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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.09367 |
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| _version_ | 1866917867205689344 |
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| author | Nie, Jiaxi Spiro, Sam |
| author_facet | Nie, Jiaxi Spiro, Sam |
| contents | Let $K_{s,t}^{(r)}$ denote the $r$-uniform hypergraph obtained from the graph $K_{s,t}$ by inserting $r-2$ new vertices inside each edge of $K_{s,t}$. We prove essentially tight bounds on the size of a largest $K_{s,t}^{(r)}$-subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$ whenever $r\ge 2s/3+2$, giving the first random Turán results for expansions that go beyond a natural "tight-tree barrier." In addition to this, our methods yield optimal supersaturation results for $K_{s,t}^{(3)}$ for sufficiently dense host hypergraphs, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09367 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random Turán Problems for $K_{s,t}$ Expansions Nie, Jiaxi Spiro, Sam Combinatorics Probability Let $K_{s,t}^{(r)}$ denote the $r$-uniform hypergraph obtained from the graph $K_{s,t}$ by inserting $r-2$ new vertices inside each edge of $K_{s,t}$. We prove essentially tight bounds on the size of a largest $K_{s,t}^{(r)}$-subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$ whenever $r\ge 2s/3+2$, giving the first random Turán results for expansions that go beyond a natural "tight-tree barrier." In addition to this, our methods yield optimal supersaturation results for $K_{s,t}^{(3)}$ for sufficiently dense host hypergraphs, which may be of independent interest. |
| title | Random Turán Problems for $K_{s,t}$ Expansions |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2412.09367 |