Hypergraph Erdős--Rogers functions with consecutive clique sizes

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.