Let $G$ be a simple graph with $n$ vertices. Let $\omega(G)$ and $\chi(G)$ denotes the clique number and chromatic number of $G$ respectively. Then



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> Does $\omega(G)\leq k$ imply $\chi(G)\leq k$ for $n\geq 3$?
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Does $\omega(G)\leq k$ imply $\chi(G)\leq k$ for $n\geq 3$?
No, even triangle free graphs are counterexample since they can have arbitrary large chromatic number.


<https://en.wikipedia.org/wiki/Triangle-free_graph#Coloring_triangle-free_graphs>


