Erdős Problem #1092

The problem

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 $\leq r$ 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 $f_r(n)\gg_r n$?

Statement quoted from erdosproblems.com/1092, maintained by Thomas Bloom. Original source: [Er76c,p.4]. Current status there: disproved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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