Turán theorems for even cycles in random hypergraph

IF 1.2 1区 数学 Q1 MATHEMATICS
Jiaxi Nie
{"title":"Turán theorems for even cycles in random hypergraph","authors":"Jiaxi Nie","doi":"10.1016/j.jctb.2024.02.002","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>F</mi></math></span> be a family of <em>r</em>-uniform hypergraphs. The random Turán number <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>,</mo><mi>F</mi><mo>)</mo></math></span> is the maximum number of edges in an <span><math><mi>F</mi></math></span>-free subgraph of <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>, where <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span> is the Erdős-Rényi random <em>r</em>-graph with parameter <em>p</em>. Let <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span> denote the <em>r</em>-uniform linear cycle of length <em>ℓ</em>. For <span><math><mi>p</mi><mo>≥</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>, Mubayi and Yepremyan showed that <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>max</mi><mo>⁡</mo><mo>{</mo><msup><mrow><mi>p</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><mo>×</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>,</mo><mi>p</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>}</mo></math></span>. This upper bound is not tight when <span><math><mi>p</mi><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>. In this paper, we close the gap for <span><math><mi>r</mi><mo>≥</mo><mn>4</mn></math></span>. More precisely, we show that <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi></mrow><mrow><mi>r</mi></mrow></msubsup><mo>)</mo><mo>=</mo><mi>Θ</mi><mo>(</mo><mi>p</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> when <span><math><mi>p</mi><mo>≥</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>r</mi><mo>+</mo><mn>2</mn><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span>. Similar results have recently been obtained independently in a different way by Mubayi and Yepremyan. For <span><math><mi>r</mi><mo>=</mo><mn>3</mn></math></span>, we significantly improve Mubayi and Yepremyan's upper bound. Moreover, we give reasonably good upper bounds for the random Turán numbers of Berge even cycles, which improve previous results of Spiro and Verstraëte.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"167 ","pages":"Pages 23-54"},"PeriodicalIF":1.2000,"publicationDate":"2024-03-01","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/S009589562400008X","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

Abstract

Let F be a family of r-uniform hypergraphs. The random Turán number ex(Gn,pr,F) is the maximum number of edges in an F-free subgraph of Gn,pr, where Gn,pr is the Erdős-Rényi random r-graph with parameter p. Let Cr denote the r-uniform linear cycle of length . For pnr+2+o(1), Mubayi and Yepremyan showed that ex(Gn,pr,C2r)max{p121×n1+r121+o(1),pnr1+o(1)}. This upper bound is not tight when pnr+2+122+o(1). In this paper, we close the gap for r4. More precisely, we show that ex(Gn,pr,C2r)=Θ(pnr1) when pnr+2+121+o(1). Similar results have recently been obtained independently in a different way by Mubayi and Yepremyan. For r=3, we significantly improve Mubayi and Yepremyan's upper bound. Moreover, we give reasonably good upper bounds for the random Turán numbers of Berge even cycles, which improve previous results of Spiro and Verstraëte.

随机超图中偶数循环的图兰定理
假设 F 是 r-uniform 超图族。随机图兰数 ex(Gn,pr,F) 是 Gn,pr 的无 F 子图中的最大边数,其中 Gn,pr 是参数为 p 的厄尔多斯-雷尼随机 r 图。对于 p≥n-r+2+o(1),Mubayi 和 Yepremyan 证明了 ex(Gn,pr,C2ℓr)≤max{p12ℓ-1×n1+r-12ℓ-1+o(1),pnr-1+o(1)}。当 p≤n-r+2+12ℓ-2+o(1) 时,这一上限并不严格。本文将缩小 r≥4 时的差距。更确切地说,我们证明了当 p≥n-r+2+12ℓ-1+o(1) 时,ex(Gn,pr,C2ℓr)=Θ(pnr-1)。最近,Mubayi 和 Yepremyan 以不同方式独立获得了类似结果。对于 r=3,我们大大改进了 Mubayi 和 Yepremyan 的上界。此外,我们还给出了 Berge 偶数循环的随机图兰数的合理上界,从而改进了 Spiro 和 Verstraëte 以前的结果。
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来源期刊
CiteScore
2.70
自引率
14.30%
发文量
99
审稿时长
6-12 weeks
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.
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