On the maximum number of $r$-cliques in graphs free of complete $r$-partite subgraphs
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913752209686528 |
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| author | Balogh, József Jiang, Suyun Luo, Haoran |
| author_facet | Balogh, József Jiang, Suyun Luo, Haoran |
| contents | We estimate the maximum possible number of cliques of size $r$ in an $n$-vertex graph free of a fixed complete $r$-partite graph $K_{s_1, s_2, \ldots, s_r}$. By viewing every $r$-clique as a hyperedge, the upper bound on the Turán number of the complete $r$-partite hypergraphs gives the upper bound $O\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. We improve this to $o\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. The main tool in our proof is the graph removal lemma. We also provide several lower bound constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16818 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the maximum number of $r$-cliques in graphs free of complete $r$-partite subgraphs Balogh, József Jiang, Suyun Luo, Haoran Combinatorics 05C35 We estimate the maximum possible number of cliques of size $r$ in an $n$-vertex graph free of a fixed complete $r$-partite graph $K_{s_1, s_2, \ldots, s_r}$. By viewing every $r$-clique as a hyperedge, the upper bound on the Turán number of the complete $r$-partite hypergraphs gives the upper bound $O\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. We improve this to $o\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. The main tool in our proof is the graph removal lemma. We also provide several lower bound constructions. |
| title | On the maximum number of $r$-cliques in graphs free of complete $r$-partite subgraphs |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2402.16818 |