Multicolor Turán numbers II: A generalization of the Ruzsa–Szemerédi theorem and new results on cliques and odd cycles

IF 0.9 3区 数学 Q2 MATHEMATICS
Benedek Kovács, Zoltán Lóránt Nagy
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Let <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mtext>ex</mtext>\n \n <mi>F</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>G</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${\\text{ex}}_{F}(n,G)$</annotation>\n </semantics></math> denote the maximum number of edge-disjoint copies of a fixed simple graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math> that can be placed on an <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math>-vertex ground set without forming a subgraph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> whose edges are from different <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math>-copies. The case when both <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> are arbitrary cliques by applying a number theoretic result due to Erdős, Frankl, and Rödl. This extension in turn decides the order of magnitude for a large family of graph pairs, which will be subquadratic, but almost quadratic. Since the linear <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n </mrow>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math>-uniform hypergraph Turán problems to determine <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msubsup>\n <mtext>ex</mtext>\n \n <mi>r</mi>\n \n <mrow>\n <mi>l</mi>\n \n <mi>i</mi>\n \n <mi>n</mi>\n </mrow>\n </msubsup>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>G</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${\\text{ex}}_{r}^{lin}(n,G)$</annotation>\n </semantics></math> form a class of the multicolor Turán problem, following the identity <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msubsup>\n <mtext>ex</mtext>\n \n <mi>r</mi>\n \n <mrow>\n <mi>l</mi>\n \n <mi>i</mi>\n \n <mi>n</mi>\n </mrow>\n </msubsup>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>G</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n \n <mo>=</mo>\n \n <msub>\n <mtext>ex</mtext>\n \n <msub>\n <mi>K</mi>\n \n <mi>r</mi>\n </msub>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>G</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> ${\\text{ex}}_{r}^{lin}(n,G)={\\text{ex}}_{{K}_{r}}(n,G)$</annotation>\n </semantics></math>, our results determine the linear hypergraph Turán numbers of every graph of girth 3 and for every <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n </mrow>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math> up to a subpolynomial factor. Furthermore, when <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is a triangle, we settle the case <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n \n <mo>=</mo>\n \n <msub>\n <mi>C</mi>\n \n <mn>5</mn>\n </msub>\n </mrow>\n </mrow>\n <annotation> $F={C}_{5}$</annotation>\n </semantics></math> and give bounds for the cases <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n \n <mo>=</mo>\n \n <msub>\n <mi>C</mi>\n \n <mrow>\n <mn>2</mn>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n </mrow>\n </mrow>\n <annotation> $F={C}_{2k+1}$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>3</mn>\n </mrow>\n </mrow>\n <annotation> $k\\ge 3$</annotation>\n </semantics></math> as well.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"629-641"},"PeriodicalIF":0.9000,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23147","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

In this paper we continue the study of a natural generalization of Turán's forbidden subgraph problem and the Ruzsa–Szemerédi problem. Let ex F ( n , G ) ${\text{ex}}_{F}(n,G)$ denote the maximum number of edge-disjoint copies of a fixed simple graph F $F$ that can be placed on an n $n$ -vertex ground set without forming a subgraph G $G$ whose edges are from different F $F$ -copies. The case when both F $F$ and G $G$ are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when F $F$ and G $G$ are arbitrary cliques by applying a number theoretic result due to Erdős, Frankl, and Rödl. This extension in turn decides the order of magnitude for a large family of graph pairs, which will be subquadratic, but almost quadratic. Since the linear r $r$ -uniform hypergraph Turán problems to determine ex r l i n ( n , G ) ${\text{ex}}_{r}^{lin}(n,G)$ form a class of the multicolor Turán problem, following the identity ex r l i n ( n , G ) = ex K r ( n , G ) ${\text{ex}}_{r}^{lin}(n,G)={\text{ex}}_{{K}_{r}}(n,G)$ , our results determine the linear hypergraph Turán numbers of every graph of girth 3 and for every r $r$ up to a subpolynomial factor. Furthermore, when G $G$ is a triangle, we settle the case F = C 5 $F={C}_{5}$ and give bounds for the cases F = C 2 k + 1 $F={C}_{2k+1}$ , k 3 $k\ge 3$ as well.

多色图兰数 II:鲁兹萨-塞梅雷迪定理的一般化以及关于小群和奇数循环的新结果
在本文中,我们将继续研究图兰的禁止子图问题和鲁兹萨-塞梅雷迪问题的自然概括。让表示一个固定简单图的边不相交副本的最大数目,这些副本可以放在一个-顶点地面集上,而不会形成一个边来自不同副本的子图。当 和 都是三角形时,基本上就可以得出鲁兹萨和塞梅雷迪的定理。我们应用厄尔多斯、弗兰克尔和罗德尔的一个数论结果,将他们的结果推广到和都是任意小块的情况。这一扩展反过来决定了一大系列图对的数量级,它们将是亚二次方的,但几乎是二次方的。由于要确定的线性均匀超图图兰问题构成了多色图兰问题的一个类别,根据同一性,我们的结果确定了每一个周长为 3 的图的线性超图图兰数,并且每一个图的线性超图图兰数都达到了亚对数因子。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Graph Theory
Journal of Graph Theory 数学-数学
CiteScore
1.60
自引率
22.20%
发文量
130
审稿时长
6-12 weeks
期刊介绍: The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
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