A proof of Frankl–Kupavskii's conjecture on edge-union condition

Pub Date : 2024-01-07 DOI:10.1002/jgt.23073

Hongliang Lu, Xuechun Zhang

{"title":"A proof of Frankl–Kupavskii's conjecture on edge-union condition","authors":"Hongliang Lu, Xuechun Zhang","doi":"10.1002/jgt.23073","DOIUrl":null,"url":null,"abstract":"<p>A 3-graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> ${\\rm{ {\\mathcal F} }}$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>U</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>s</mi>\n \n <mo>,</mo>\n \n <mn>2</mn>\n \n <mi>s</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $U(s,2s+1)$</annotation>\n </semantics></math> if for any <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>s</mi>\n </mrow>\n </mrow>\n <annotation> $s$</annotation>\n </semantics></math> edges <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>e</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>,</mo>\n \n <mi>…</mi>\n \n <mo>,</mo>\n \n <msub>\n <mi>e</mi>\n \n <mi>s</mi>\n </msub>\n \n <mo>∈</mo>\n \n <mi>E</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>F</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${e}_{1},\\ldots ,{e}_{s}\\in E({\\rm{ {\\mathcal F} }})$</annotation>\n </semantics></math>, <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mo>∣</mo>\n \n <msub>\n <mi>e</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>∪</mo>\n \n <mtext>⋯</mtext>\n \n <mo>∪</mo>\n \n <msub>\n <mi>e</mi>\n \n <mi>s</mi>\n </msub>\n \n <mo>∣</mo>\n \n <mo>≤</mo>\n \n <mn>2</mn>\n \n <mi>s</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n <annotation> $| {e}_{1}\\cup \\cdots \\cup {e}_{s}| \\le 2s+1$</annotation>\n </semantics></math>. Frankl and Kupavskii proposed the following conjecture: For any 3-graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> ${\\rm{ {\\mathcal F} }}$</annotation>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> vertices, if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> ${\\rm{ {\\mathcal F} }}$</annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>U</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>s</mi>\n \n <mo>,</mo>\n \n <mn>2</mn>\n \n <mi>s</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $U(s,2s+1)$</annotation>\n </semantics></math>, then\n\n </p><p>In this paper, we confirm Frankl and Kupavskii's conjecture.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23073","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

Abstract

A 3-graph $F$ is $U (s, 2 s + 1)$ if for any $s$ edges $e_{1}, \dots, e_{s} \in E (F)$ , $∣ e_{1} \cup ⋯ \cup e_{s} ∣ \leq 2 s + 1$ . Frankl and Kupavskii proposed the following conjecture: For any 3-graph $F$ with $n$ vertices, if $F$ is $U (s, 2 s + 1)$ , then

In this paper, we confirm Frankl and Kupavskii's conjecture.

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弗兰克尔-库帕夫斯基关于边缘联合条件猜想的证明

一个 3 图 F${{rm{ {\mathcal F}U(s,2s+1)$U(s,2s+1)$ If for any s$s$ edges e1,...,es∈E(F)${e}_{1},\ldots ,{e}_{s}\in E({\rm{ {mathcal F} }})$, ∣e1∪⋯∪es∣≤2s+1$| {e}_{1}\cup \cdots \cup {e}_{s}| \le 2s+1$.弗兰克尔和库帕夫斯基提出了以下猜想：对于任意 3 图 F${\rm{ {\mathcal F}}$ 有 n$n$ 个顶点，如果 F${\rm{ {\mathcal F}}}$ 是 U(s,2s+1)$U(s,2s+1)$, 那么

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