Prove that any graph $G$ with $n$ vertices and $ \chi(G)=k$ has a subgraph $H$ such that $ H \simeq \overline{K_p}$ where $p=n/k$ and $K_p$ is the complete graph with $n/k$ vertices.
My attempt: Because $ \chi(G)=k$ it must be $G \subseteq K_{p_1 p_2 \cdots p_k} $ where $\displaystyle{\sum_{j=1}^{k} p_j =n}$.
Can I consider now that $ p_j =n/k$ for all j?
If no then the other cases is to have $ p_j >n/k$ for some $j \in \{1, \cdots ,k\}$.
But now how can I continue?