姆齐数的研究由来已久，可以追溯到拉姆齐理论早期 Erdős 和 Szekeres 的著名工作。在本文中，我们获得了这一领域的若干结果，确立了 Mubayi 和 Suk 的两个猜想，并改进了 Balko、Cibulka、Král 和 Kynčl 的边界。例如，在图的情况下，我们证明了长度为 n 的固定簇与单调路径的固定幂的有序拉姆齐数总是与 n 成线性关系。此外，在 3 图的情况下，我们证明了长度为 n 的固定簇与紧密单调路径的有序拉姆齐数总是与 n 成多项式关系。作为副产品，我们还得到了厄多斯和拉多著名的 Canonical Ramsey Theorem 的彩色单调版本，这可能会引起人们的兴趣。
Ramsey Problems for Monotone Paths in Graphs and Hypergraphs
The study of ordered Ramsey numbers of monotone paths for graphs and hypergraphs has a long history, going back to the celebrated work by Erdős and Szekeres in the early days of Ramsey theory. In this paper we obtain several results in this area, establishing two conjectures of Mubayi and Suk and improving bounds due to Balko, Cibulka, Král and Kynčl. For example, in the graph case, we show that the ordered Ramsey number for a fixed clique versus a fixed power of a monotone path of length n is always linear in n. Also, in the 3-graph case, we show that the ordered Ramsey number for a fixed clique versus a tight monotone path of length n is always polynomial in n. As a by-product, we also obtain a color-monotone version of the well-known Canonical Ramsey Theorem of Erdős and Rado, which could be of independent interest.
期刊介绍：
COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are
- Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups).
- Combinatorial optimization.
- Combinatorial aspects of geometry and number theory.
- Algorithms in combinatorics and related fields.
- Computational complexity theory.
- Randomization and explicit construction in combinatorics and algorithms.
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
