New bounds of two hypergraph Ramsey problems
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912093387620352 |
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| author | Fan, Chunchao Hu, Xinyu Lin, Qizhong Lu, Xin |
| author_facet | Fan, Chunchao Hu, Xinyu Lin, Qizhong Lu, Xin |
| contents | We focus on two hypergraph Ramsey problems. First, we consider the Erdős-Hajnal function $r_k(k+1,t;n)$. In 1972, Erdős and Hajnal conjectured that the tower growth rate of $r_k(k+1,t;n)$ is $t-1$ for each $2\le t\le k$. To finish this conjecture, it remains to show that the tower growth rate of $r_4(5,4;n)$ is three. We prove a superexponential lower bound for $r_4(5,4;n)$, which improves the previous best lower bound $r_4(5,4;n)\geq 2^{Ω(n^2)}$ from Mubayi and Suk (\emph{J. Eur. Math. Soc., 2020}). Second, we prove an upper bound for the hypergraph Erdős-Rogers function $f^{(k)}_{k+1,k+2}(N)$ that is an iterated $(k-3)$-fold logarithm in $N$ for each $k\geq 5$. This improves the previous upper bound that is an iterated $(k-13)$-fold logarithm in $N$ for $k\ge14$ due to Mubayi and Suk (\emph{J. London Math. Soc., 2018}), in which they conjectured that $f^{(k)}_{k+1,k+2}(N)$ is an iterated $(k-2)$-fold logarithm in $N$ for each $k\ge3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22019 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New bounds of two hypergraph Ramsey problems Fan, Chunchao Hu, Xinyu Lin, Qizhong Lu, Xin Combinatorics We focus on two hypergraph Ramsey problems. First, we consider the Erdős-Hajnal function $r_k(k+1,t;n)$. In 1972, Erdős and Hajnal conjectured that the tower growth rate of $r_k(k+1,t;n)$ is $t-1$ for each $2\le t\le k$. To finish this conjecture, it remains to show that the tower growth rate of $r_4(5,4;n)$ is three. We prove a superexponential lower bound for $r_4(5,4;n)$, which improves the previous best lower bound $r_4(5,4;n)\geq 2^{Ω(n^2)}$ from Mubayi and Suk (\emph{J. Eur. Math. Soc., 2020}). Second, we prove an upper bound for the hypergraph Erdős-Rogers function $f^{(k)}_{k+1,k+2}(N)$ that is an iterated $(k-3)$-fold logarithm in $N$ for each $k\geq 5$. This improves the previous upper bound that is an iterated $(k-13)$-fold logarithm in $N$ for $k\ge14$ due to Mubayi and Suk (\emph{J. London Math. Soc., 2018}), in which they conjectured that $f^{(k)}_{k+1,k+2}(N)$ is an iterated $(k-2)$-fold logarithm in $N$ for each $k\ge3$. |
| title | New bounds of two hypergraph Ramsey problems |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.22019 |