{"title":"λ=3的所有λ-设计的测定","authors":"William G. Bridges Jr. , Earl S. Kramer","doi":"10.1016/S0021-9800(70)80029-1","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>S</em><sub>1</sub>, …, <em>S<sub>n</sub></em>, <em>n</em>>1, be subsets of an <em>n</em>-set <em>S</em> where |<em>S<sub>i</sub></em>|>λ≥1 and |<em>S<sub>i</sub></em>∩<em>S<sub>j</sub></em>|=λ for <em>i≠j</em>. Then our configuration is either a symmetric block design, with possible degeneracies, or what Ryser [3] has called a λ-design. A λ-design has the remarkable property, established by Ryser [3], that each element of <em>S</em> occurs either <em>r</em><sub>1</sub> or <em>r</em><sub>2</sub> times among the sets <em>S<sub>i</sub></em>, …, <em>S<sub>n</sub></em> and <em>r<sub>1</sub>+r<sub>2</sub>=n+1</em>. The 1-designs are completely known and so is the unique 2-design. The present paper establishes that there are exactly three 3-designs.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"8 4","pages":"Pages 343-349"},"PeriodicalIF":0.0000,"publicationDate":"1970-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80029-1","citationCount":"15","resultStr":"{\"title\":\"The determination of all λ-designs with λ=3\",\"authors\":\"William G. Bridges Jr. , Earl S. Kramer\",\"doi\":\"10.1016/S0021-9800(70)80029-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <em>S</em><sub>1</sub>, …, <em>S<sub>n</sub></em>, <em>n</em>>1, be subsets of an <em>n</em>-set <em>S</em> where |<em>S<sub>i</sub></em>|>λ≥1 and |<em>S<sub>i</sub></em>∩<em>S<sub>j</sub></em>|=λ for <em>i≠j</em>. Then our configuration is either a symmetric block design, with possible degeneracies, or what Ryser [3] has called a λ-design. A λ-design has the remarkable property, established by Ryser [3], that each element of <em>S</em> occurs either <em>r</em><sub>1</sub> or <em>r</em><sub>2</sub> times among the sets <em>S<sub>i</sub></em>, …, <em>S<sub>n</sub></em> and <em>r<sub>1</sub>+r<sub>2</sub>=n+1</em>. The 1-designs are completely known and so is the unique 2-design. The present paper establishes that there are exactly three 3-designs.</p></div>\",\"PeriodicalId\":100765,\"journal\":{\"name\":\"Journal of Combinatorial Theory\",\"volume\":\"8 4\",\"pages\":\"Pages 343-349\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1970-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80029-1\",\"citationCount\":\"15\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021980070800291\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021980070800291","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Let S1, …, Sn, n>1, be subsets of an n-set S where |Si|>λ≥1 and |Si∩Sj|=λ for i≠j. Then our configuration is either a symmetric block design, with possible degeneracies, or what Ryser [3] has called a λ-design. A λ-design has the remarkable property, established by Ryser [3], that each element of S occurs either r1 or r2 times among the sets Si, …, Sn and r1+r2=n+1. The 1-designs are completely known and so is the unique 2-design. The present paper establishes that there are exactly three 3-designs.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
