Hypergraph Erdős--Rogers functions with consecutive clique sizes

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.

This version incorporates several detailed refinements. 20 pages