New bounds of two hypergraph Ramsey problems

Abstract

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 $k\ge3$.

Figures (1)

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Theorems & Definitions (16)

• Definition 3.1
• Conjecture 3.2: Erdős and Hajnal E-H-Con
• Theorem 3.3: Mubayi and Suk M-S-1M-S-3
• Theorem 3.4
• Corollary 3.5
• Theorem 3.6: Fox and He F-H
• Theorem 3.7
• Definition 4.1
• Theorem 4.2: Mubayi and Suk M-S-2
• Conjecture 4.3: Mubayi and Suk M-S-2