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Let $G$ be a graph with chromatic number $k$ containing no $K_k$. If $a,b\geq 2$ and $a+b=k+1$ then must there exist two disjoint subgraphs of $G$ with chromatic numbers $\geq a$ and $\geq b$ respectively?
This property is sometimes called being $(a,b)$-splittable. A question of Erdős and Lovász (often called the Erdős-Lovász Tihany conjecture). Erdős [Er68b] originally asked about $a=b=3$ which was proved by Brown and Jung [BrJu69] (who in fact prove that $G$ must contain two vertex disjoint odd cycles).

Balogh, Kostochka, Prince, and Stiebitz [BKPS09] have proved the full conjecture for quasi-line graphs and graphs with independence number $2$.

See also the entry in the graphs problem collection.