David J. Fernández-Bretón, Eliseo Sarmiento Rosales, Germán Vera
{"title":"Owings-like theorems for infinitely many colours or finite monochromatic sets","authors":"David J. Fernández-Bretón, Eliseo Sarmiento Rosales, Germán Vera","doi":"10.1016/j.apal.2024.103495","DOIUrl":null,"url":null,"abstract":"<div><p>Inspired by Owings's problem, we investigate whether, for a given an Abelian group <em>G</em> and cardinal numbers <span><math><mi>κ</mi><mo>,</mo><mi>θ</mi></math></span>, every colouring <span><math><mi>c</mi><mo>:</mo><mi>G</mi><mo>⟶</mo><mi>θ</mi></math></span> yields a subset <span><math><mi>X</mi><mo>⊆</mo><mi>G</mi></math></span> with <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><mi>κ</mi></math></span> such that <span><math><mi>X</mi><mo>+</mo><mi>X</mi></math></span> is monochromatic. (Owings's problem asks this for <span><math><mi>G</mi><mo>=</mo><mi>Z</mi></math></span>, <span><math><mi>θ</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>κ</mi><mo>=</mo><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>; this is known to be false for the same <em>G</em> and <em>κ</em> but <span><math><mi>θ</mi><mo>=</mo><mn>3</mn></math></span>.) We completely settle the question for <em>κ</em> and <em>θ</em> both finite (by obtaining sufficient and necessary conditions for a positive answer) and for <em>κ</em> and <em>θ</em> both infinite (with a negative answer). Also, in the case where <em>θ</em> is infinite but <em>κ</em> is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 10","pages":"Article 103495"},"PeriodicalIF":0.6000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Pure and Applied Logic","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S016800722400099X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
Abstract
Inspired by Owings's problem, we investigate whether, for a given an Abelian group G and cardinal numbers , every colouring yields a subset with such that is monochromatic. (Owings's problem asks this for , and ; this is known to be false for the same G and κ but .) We completely settle the question for κ and θ both finite (by obtaining sufficient and necessary conditions for a positive answer) and for κ and θ both infinite (with a negative answer). Also, in the case where θ is infinite but κ is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer.
期刊介绍:
The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.