Large cliques in graphs with forbidden semi-induced structures
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915622315622400 |
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| author | Chen, Nannan Ma, Yulai Yang, Fan |
| author_facet | Chen, Nannan Ma, Yulai Yang, Fan |
| contents | In 2022, Holmsen showed that any graph with at least \( c \binom{n}{r} \) \(r\)-cliques but no induced complete $r$-partite graph $K_{2,\ldots, 2}$ must contain a clique of order \(Ω(c^{2^{r-1}} n)\). In this paper, we study graphs forbidding semi-induced substructures and show that every $n$-vertex graph $G$ containing at least $c\binom{n}{r}$ copies of $K_r$ (for some constant $c>0$) and forbidding semi-induced substructures, related to $K_{2,\ldots, 2}$, must contain a clique of order $Ω(cn)$. Our result strengthens Holmsen's bound by improving the dependence on $c$ from $c^{2^{r-1}}$ to linear in $c$ with bounded number of forbidden structures. Furthermore, our approach is naturally linked to the notion of VC-dimension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_13073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large cliques in graphs with forbidden semi-induced structures Chen, Nannan Ma, Yulai Yang, Fan Combinatorics In 2022, Holmsen showed that any graph with at least \( c \binom{n}{r} \) \(r\)-cliques but no induced complete $r$-partite graph $K_{2,\ldots, 2}$ must contain a clique of order \(Ω(c^{2^{r-1}} n)\). In this paper, we study graphs forbidding semi-induced substructures and show that every $n$-vertex graph $G$ containing at least $c\binom{n}{r}$ copies of $K_r$ (for some constant $c>0$) and forbidding semi-induced substructures, related to $K_{2,\ldots, 2}$, must contain a clique of order $Ω(cn)$. Our result strengthens Holmsen's bound by improving the dependence on $c$ from $c^{2^{r-1}}$ to linear in $c$ with bounded number of forbidden structures. Furthermore, our approach is naturally linked to the notion of VC-dimension. |
| title | Large cliques in graphs with forbidden semi-induced structures |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.13073 |