{"title":"奇特的向日葵","authors":"Peter Frankl , János Pach , Dömötör Pálvölgyi","doi":"10.1016/j.jcta.2024.105889","DOIUrl":null,"url":null,"abstract":"<div><p>Extending the notion of sunflowers, we call a family of at least two sets an <em>odd-sunflower</em> if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erdős–Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant <span><math><mi>μ</mi><mo><</mo><mn>2</mn></math></span> such that every family of subsets of an <em>n</em>-element set that contains no odd-sunflower consists of at most <span><math><msup><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> sets. We construct such families of size at least <span><math><msup><mrow><mn>1.5021</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We also characterize minimal odd-sunflowers of triples.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"206 ","pages":"Article 105889"},"PeriodicalIF":0.9000,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0097316524000281/pdfft?md5=4524ddd068e6ba4b9569281736257e67&pid=1-s2.0-S0097316524000281-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Odd-sunflowers\",\"authors\":\"Peter Frankl , János Pach , Dömötör Pálvölgyi\",\"doi\":\"10.1016/j.jcta.2024.105889\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Extending the notion of sunflowers, we call a family of at least two sets an <em>odd-sunflower</em> if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erdős–Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant <span><math><mi>μ</mi><mo><</mo><mn>2</mn></math></span> such that every family of subsets of an <em>n</em>-element set that contains no odd-sunflower consists of at most <span><math><msup><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> sets. We construct such families of size at least <span><math><msup><mrow><mn>1.5021</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>. We also characterize minimal odd-sunflowers of triples.</p></div>\",\"PeriodicalId\":50230,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series A\",\"volume\":\"206 \",\"pages\":\"Article 105889\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-03-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0097316524000281/pdfft?md5=4524ddd068e6ba4b9569281736257e67&pid=1-s2.0-S0097316524000281-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series A\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0097316524000281\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524000281","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
摘要
扩展向日葵的概念,如果底层集合的每个元素都包含在奇数个集合中或不包含在任何一个集合中,我们就称至少有两个集合的族为奇数向日葵。根据最近由纳斯伦德和萨温证明的厄尔多斯-塞梅雷迪猜想,存在一个常数 μ<2,使得不包含奇数向日葵的 n 元素集合的每个子集族至多由 μn 个集合组成。我们构建的这种族的大小至少为 1.5021n。我们还描述了三元组的最小奇数太阳花的特征。
Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erdős–Szemerédi conjecture, recently proved by Naslund and Sawin, that there is a constant such that every family of subsets of an n-element set that contains no odd-sunflower consists of at most sets. We construct such families of size at least . We also characterize minimal odd-sunflowers of triples.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.