Polynomial Erdős–Hajnal bound for graphs of bounded VC-dimension

Let $d\geq 2$. A graph has VC-dimension at most $d$ when its VC-dimension is bounded by $d$. Bounded-VC-dimension Erdős–Hajnal conjecture. There exists a constant $\varepsilon(d)$ such that every graph on $n$ vertices with VC-dimension at most $d$ contains a clique or an independent set of size $n^{\varepsilon(d)}$.

This is a bounded-VC-dimension analogue of the Erdős–Hajnal conjecture. The paper records weaker lower bounds, including $e^{(\log n)^{1-o(1)}}$, and states that the polynomial bound remains open.