Hypergraph Erdős--Rogers functions with consecutive clique sizes

Abstract

For integers \(k\le s<t\), let \(f^{(k)}_{s,t}(n)\) denote 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 give an affirmative answer to a problem of Conlon, Fox and Sudakov by proving that, for every fixed \(s\ge4\), \[ f^{(4)}_{s,s+1}(n)=(\log n)^{o(1)} . \] The key input is a new \(3\)-uniform estimate: for every fixed \(s\ge3\), \(f^{(3)}_{s,s+1}(n)=O(\frac{\log n}{\log\log n})\). This improves the logarithmic upper bound of Dudek and Mubayi. The proof combines hypergraph containers with a probabilistic construction. As a further consequence, for every fixed \(k\ge5\) there exists a constant $C_k>0$ such that \(f^{(k)}_{k+1,k+2}(n)\le\exp\left(C_k\frac{\log_{(k-2)} n}{\log_{(k-1)} n}\right)\). This gives the first upper bound of the form $(\log_{(k-3)} n)^{o(1)}$ and makes substantial progress towards a conjecture of Mubayi and Suk.