Ramsey numbers and the Zarankiewicz problem
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917648172843008 |
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| author | Conlon, David Mattheus, Sam Mubayi, Dhruv Verstraëte, Jacques |
| author_facet | Conlon, David Mattheus, Sam Mubayi, Dhruv Verstraëte, Jacques |
| contents | Building on recent work of Mattheus and Verstraëte, we establish a general connection between Ramsey numbers of the form $r(F,t)$ for $F$ a fixed graph and a variant of the Zarankiewicz problem asking for the maximum number of 1s in an $m$ by $n$ $0/1$-matrix that does not have any matrix from a fixed finite family $\mathcal{L}(F)$ derived from $F$ as a submatrix. As an application, we give new lower bounds for the Ramsey numbers $r(C_5,t)$ and $r(C_7,t)$, namely, $r(C_5,t) = \tildeΩ(t^{\frac{10}{7}})$ and $r(C_7,t) = \tildeΩ(t^{\frac{5}{4}})$. We also show how the truth of a plausible conjecture about Zarankiewicz numbers would allow an approximate determination of $r(C_{2\ell+1}, t)$ for any fixed integer $\ell \geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08694 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ramsey numbers and the Zarankiewicz problem Conlon, David Mattheus, Sam Mubayi, Dhruv Verstraëte, Jacques Combinatorics Building on recent work of Mattheus and Verstraëte, we establish a general connection between Ramsey numbers of the form $r(F,t)$ for $F$ a fixed graph and a variant of the Zarankiewicz problem asking for the maximum number of 1s in an $m$ by $n$ $0/1$-matrix that does not have any matrix from a fixed finite family $\mathcal{L}(F)$ derived from $F$ as a submatrix. As an application, we give new lower bounds for the Ramsey numbers $r(C_5,t)$ and $r(C_7,t)$, namely, $r(C_5,t) = \tildeΩ(t^{\frac{10}{7}})$ and $r(C_7,t) = \tildeΩ(t^{\frac{5}{4}})$. We also show how the truth of a plausible conjecture about Zarankiewicz numbers would allow an approximate determination of $r(C_{2\ell+1}, t)$ for any fixed integer $\ell \geq 2$. |
| title | Ramsey numbers and the Zarankiewicz problem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2307.08694 |