Problem 1092

Erdős Problem #1092: declared status 'disproved'. Formalized: yes. Let $f_r(n)$ be maximal such that, if a graph $G$ has the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic number $r$ and a graph with $\leq f_r(m)$ edges, then $G$ has chromatic number $\leq r+1$. Is it true that $f_2(n) \gg n$? More generally, is $f_r(n)\gg_r n$? Current best: Tang notes in the comments that a construction of R\"{o}dl [Ro82] disproves the first question, so that $f_2(n)\not\gg n$. Prize: no. Tags: chromatic number, geometry.