Erd\H{o}s-Hajnal properties for powers of sparse graphs

We prove that for every nowhere dense class of graphs $\mathcal{C}$, positive integer $d$, and $\epsilon >0$, the following holds: in every $n$-vertex graph $G$ from $\mathcal{C}$ one can find two disjoint vertex subsets $A,B\subseteq V(G)$ such that $\mathrm{\mid }A\mathrm{\mid }\ge (1\mathrm{/}2-\epsilon )\cdot n$ and $\mathrm{\mid }B\mathrm{\mid }=\mathrm{\Omega }(n^{1-\epsilon })$ and either $\mathrm{dist}(a,b)\le d$ for all $a\in A$ and $b\in B$, or $\mathrm{dist}(a,b)>d$ for all $a\in A$ and $b\in B$. We also show some stronger variants of this statement, including a generalization to the setting of First-Order interpretations of nowhere dense graph classes.