{"title":"通过熵方法解决克鲁斯卡尔-卡托纳类型问题","authors":"Ting-Wei Chao , Hung-Hsun Hans Yu","doi":"10.1016/j.jctb.2024.08.003","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call problems of this type Kruskal–Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a 3-edge-colored graph with <em>R</em> red, <em>G</em> green, <em>B</em> blue edges, the number of rainbow triangles is at most <span><math><msqrt><mrow><mn>2</mn><mi>R</mi><mi>G</mi><mi>B</mi></mrow></msqrt></math></span>, which is sharp. Second, we give a generalization of the Kruskal–Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"169 ","pages":"Pages 480-506"},"PeriodicalIF":1.2000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Kruskal–Katona-type problems via the entropy method\",\"authors\":\"Ting-Wei Chao , Hung-Hsun Hans Yu\",\"doi\":\"10.1016/j.jctb.2024.08.003\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call problems of this type Kruskal–Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a 3-edge-colored graph with <em>R</em> red, <em>G</em> green, <em>B</em> blue edges, the number of rainbow triangles is at most <span><math><msqrt><mrow><mn>2</mn><mi>R</mi><mi>G</mi><mi>B</mi></mrow></msqrt></math></span>, which is sharp. Second, we give a generalization of the Kruskal–Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality.</p></div>\",\"PeriodicalId\":54865,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series B\",\"volume\":\"169 \",\"pages\":\"Pages 480-506\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-08-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series B\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0095895624000698\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series B","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0095895624000698","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
在本文中,我们研究了几个极值组合问题,这些问题要求在给定边数的情况下,求出固定子图的最大副本数。我们称这类问题为 Kruskal-Katona-type 问题。本文将要讨论的大多数问题都与关节问题有关。本文有两个主要结果。首先,我们证明了在一个红边为 R、绿边为 G、蓝边为 B 的三边彩色图中,彩虹三角形的数量最多为 2RGB,这是一个尖锐的结果。其次,我们给出了对 Kruskal-Katona 定理的概括,其中隐含了许多之前的概括。这两个论证都使用了熵方法,主要的创新在于一个更巧妙的论证,改进了希勒不等式给出的界限。
Kruskal–Katona-type problems via the entropy method
In this paper, we investigate several extremal combinatorics problems that ask for the maximum number of copies of a fixed subgraph given the number of edges. We call problems of this type Kruskal–Katona-type problems. Most of the problems that will be discussed in this paper are related to the joints problem. There are two main results in this paper. First, we prove that, in a 3-edge-colored graph with R red, G green, B blue edges, the number of rainbow triangles is at most , which is sharp. Second, we give a generalization of the Kruskal–Katona theorem that implies many other previous generalizations. Both arguments use the entropy method, and the main innovation lies in a more clever argument that improves bounds given by Shearer's inequality.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.