{"title":"Z5n 中的大无和集","authors":"Vsevolod F. Lev","doi":"10.1016/j.jcta.2024.105865","DOIUrl":null,"url":null,"abstract":"<div><p>It is well-known that for a prime <span><math><mi>p</mi><mo>≡</mo><mn>2</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span> and integer <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span><span>, the maximum possible size of a sum-free subset of the elementary abelian group </span><span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mspace></mspace><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>. However, the matching stability result is known for <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> only. We consider the first open case <span><math><mi>p</mi><mo>=</mo><mn>5</mn></math></span> showing that if <span><math><mi>A</mi><mo>⊆</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is a sum-free subset with <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>></mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, then there are a subgroup <span><math><mi>H</mi><mo><</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> of size <span><math><mo>|</mo><mi>H</mi><mo>|</mo><mo>=</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and an element <span><math><mi>e</mi><mo>∉</mo><mi>H</mi></math></span> such that <span><math><mi>A</mi><mo>⊆</mo><mo>(</mo><mi>e</mi><mo>+</mo><mi>H</mi><mo>)</mo><mo>∪</mo><mo>(</mo><mo>−</mo><mi>e</mi><mo>+</mo><mi>H</mi><mo>)</mo></math></span>.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Large sum-free sets in Z5n\",\"authors\":\"Vsevolod F. Lev\",\"doi\":\"10.1016/j.jcta.2024.105865\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>It is well-known that for a prime <span><math><mi>p</mi><mo>≡</mo><mn>2</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span> and integer <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span><span>, the maximum possible size of a sum-free subset of the elementary abelian group </span><span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mspace></mspace><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>. However, the matching stability result is known for <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> only. We consider the first open case <span><math><mi>p</mi><mo>=</mo><mn>5</mn></math></span> showing that if <span><math><mi>A</mi><mo>⊆</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is a sum-free subset with <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>></mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⋅</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, then there are a subgroup <span><math><mi>H</mi><mo><</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>5</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> of size <span><math><mo>|</mo><mi>H</mi><mo>|</mo><mo>=</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> and an element <span><math><mi>e</mi><mo>∉</mo><mi>H</mi></math></span> such that <span><math><mi>A</mi><mo>⊆</mo><mo>(</mo><mi>e</mi><mo>+</mo><mi>H</mi><mo>)</mo><mo>∪</mo><mo>(</mo><mo>−</mo><mi>e</mi><mo>+</mo><mi>H</mi><mo>)</mo></math></span>.</p></div>\",\"PeriodicalId\":50230,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series A\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-02-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"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/S0097316524000049\",\"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/S0097316524000049","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
It is well-known that for a prime and integer , the maximum possible size of a sum-free subset of the elementary abelian group is . However, the matching stability result is known for only. We consider the first open case showing that if is a sum-free subset with , then there are a subgroup of size and an element such that .
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.