A note on the Kővári--Sós--Turán theorem for stable hypergraphs

In this note, we prove a stronger version of the Kővári-Sós-Turán theorem for partition-wise $k$-stable $r$-hypergraphs. More precisely, we show that for every $k,r\in {\mathbb{N}}_{\ge 2}$ there is $\eta =\eta (r,k)>0$ such that if $H=(V;E)$ is a partition-wise $k$-stable $r$-uniform $K_{t,\dots ,t}^{(r)}$-free hypergraph with $\mathrm{\mid }V\mathrm{\mid }=n$, then $\mathrm{\mid }E\mathrm{\mid }=O_{r,k,t}(n^{r-\eta })$. Crucially, $\eta$ is independent of $t$. The proof follows the pseudofinite regime used by Chernikov and Starchenko to prove an analogous version of Erdős-Hajnal for stable hypergraphs.