{"title":"集合论中的库尔维数","authors":"Jindřich Zapletal","doi":"10.1016/j.apal.2023.103299","DOIUrl":null,"url":null,"abstract":"<div><p>For every number <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, let <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the hypergraph on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> of arity four consisting of all non-degenerate Euclidean rectangles. It is consistent with ZF+DC set theory that the chromatic number of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is countable while that of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is not.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"174 9","pages":"Article 103299"},"PeriodicalIF":0.6000,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Krull dimension in set theory\",\"authors\":\"Jindřich Zapletal\",\"doi\":\"10.1016/j.apal.2023.103299\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>For every number <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, let <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the hypergraph on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> of arity four consisting of all non-degenerate Euclidean rectangles. It is consistent with ZF+DC set theory that the chromatic number of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is countable while that of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is not.</p></div>\",\"PeriodicalId\":50762,\"journal\":{\"name\":\"Annals of Pure and Applied Logic\",\"volume\":\"174 9\",\"pages\":\"Article 103299\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annals of Pure and Applied Logic\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0168007223000568\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"LOGIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Pure and Applied Logic","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168007223000568","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"LOGIC","Score":null,"Total":0}
For every number , let be the hypergraph on of arity four consisting of all non-degenerate Euclidean rectangles. It is consistent with ZF+DC set theory that the chromatic number of is countable while that of is not.
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.
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