{"title":"随机超图中的分割和分叉","authors":"","doi":"10.1016/j.apal.2024.103521","DOIUrl":null,"url":null,"abstract":"<div><div>We investigate the class of <em>m</em>-hypergraphs in which substructures with <em>l</em> elements have more than <em>s</em> subsets of size <em>m</em> that do not form a hyperedge. The class has a (unique) Fraïssé limit, if <span><math><mn>0</mn><mo>≤</mo><mi>s</mi><mo><</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. We show that the theory of the Fraïssé limit has <em>SU</em>-rank one if <span><math><mn>0</mn><mo>≤</mo><mi>s</mi><mo><</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>, and dividing and forking will be different concepts in the theory if <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>≤</mo><mi>s</mi><mo><</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Dividing and forking in random hypergraphs\",\"authors\":\"\",\"doi\":\"10.1016/j.apal.2024.103521\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We investigate the class of <em>m</em>-hypergraphs in which substructures with <em>l</em> elements have more than <em>s</em> subsets of size <em>m</em> that do not form a hyperedge. The class has a (unique) Fraïssé limit, if <span><math><mn>0</mn><mo>≤</mo><mi>s</mi><mo><</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. We show that the theory of the Fraïssé limit has <em>SU</em>-rank one if <span><math><mn>0</mn><mo>≤</mo><mi>s</mi><mo><</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>, and dividing and forking will be different concepts in the theory if <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>≤</mo><mi>s</mi><mo><</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>.</div></div>\",\"PeriodicalId\":50762,\"journal\":{\"name\":\"Annals of Pure and Applied Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-09-24\",\"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/S0168007224001258\",\"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/S0168007224001258","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
摘要
我们研究了 m-hypergraphs 类，在这类图中，有 l 个元素的子结构有多于 s 个大小为 m 的子集不构成一个 hyperedge。如果0≤s<(l-2m-2)，则该类图具有（唯一的）弗雷泽极限。我们证明，如果 0≤s<(l-3m-3), 那么弗拉伊塞极限理论具有 SU-rank one，如果 (l-3m-3)≤s<(l-2m-2), 那么分割和分叉在理论中将是不同的概念。
We investigate the class of m-hypergraphs in which substructures with l elements have more than s subsets of size m that do not form a hyperedge. The class has a (unique) Fraïssé limit, if . We show that the theory of the Fraïssé limit has SU-rank one if , and dividing and forking will be different concepts in the theory if .
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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