{"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 be a family of r-uniform hypergraphs. The random Turán number is the maximum number of edges in an -free subgraph of , where is the Erdős-Rényi random r-graph with parameter p. Let denote the r-uniform linear cycle of length ℓ. For , Mubayi and Yepremyan showed that . This upper bound is not tight when . In this paper, we close the gap for . More precisely, we show that when . Similar results have recently been obtained independently in a different way by Mubayi and Yepremyan. For , 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.
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.