Large monochromatic components in expansive hypergraphs

Deepak Bal, Louis DeBiasio
{"title":"Large monochromatic components in expansive hypergraphs","authors":"Deepak Bal, Louis DeBiasio","doi":"10.1017/s096354832400004x","DOIUrl":null,"url":null,"abstract":"<p>A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$r$</span></span></img></span></span>-colouring of the complete <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$k$</span></span></img></span></span>-uniform hypergraph <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$K_n^k$</span></span></img></span></span> when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$k\\geq 2$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$k\\in \\{r-1,r\\}$</span></span></img></span></span>. We prove a result which says that if one replaces <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$K_n^k$</span></span></img></span></span> in Gyárfás’ theorem by any ‘expansive’ <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$k$</span></span></img></span></span>-uniform hypergraph on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$n$</span></span></img></span></span> vertices (that is, a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$k$</span></span></img></span></span>-uniform hypergraph <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$G$</span></span></img></span></span> on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline11.png\"><span data-mathjax-type=\"texmath\"><span>$n$</span></span></img></span></span> vertices in which <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline12.png\"><span data-mathjax-type=\"texmath\"><span>$e(V_1, \\ldots, V_k)\\gt 0$</span></span></img></span></span> for all disjoint sets <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline13.png\"><span data-mathjax-type=\"texmath\"><span>$V_1, \\ldots, V_k\\subseteq V(G)$</span></span></img></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline14.png\"><span data-mathjax-type=\"texmath\"><span>$|V_i|\\gt \\alpha$</span></span></img></span></span> for all <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline15.png\"><span data-mathjax-type=\"texmath\"><span>$i\\in [k]$</span></span></img></span></span>), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline16.png\"><span data-mathjax-type=\"texmath\"><span>$r$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline17.png\"/><span data-mathjax-type=\"texmath\"><span>$\\alpha$</span></span></span></span>). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms.</p><p>Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline18.png\"/><span data-mathjax-type=\"texmath\"><span>$r$</span></span></span></span>-partite <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline19.png\"/><span data-mathjax-type=\"texmath\"><span>$r$</span></span></span></span>-uniform hypergraph <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline20.png\"/><span data-mathjax-type=\"texmath\"><span>$H$</span></span></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline21.png\"/><span data-mathjax-type=\"texmath\"><span>$n$</span></span></span></span> edges in which every set of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline22.png\"/><span data-mathjax-type=\"texmath\"><span>$k$</span></span></span></span> edges has a common intersection. In this language, our result says that if one replaces the condition that every set of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline23.png\"/><span data-mathjax-type=\"texmath\"><span>$k$</span></span></span></span> edges has a common intersection with the condition that for every collection of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline24.png\"/><span data-mathjax-type=\"texmath\"><span>$k$</span></span></span></span> disjoint sets <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline25.png\"/><span data-mathjax-type=\"texmath\"><span>$E_1, \\ldots, E_k\\subseteq E(H)$</span></span></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline26.png\"/><span data-mathjax-type=\"texmath\"><span>$|E_i|\\gt \\alpha$</span></span></span></span>, there exists <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline27.png\"/><span data-mathjax-type=\"texmath\"><span>$(e_1, \\ldots, e_k)\\in E_1\\times \\cdots \\times E_k$</span></span></span></span> such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline28.png\"/><span data-mathjax-type=\"texmath\"><span>$e_1\\cap \\cdots \\cap e_k\\neq \\emptyset$</span></span></span></span>, then the smallest possible maximum degree of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline29.png\"/><span data-mathjax-type=\"texmath\"><span>$H$</span></span></span></span> is essentially the same (within a small error term depending on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline30.png\"/><span data-mathjax-type=\"texmath\"><span>$r$</span></span></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240228163240574-0578:S096354832400004X:S096354832400004X_inline31.png\"/><span data-mathjax-type=\"texmath\"><span>$\\alpha$</span></span></span></span>). We prove our results in this dual setting.</p>","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorics, Probability and Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/s096354832400004x","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 result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary Abstract Image$r$-colouring of the complete Abstract Image$k$-uniform hypergraph Abstract Image$K_n^k$ when Abstract Image$k\geq 2$ and Abstract Image$k\in \{r-1,r\}$. We prove a result which says that if one replaces Abstract Image$K_n^k$ in Gyárfás’ theorem by any ‘expansive’ Abstract Image$k$-uniform hypergraph on Abstract Image$n$ vertices (that is, a Abstract Image$k$-uniform hypergraph Abstract Image$G$ on Abstract Image$n$ vertices in which Abstract Image$e(V_1, \ldots, V_k)\gt 0$ for all disjoint sets Abstract Image$V_1, \ldots, V_k\subseteq V(G)$ with Abstract Image$|V_i|\gt \alpha$ for all Abstract Image$i\in [k]$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on Abstract Image$r$ and Abstract Image$\alpha$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms.

Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary Abstract Image$r$-partite Abstract Image$r$-uniform hypergraph Abstract Image$H$ with Abstract Image$n$ edges in which every set of Abstract Image$k$ edges has a common intersection. In this language, our result says that if one replaces the condition that every set of Abstract Image$k$ edges has a common intersection with the condition that for every collection of Abstract Image$k$ disjoint sets Abstract Image$E_1, \ldots, E_k\subseteq E(H)$ with Abstract Image$|E_i|\gt \alpha$, there exists Abstract Image$(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$ such that Abstract Image$e_1\cap \cdots \cap e_k\neq \emptyset$, then the smallest possible maximum degree of Abstract Image$H$ is essentially the same (within a small error term depending on Abstract Image$r$ and Abstract Image$\alpha$). We prove our results in this dual setting.

扩张超图中的大型单色成分
Gyárfás [12] 的一个结果精确地确定了当 $k\geq 2$ 和 $kin\{r-1,r\}$ 时,完整 $k$ Uniform 超图 $K_n^k$ 的任意 $r$ 着色中最大单色分量的大小。我们证明了这样一个结果:如果把 Gyárfás 定理中的 $K_n^k$ 替换为任何在 $n$ 顶点上的 "扩张性"$k$-匀速超图(即在 $n$ 顶点上的 $k$ 匀速超图 $G$,其中 $e(V_1, \ldots、V_k)\gt 0$ 对于所有不相邻的集合 $V_1, \ldots, V_ksubseteq V(G)$,对于 [k]$ 中的所有 $i\ 都是 $|V_i|\gt\alpha$),那么我们会得到一个大小基本相同的最大单色分量(在取决于 $r$ 和 $\alpha$ 的小误差范围内)。Gyárfás 的结果等同于一个对偶问题,即确定一个具有 $n$ 边的任意 $r$ 部分 $r$-Uniform 超图 $H$ 的最小最大度,其中每组 $k$ 边都有一个共同交集。用这种语言表达,我们的结果就是说,如果把每组 $k$ 边有共同交集的条件换成这样的条件:对于每组 $k$ 不相交的集合 $E_1, \ldots, E_k\subseteq E(H)$ 带有 $|E_i|gt \alpha$、存在 $(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$,使得 $e_1\cap \cdots \cap e_k\neq \emptyset$,那么 $H$ 的最小可能最大度基本上是相同的(在取决于 $r$ 和 $\alpha$ 的小误差范围内)。我们将在这种对偶设置中证明我们的结果。
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