Counting induced subgraphs with given intersection sizes
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866916957265068032 |
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| author | Zhang, Haixiang Wang, Yichen Zhao, Xiamiao Lu, Mei |
| author_facet | Zhang, Haixiang Wang, Yichen Zhao, Xiamiao Lu, Mei |
| contents | Let $F$ be a graph of order $r$. In this paper, we study the maximum number of induced copies of $F$ with restricted intersections, which highlights the motivation from extremal set theory. Let $L=\{\ell_1,\dots,\ell_s\}\subseteq[0,r-1]$ be an integer set with $s\not\in\{1,r\}$. Let $Ψ_r(n,F,L)$ be the maximum number of induced copies of $F$ in an $n$-vertex graph, where the induced copies of $F$ are $L$-intersecting as a family of $r$-subsets, i.e., for any two induced copies of $F$, the size of their intersection is in $L$. Helliar and Liu initiated a study of the function $Ψ_r(n,K_r,L)$. Very recently, Zhao and Zhang improved their result and showed that $Ψ_r(n,K_r,L)=Θ_{r,L}(n^{s})$ if and only if $\ell_1,\dots,\ell_s,r$ form an arithmetic progression. In this paper, we show that $Ψ_r(n,F,L)=o_{r,L}(n^{s})$ when $\ell_1,\dots,\ell_s,r$ do not form an arithmetic progression. We study the asymptotical result of $Ψ_r(n,C_r,L)$, and determined the asymptotically optimal result when $\ell_1,\dots,\ell_s,r$ form an arithmetic progression and take certain values. We also study the generalized Turán problem, determining the maximum number of $H$, where the copies of $H$ are $L$-intersecting as a family of $r$-subsets. The entropy method is used to prove our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15466 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting induced subgraphs with given intersection sizes Zhang, Haixiang Wang, Yichen Zhao, Xiamiao Lu, Mei Combinatorics 05C35 Let $F$ be a graph of order $r$. In this paper, we study the maximum number of induced copies of $F$ with restricted intersections, which highlights the motivation from extremal set theory. Let $L=\{\ell_1,\dots,\ell_s\}\subseteq[0,r-1]$ be an integer set with $s\not\in\{1,r\}$. Let $Ψ_r(n,F,L)$ be the maximum number of induced copies of $F$ in an $n$-vertex graph, where the induced copies of $F$ are $L$-intersecting as a family of $r$-subsets, i.e., for any two induced copies of $F$, the size of their intersection is in $L$. Helliar and Liu initiated a study of the function $Ψ_r(n,K_r,L)$. Very recently, Zhao and Zhang improved their result and showed that $Ψ_r(n,K_r,L)=Θ_{r,L}(n^{s})$ if and only if $\ell_1,\dots,\ell_s,r$ form an arithmetic progression. In this paper, we show that $Ψ_r(n,F,L)=o_{r,L}(n^{s})$ when $\ell_1,\dots,\ell_s,r$ do not form an arithmetic progression. We study the asymptotical result of $Ψ_r(n,C_r,L)$, and determined the asymptotically optimal result when $\ell_1,\dots,\ell_s,r$ form an arithmetic progression and take certain values. We also study the generalized Turán problem, determining the maximum number of $H$, where the copies of $H$ are $L$-intersecting as a family of $r$-subsets. The entropy method is used to prove our results. |
| title | Counting induced subgraphs with given intersection sizes |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2509.15466 |