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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.16229 |
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| _version_ | 1866913747980779520 |
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| author | Zhao, Yuhao Zhang, Xiande |
| author_facet | Zhao, Yuhao Zhang, Xiande |
| contents | We study the generalized Turán problem regarding cliques with restricted intersections, which highlights the motivation from extremal set theory. Let $L=\{\ell_1,\dots,\ell_s\}\subset [0,r-1]$ be a fixed integer set with $|L|\notin \{1,r\}$ and $\ell_1<\dots<\ell_s$, and let $Ψ_r(n,L)$ denote the maximum number of $r$-cliques in an $n$-vertex graph whose $r$-cliques are $L$-intersecting as a family of $r$-subsets. Helliar and Liu recently initiated the systematic study of the function $Ψ_r(n,L)$ and showed that $Ψ_r(n,L)\le \left(1-\frac{1}{3r}\right) \prod_{\ell\in L}\frac{n-\ell}{r-\ell}$ for large $n$, improving the trivial bound from the Deza--Erdős--Frankl theorem by a factor of $1-\frac{1}{3r}$. In this article, we improve their result by showing that as $n$ goes to infinity $Ψ_r(n,L)=Θ_{r,L}(n^{|L|})$ if and only if $\ell_1,\dots,\ell_s,r$ form an arithmetic progression and fully determining the corresponding exact values of $Ψ_r(n,L)$ for sufficiently large $n$ in this case. Moreover, when $L=[t,r-1]$, for the generalized Turán extension of the Erdős--Ko--Rado theorem given by Helliar and Liu, we show a Hilton--Milner-type stability result. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting cliques with prescribed intersection sizes Zhao, Yuhao Zhang, Xiande Combinatorics We study the generalized Turán problem regarding cliques with restricted intersections, which highlights the motivation from extremal set theory. Let $L=\{\ell_1,\dots,\ell_s\}\subset [0,r-1]$ be a fixed integer set with $|L|\notin \{1,r\}$ and $\ell_1<\dots<\ell_s$, and let $Ψ_r(n,L)$ denote the maximum number of $r$-cliques in an $n$-vertex graph whose $r$-cliques are $L$-intersecting as a family of $r$-subsets. Helliar and Liu recently initiated the systematic study of the function $Ψ_r(n,L)$ and showed that $Ψ_r(n,L)\le \left(1-\frac{1}{3r}\right) \prod_{\ell\in L}\frac{n-\ell}{r-\ell}$ for large $n$, improving the trivial bound from the Deza--Erdős--Frankl theorem by a factor of $1-\frac{1}{3r}$. In this article, we improve their result by showing that as $n$ goes to infinity $Ψ_r(n,L)=Θ_{r,L}(n^{|L|})$ if and only if $\ell_1,\dots,\ell_s,r$ form an arithmetic progression and fully determining the corresponding exact values of $Ψ_r(n,L)$ for sufficiently large $n$ in this case. Moreover, when $L=[t,r-1]$, for the generalized Turán extension of the Erdős--Ko--Rado theorem given by Helliar and Liu, we show a Hilton--Milner-type stability result. |
| title | Counting cliques with prescribed intersection sizes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.16229 |