集合的相交族通常是平凡的

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-09-20 DOI:10.1016/j.jctb.2023.09.002

József Balogh , Ramon I. Garcia , Lina Li , Adam Zsolt Wagner

引用次数: 4

摘要

一个[n]的子集族是相交的，如果它的每一对集合都相交。确定大型相交族的结构是极值组合学中的一个中心问题。Frankl–Kupavskii和Balogh–Das–Liu–Sharifzadeh–Tran证明，对于n≥2k+ckln⁡k、几乎所有k一致相交族都是恒星。改进了它们的结果，我们证明了对于n≥2k+100ln，同样的结论成立⁡k.我们的证明使用了图容器方法和Das–Tran移除引理。

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Intersecting families of sets are typically trivial

A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl–Kupavskii and Balogh–Das–Liu–Sharifzadeh–Tran showed that for $n \geq 2 k + c \sqrt{k \ln k}$ , almost all k-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for $n \geq 2 k + 100 \ln k$ . Our proof uses, among others, the graph container method and the Das–Tran removal lemma.

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.