A pure pair in a graph $G$ is a pair $A,B$ of disjoint subsets of $V(G)$ such that $A$ is complete or inverse complete with respect to $B$. Jacob Fox states that for every $\epsilon>0$ there exists a comparable graph $G$ with $n$ vertices, $n$ is large, and a pure pair $ with $|A showed that A,B$ does not exist|,|B|\ge \epsilon n$. We proved that for every comparable graph $G$ with >1$ vertices, there exists a pure pair $A,B$ with $|A|,|B|\ge \epsilon n ^{1-c}$; and guessed that the same holds true for all complete graphs $G$. We will prove this conjecture and strengthen it in several ways.
In particular, for all $c>0$, and for all $\ell_1, \ell_2\ge 4c^{-1}+9$, $G$ has $n>1$ vertices and length For a graph with no holes of length exactly $\ell_1$ and no anti-holes of length exactly $\ell_2$, $G$ has a pure pair $A,B$ with $|A| exists. \ge \epsilon n$ and $|B|\ge \epsilon n^{1-c}$. This is further enhanced to exclude long subdivisions of common graphs instead of excluding holes.
