Multicolor Erdős–Hajnal conjecture

Fix integers $k,m a\geq 2$ and an $m$-coloring $\chi$ of the edges of the complete graph $K_k$. A coloring of a complete graph contains a set of vertices whose edges are colored according to $\chi$ if the induced edge-coloring agrees with $\chi$ up to the relevant vertex correspondence.

Multicolor Erdős–Hajnal conjecture. There exists $\varepsilon>0$ such that every coloring of the edges of $K_n$ contains either $k$ vertices whose edges are colored according to $\chi$, or $n^\varepsilon$ vertices whose edges are colored with at most $m-1$ colors.

This conjecture is used conditionally to prove that many grid subgraphs have polynomial Ramsey growth; the supplied text gives no resolution of the conjecture.