{"title":"Erdős–Moser and IΣ2","authors":"Henry Towsner, Keita Yokoyama","doi":"10.1007/s11856-024-2643-8","DOIUrl":null,"url":null,"abstract":"<p>The first-order part of Ramsey’s theorem for pairs with an arbitrary number of colors is known to be precisely <i>B</i>Σ<span>\n<sup>0</sup><sub>3</sub>\n</span>. We compare this to the known division of Ramsey’s theorem for pairs into the weaker principles, EM (the Erdős–Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond <i>I</i>Σ<span>\n<sup>0</sup><sub>2</sub>\n</span> is entirely due to the arbitrary color analog of ADS.</p><p>Specifically, we show that ADS for an arbitrary number of colors implies <i>B</i>Σ<span>\n<sup>0</sup><sub>3</sub>\n</span> while EM for an arbitrary number of colors is Π<span>\n<sup>1</sup><sub>1</sub>\n</span>-conservative over <i>I</i>Σ<span>\n<sup>0</sup><sub>2</sub>\n</span> and it does not imply <i>I</i>Σ<span>\n<sup>0</sup><sub>2</sub>\n</span>.</p>","PeriodicalId":14661,"journal":{"name":"Israel Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Israel Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11856-024-2643-8","RegionNum":2,"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
The first-order part of Ramsey’s theorem for pairs with an arbitrary number of colors is known to be precisely BΣ03. We compare this to the known division of Ramsey’s theorem for pairs into the weaker principles, EM (the Erdős–Moser principle) and ADS (the ascending-descending sequence principle): we show that the additional strength beyond IΣ02 is entirely due to the arbitrary color analog of ADS.
Specifically, we show that ADS for an arbitrary number of colors implies BΣ03 while EM for an arbitrary number of colors is Π11-conservative over IΣ02 and it does not imply IΣ02.
期刊介绍：
The Israel Journal of Mathematics is an international journal publishing high-quality original research papers in a wide spectrum of pure and applied mathematics. The prestigious interdisciplinary editorial board reflects the diversity of subjects covered in this journal, including set theory, model theory, algebra, group theory, number theory, analysis, functional analysis, ergodic theory, algebraic topology, geometry, combinatorics, theoretical computer science, mathematical physics, and applied mathematics.
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