Hypergraph Erdős--Rogers functions with consecutive clique sizes

For integers $k\le s<t$, let $f_{s,t}^{(k)}(n)$ 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 $s\ge 4$, $$f_{s,s+1}^{(4)}(n)=(\log n{)}^{o(1)}\mathrm{.}$$ The key input is a new $3$-uniform estimate: for every fixed $s\ge 3$, $f_{s,s+1}^{(3)}(n)=O(\frac{\log n}{\log \log n})$. 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 $k\ge 5$ there exists a constant $C_k>0$ such that $f_{k+1,k+2}^{(k)}(n)\le \exp \left(C_k\frac{{\log }_{(k-2)}n}{{\log }_{(k-1)}n}\right)$. 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.