Erdős Problem #1092

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 $f_r(n)\gg_r n$?

#1092: [Er76c,p.4]

graph theory | chromatic number

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A conjecture of Erdős, Hajnal, and Szemerédi. This seems to be closely related to, but distinct from, [744].

Tang notes in the comments that a construction of Rödl [Ro82] disproves the first question, so that $f_2(n)\not\gg n$.

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This page was last edited 06 December 2025.

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Additional thanks to: Quanyu Tang

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