Problem 628

Erdős Problem #628 [status: falsifiable; formalized: no]. 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? Prize: no. Tags: chromatic number, graph theory.