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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.

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