{"title":"紧循环的极值数","authors":"B. Sudakov, István Tomon","doi":"10.1093/IMRN/RNAA396","DOIUrl":null,"url":null,"abstract":"A tight cycle in an $r$-uniform hypergraph $\\mathcal{H}$ is a sequence of $\\ell\\geq r+1$ vertices $x_1,\\dots,x_{\\ell}$ such that all $r$-tuples $\\{x_{i},x_{i+1},\\dots,x_{i+r-1}\\}$ (with subscripts modulo $\\ell$) are edges of $\\mathcal{H}$. \nAn old problem of V. Sos, also posed independently by J. Verstraete, asks for the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices which has no tight cycle. Although this is a very basic question, until recently, no good upper bounds were known for this problem for $r\\geq 3$. Here we prove that the answer is at most $n^{r-1+o(1)}$, which is tight up to the $o(1)$ error term. \nOur proof is based on finding robust expanders in the line graph of $\\mathcal{H}$ together with certain density increment type arguments.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"125 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"15","resultStr":"{\"title\":\"The Extremal Number of Tight Cycles\",\"authors\":\"B. Sudakov, István Tomon\",\"doi\":\"10.1093/IMRN/RNAA396\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A tight cycle in an $r$-uniform hypergraph $\\\\mathcal{H}$ is a sequence of $\\\\ell\\\\geq r+1$ vertices $x_1,\\\\dots,x_{\\\\ell}$ such that all $r$-tuples $\\\\{x_{i},x_{i+1},\\\\dots,x_{i+r-1}\\\\}$ (with subscripts modulo $\\\\ell$) are edges of $\\\\mathcal{H}$. \\nAn old problem of V. Sos, also posed independently by J. Verstraete, asks for the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices which has no tight cycle. Although this is a very basic question, until recently, no good upper bounds were known for this problem for $r\\\\geq 3$. Here we prove that the answer is at most $n^{r-1+o(1)}$, which is tight up to the $o(1)$ error term. \\nOur proof is based on finding robust expanders in the line graph of $\\\\mathcal{H}$ together with certain density increment type arguments.\",\"PeriodicalId\":8442,\"journal\":{\"name\":\"arXiv: Combinatorics\",\"volume\":\"125 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"15\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Combinatorics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1093/IMRN/RNAA396\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1093/IMRN/RNAA396","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell\geq r+1$ vertices $x_1,\dots,x_{\ell}$ such that all $r$-tuples $\{x_{i},x_{i+1},\dots,x_{i+r-1}\}$ (with subscripts modulo $\ell$) are edges of $\mathcal{H}$.
An old problem of V. Sos, also posed independently by J. Verstraete, asks for the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices which has no tight cycle. Although this is a very basic question, until recently, no good upper bounds were known for this problem for $r\geq 3$. Here we prove that the answer is at most $n^{r-1+o(1)}$, which is tight up to the $o(1)$ error term.
Our proof is based on finding robust expanders in the line graph of $\mathcal{H}$ together with certain density increment type arguments.