Hypergraph Erdős--Rogers functions with consecutive clique sizes

Qizhong Lin; Lin Niu

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arXiv · 2607.10111

Hypergraph Erdős--Rogers functions with consecutive clique sizes

Qizhong Lin·Lin Niu

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Abstract

For integers $k\le s<t$, the hypergraph Erdős--Rogers function $f^{(k)}_{s,t}(n)$ is the largest integer $m$ such that every $n$-vertex $K_t^{(k)}$-free $k$-graph contains a set of $m$ vertices spanning no copy of $K_s^{(k)}$. We prove that, for every fixed $s\ge4$, \[ f^{(4)}_{s,s+1}(n)=(\log n)^{o(1)}, \] thereby resolving a problem posed by Conlon, Fox and Sudakov. The key input is a new $3$-uniform estimate: for every fixed $s\ge3$, $f^{(3)}_{s,s+1}(n)=O(\frac{\log n}{\log\log n})$, which improves the logarithmic upper bound of Dudek and Mubayi. The proof develops a probabilistic pair-coloring construction based on a robust auxiliary palette and hypergraph containers. As a further consequence, we obtain $f^{(k)}_{k+1,k+2}(n)=(\log_{(k-3)} n)^{o(1)}$ for every fixed $k\ge5$, making substantial progress towards a conjecture of Mubayi and Suk.

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BibTeX RIS

Qizhong Lin, Lin Niu. 2026-07-26. Hypergraph Erdős--Rogers functions with consecutive clique sizes. https://arxiv.org/abs/2607.10111

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