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?

1∀ (V : Type u_1) [Fintype V] (G : SimpleGraph V) (k : ℕ),2  G.chromaticNumber = ↑k →3    G.CliqueFree k →4      ∀ (a b : ℕ),5        a ≥ 2 →6          b ≥ 2 →7            a + b = k + 1 →8              ∃ s, (SimpleGraph.induce s G).chromaticNumber ≥ ↑a ∧ (SimpleGraph.induce sᶜ G).chromaticNumber ≥ ↑b