Graph with any rational density and no rich subsets of linear size
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| Format: | Preprint |
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2024
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| _version_ | 1866910340318494720 |
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| author | Im, Seonghyuk Jiang, Suyun Liu, Hong Tran, Tuan |
| author_facet | Im, Seonghyuk Jiang, Suyun Liu, Hong Tran, Tuan |
| contents | A well-known application of the dependent random choice asserts that any $n$-vertex graph $G$ with positive edge density contains a `rich' vertex subset $U$ of size $n^{1-o(1)}$ such that every pair of vertices in $U$ has at least $n^{1-o(1)}$ common neighbors. In 2003, using a beautiful construction on hypercube, Kostochka and Sudakov showed that this is tight: one cannot remove the $o(1)$ terms even if the edge density of $G$ is $1/2$. In this paper, we generalize their result from pairs to tuples. To be precise, we show that given every pair of positive integers $p<q$, there is an $n$-vertex graph $G$ for all sufficiently large $n$ with edge density $p/q$ such that any vertex subset $U$ of size $Ω(n)$ contains $q$ vertices, any $p+1$ of which have $o(n)$ common neighbors. The edge density $p/q$ is best possible. Our construction uses isoperimetry and concentration of measure on high dimensional complex spheres. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_13825 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Graph with any rational density and no rich subsets of linear size Im, Seonghyuk Jiang, Suyun Liu, Hong Tran, Tuan Combinatorics A well-known application of the dependent random choice asserts that any $n$-vertex graph $G$ with positive edge density contains a `rich' vertex subset $U$ of size $n^{1-o(1)}$ such that every pair of vertices in $U$ has at least $n^{1-o(1)}$ common neighbors. In 2003, using a beautiful construction on hypercube, Kostochka and Sudakov showed that this is tight: one cannot remove the $o(1)$ terms even if the edge density of $G$ is $1/2$. In this paper, we generalize their result from pairs to tuples. To be precise, we show that given every pair of positive integers $p<q$, there is an $n$-vertex graph $G$ for all sufficiently large $n$ with edge density $p/q$ such that any vertex subset $U$ of size $Ω(n)$ contains $q$ vertices, any $p+1$ of which have $o(n)$ common neighbors. The edge density $p/q$ is best possible. Our construction uses isoperimetry and concentration of measure on high dimensional complex spheres. |
| title | Graph with any rational density and no rich subsets of linear size |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.13825 |