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) [DecidableRel G.Adj],2  G.indepNum = 2 →3    ∀ (k : ℕ),4      G.chromaticNumber = ↑k →5        G.CliqueFree k →6          ∀ (a b : ℕ),7            a ≥ 2 →8              b ≥ 2 →9                a + b = k + 1 →10                  ∃ s, (SimpleGraph.induce s G).chromaticNumber ≥ ↑a ∧ (SimpleGraph.induce sᶜ G).chromaticNumber ≥ ↑b