{"title":"完整超图着色中的大型单色成分","authors":"Lyuben Lichev , Sammy Luo","doi":"10.1016/j.jcta.2024.105867","DOIUrl":null,"url":null,"abstract":"<div><p>Gyárfás famously showed that in every <em>r</em>-coloring of the edges of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, there is a monochromatic connected component with at least <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span> vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for <em>k</em><span>-uniform hypergraphs<span>. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of </span></span><span><math><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> as the number of colors grows. We further generalize these questions to ask about counts of vertex <em>s</em>-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Large monochromatic components in colorings of complete hypergraphs\",\"authors\":\"Lyuben Lichev , Sammy Luo\",\"doi\":\"10.1016/j.jcta.2024.105867\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Gyárfás famously showed that in every <em>r</em>-coloring of the edges of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, there is a monochromatic connected component with at least <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span> vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for <em>k</em><span>-uniform hypergraphs<span>. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of </span></span><span><math><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> as the number of colors grows. We further generalize these questions to ask about counts of vertex <em>s</em>-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.</p></div>\",\"PeriodicalId\":50230,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series A\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-02-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series A\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0097316524000062\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524000062","RegionNum":2,"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
摘要
Gyárfás 的著名研究表明,在完整图 Kn 的边的每 r 种着色中,都存在一个至少有 nr-1 个顶点的单色连通部分。最近,Conlon、Tyomkyn 和第二位作者的一项研究解决了具有许多边的单色连通成分的类似问题。在本文中,我们研究了这些问题在 k-uniform 超图中的推广。在将连通性定义扩展到更高均匀性的广泛范围内,我们为最大单色成分的大小提供了上界和下界,随着颜色数量的增加,上界和下界的紧密程度可达 1+o(1)。我们进一步将这些问题推广到大型单色分量边缘中包含的顶点 s 集的计数。最后,我们将针对两种颜色的特殊情况给出更精确的结果。
Large monochromatic components in colorings of complete hypergraphs
Gyárfás famously showed that in every r-coloring of the edges of the complete graph , there is a monochromatic connected component with at least vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for k-uniform hypergraphs. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of as the number of colors grows. We further generalize these questions to ask about counts of vertex s-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.