{"title":"Turán numbers of general star forests in hypergraphs","authors":"Lin-Peng Zhang , Hajo Broersma , Ligong Wang","doi":"10.1016/j.disc.2024.114219","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>F</mi></math></span> be a family of <em>r</em>-uniform hypergraphs, and let <em>H</em> be an <em>r</em>-uniform hypergraph. Then <em>H</em> is called <span><math><mi>F</mi></math></span>-free if it does not contain any member of <span><math><mi>F</mi></math></span> as a subhypergraph. The Turán number of <span><math><mi>F</mi></math></span>, denoted by <span><math><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span>, is the maximum number of hyperedges in an <span><math><mi>F</mi></math></span>-free <em>n</em>-vertex <em>r</em>-uniform hypergraph. Our current results are motivated by earlier results on Turán numbers of star forests and hypergraph star forests. In particular, Lidický et al. (2013) <span><span>[17]</span></span> determined the Turán number <span><math><mrow><mi>ex</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math></span> of a star forest <em>F</em> for sufficiently large <em>n</em>. Recently, Khormali and Palmer (2022) <span><span>[13]</span></span> generalized the above result to three different well-studied hypergraph settings (the expansions of a graph, linear hypergraphs and Berge hypergraphs), but restricted to the case that all stars in the hypergraph star forests are identical. We further generalize these results to general star forests in hypergraphs.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003509/pdfft?md5=f55a8417dd66a400951a48477694c9f9&pid=1-s2.0-S0012365X24003509-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003509","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
Let be a family of r-uniform hypergraphs, and let H be an r-uniform hypergraph. Then H is called -free if it does not contain any member of as a subhypergraph. The Turán number of , denoted by , is the maximum number of hyperedges in an -free n-vertex r-uniform hypergraph. Our current results are motivated by earlier results on Turán numbers of star forests and hypergraph star forests. In particular, Lidický et al. (2013) [17] determined the Turán number of a star forest F for sufficiently large n. Recently, Khormali and Palmer (2022) [13] generalized the above result to three different well-studied hypergraph settings (the expansions of a graph, linear hypergraphs and Berge hypergraphs), but restricted to the case that all stars in the hypergraph star forests are identical. We further generalize these results to general star forests in hypergraphs.
设 F 是一个 r-Uniform 超图族,设 H 是一个 r-Uniform 超图。如果 H 的子超图不包含 F 的任何成员,则称 H 为无 F 超图。F 的图兰数(用 exr(n,F) 表示)是无 F n 顶点 r-uniform 超图中超图的最大数目。我们目前的结果是受早先关于星形森林和超图星形森林的图兰数结果的启发。最近,Khormali 和 Palmer(2022 年)[13] 将上述结果推广到三种不同的、研究得很透彻的超图环境(图的展开、线性超图和 Berge 超图),但仅限于超图星形林中所有星形都相同的情况。我们将这些结果进一步推广到超图中的一般星形林。