On non-empty cross-t-intersecting families

IF 0.9 2区 数学 Q2 MATHEMATICS Journal of Combinatorial Theory Series A Pub Date : 2024-09-24 DOI:10.1016/j.jcta.2024.105960
Anshui Li , Huajun Zhang
{"title":"On non-empty cross-t-intersecting families","authors":"Anshui Li ,&nbsp;Huajun Zhang","doi":"10.1016/j.jcta.2024.105960","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be families of <em>k</em>-element subsets of a <em>n</em>-element set. We call them cross-<em>t</em>-intersecting if <span><math><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>|</mo><mo>≥</mo><mi>t</mi></math></span> for any <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> with <span><math><mi>i</mi><mo>≠</mo><mi>j</mi></math></span>. In this paper we will prove that, for <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mo>−</mo><mi>t</mi><mo>+</mo><mn>1</mn></math></span>, if <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> are non-empty cross-<em>t</em>-intersecting families, then<span><span><span><math><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi></mrow></munder><mo>|</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo><mo>≤</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><munder><mo>∑</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>t</mi><mo>−</mo><mn>1</mn></mrow></munder><mrow><mo>(</mo><mtable><mtr><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mi>i</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mi>i</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>m</mi><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>}</mo><mo>,</mo></math></span></span></span> where <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span> is the size of the maximum <em>t</em>-intersecting family of <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></math></span>. Moreover, the extremal families attaining the upper bound are characterized.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"210 ","pages":"Article 105960"},"PeriodicalIF":0.9000,"publicationDate":"2024-09-24","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/S0097316524000992","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

Abstract

Let A1,A2,,Am be families of k-element subsets of a n-element set. We call them cross-t-intersecting if |AiAj|t for any AiAi and AjAj with ij. In this paper we will prove that, for n2kt+1, if A1,A2,,Am are non-empty cross-t-intersecting families, then1im|Ai|max{(nk)1it1(ki)(nkki)+m1,mM(n,k,t)}, where M(n,k,t) is the size of the maximum t-intersecting family of ([n]k). Moreover, the extremal families attaining the upper bound are characterized.
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
关于非空交叉相交族
设 A1,A2,...,Am 是 n 元素集合的 k 元素子集族。对于任意 Ai∈Ai 和 Aj∈Aj 且 i≠j 的情况,如果|Ai∩Aj|≥t,我们称它们为交叉-t-交集。本文将证明,对于 n≥2k-t+1,如果 A1,A2,...,Am 是非空跨 t 交集族,则∑1≤i≤m|Ai|≤max{(nk)-∑1≤i≤t-1(ki)(n-kk-i)+m-1,mM(n,k,t)},其中 M(n,k,t) 是 ([n]k) 的最大 t 交集族的大小。此外,还描述了达到上限的极值族的特征。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
CiteScore
2.90
自引率
9.10%
发文量
94
审稿时长
12 months
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.
期刊最新文献
The degree of functions in the Johnson and q-Johnson schemes Sequence reconstruction problem for deletion channels: A complete asymptotic solution Editorial Board A classification of the flag-transitive 2-(v,k,2) designs Non-empty pairwise cross-intersecting families
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1