{"title":"布尔立方中的章鱼:具有成对小交集的族,第二部分","authors":"Andrey Kupavskii , Fedor Noskov","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":"348 2","pages":"Article 114280"},"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\":\"Andrey Kupavskii , Fedor Noskov\",\"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\":\"348 2\",\"pages\":\"Article 114280\"},\"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。
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 dimensional facets is bounded by ? 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 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 be families of subsets of . We suppose that for distinct and arbitrary we have . We are interested in the maximal value of and the structure of the extremal example.
In the previous paper on the topic, the authors found the asymptotics of this product for constant ℓ 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.
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.