布尔立方中的章鱼:具有成对小交集的族,第二部分

IF 0.7 3区 数学 Q2 MATHEMATICS
{"title":"布尔立方中的章鱼:具有成对小交集的族,第二部分","authors":"","doi":"10.1016/j.disc.2024.114280","DOIUrl":null,"url":null,"abstract":"<div><div>The problem we consider originally arises from 2-level polytope theory. This class of polytopes generalizes a number of other polytope families. One of the important questions in this field can be formulated as follows: is it true for a <em>d</em>-dimensional 2-level polytope that the product of the number of its vertices and the number of its <span><math><mi>d</mi><mo>−</mo><mn>1</mn></math></span> dimensional facets is bounded by <span><math><mi>d</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>? Recently, Kupavskii and Weltge <span><span>[9]</span></span> settled this question in positive. A key element in their proof is a more general result for families of vectors in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> such that the scalar product between any two vectors from different families is either 0 or 1.</div><div>Peter Frankl noted that, when restricted to the Boolean cube, the solution boils down to an elegant application of the Harris–Kleitman correlation inequality. Meanwhile, this problem becomes much more sophisticated when we consider several families.</div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> be families of subsets of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. We suppose that for distinct <span><math><mi>k</mi><mo>,</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> and arbitrary <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span> we have <span><math><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo><mo>⩽</mo><mi>m</mi></math></span>. We are interested in the maximal value of <span><math><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>…</mo><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>|</mo></math></span> and the structure of the extremal example.</div><div>In the previous paper on the topic, the authors found the asymptotics of this product for constant <em>ℓ</em> and <em>m</em> as <em>n</em> tends to infinity. However, the possible structure of the families from the extremal example turned out to be very complicated. In this paper, we obtain a strong structural result for the extremal families.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Octopuses in the Boolean cube: Families with pairwise small intersections, part II\",\"authors\":\"\",\"doi\":\"10.1016/j.disc.2024.114280\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>The problem we consider originally arises from 2-level polytope theory. This class of polytopes generalizes a number of other polytope families. One of the important questions in this field can be formulated as follows: is it true for a <em>d</em>-dimensional 2-level polytope that the product of the number of its vertices and the number of its <span><math><mi>d</mi><mo>−</mo><mn>1</mn></math></span> dimensional facets is bounded by <span><math><mi>d</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>? Recently, Kupavskii and Weltge <span><span>[9]</span></span> settled this question in positive. A key element in their proof is a more general result for families of vectors in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> such that the scalar product between any two vectors from different families is either 0 or 1.</div><div>Peter Frankl noted that, when restricted to the Boolean cube, the solution boils down to an elegant application of the Harris–Kleitman correlation inequality. Meanwhile, this problem becomes much more sophisticated when we consider several families.</div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> be families of subsets of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. We suppose that for distinct <span><math><mi>k</mi><mo>,</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> and arbitrary <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span> we have <span><math><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo><mo>⩽</mo><mi>m</mi></math></span>. We are interested in the maximal value of <span><math><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>…</mo><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>|</mo></math></span> and the structure of the extremal example.</div><div>In the previous paper on the topic, the authors found the asymptotics of this product for constant <em>ℓ</em> and <em>m</em> as <em>n</em> tends to infinity. However, the possible structure of the families from the extremal example turned out to be very complicated. In this paper, we obtain a strong structural result for the extremal families.</div></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-10-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X24004114\",\"RegionNum\":3,\"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":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004114","RegionNum":3,"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

我们所考虑的问题最初源于 2 级多面体理论。这一类多面体概括了许多其他多面体族。该领域的一个重要问题可以表述如下:对于一个 d 维 2 层多面体,其顶点数与 d-1 维面数的乘积是否真的以 d2d-1 为界?最近,Kupavskii 和 Weltge [9] 从正面解决了这个问题。彼得-弗兰克尔(Peter Frankl)指出,当局限于布尔立方体时,解决方法可以归结为哈里斯-克莱特曼相关不等式的优雅应用。让 F1、......、Fℓ 分别是 {1,......,n} 的子集族。我们假设,对于不同的 k,k′ 和任意的 F1∈Fk,F2∈Fk′,我们有 |F1∩F2|⩽m。我们感兴趣的是|F1|...|Fℓ|的最大值和极值实例的结构。在上一篇相关论文中,作者发现了当 n 趋于无穷大时,常数 ℓ 和 m 的该积的渐近线。然而,从极值示例中得出的族的可能结构却非常复杂。在本文中,我们得到了极值族的强结构结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Octopuses in the Boolean cube: Families with pairwise small intersections, part II
The problem we consider originally arises from 2-level polytope theory. This class of polytopes generalizes a number of other polytope families. One of the important questions in this field can be formulated as follows: is it true for a d-dimensional 2-level polytope that the product of the number of its vertices and the number of its d1 dimensional facets is bounded by d2d1? Recently, Kupavskii and Weltge [9] settled this question in positive. A key element in their proof is a more general result for families of vectors in Rd such that the scalar product between any two vectors from different families is either 0 or 1.
Peter Frankl noted that, when restricted to the Boolean cube, the solution boils down to an elegant application of the Harris–Kleitman correlation inequality. Meanwhile, this problem becomes much more sophisticated when we consider several families.
Let F1,,F be families of subsets of {1,,n}. We suppose that for distinct k,k and arbitrary F1Fk,F2Fk we have |F1F2|m. We are interested in the maximal value of |F1||F| and the structure of the extremal example.
In the previous paper on the topic, the authors found the asymptotics of this product for constant and m as n tends to infinity. However, the possible structure of the families from the extremal example turned out to be very complicated. In this paper, we obtain a strong structural result for the extremal families.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Discrete Mathematics
Discrete Mathematics 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
424
审稿时长
6 months
期刊介绍: Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
×
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
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
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 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信