The palette-size Erdős–Hajnal conjecture for edge-colourings

Let $k$, $s'$, and $s$ be integers with $s\geq s'$. Let $T=(t_1,\ldots,t_{s'})$ be a tuple of nonnegative integers satisfying

$$\sum_{i=1}^{s'}t_i=\binom{k}{2}.$$

A colouring $c$ of $K_n$ in colours from $[s]$ avoids the palette $T$ if there is no clique of size $k$ in which exactly $t_i$ edges have colour $i$ for every $i=1,\ldots,s'$. Let $h_s(c)$ denote the largest vertex set inducing fewer than all $s$ colours.

Size-version of the Erdős–Hajnal conjecture. There is a positive constant $\epsilon$ such that every colouring $c$ of $K_n$ in colours from $[s]$ avoiding $T$ satisfies

$$h_s(c)\geq n^\epsilon.$$

The conjecture is a weaker size-version of the Erdős–Hajnal conjecture for palettes rather than specific forbidden coloured patterns. It is proposed after the corresponding assertion fails for hypergraphs of uniformity at least three; its status is open.