# If a graph G has chromatic number 3, then G contains a subgraph isomorphic to $K_3$?

Is there a theorem that states if a graph has $\chi(G) = n$ then it also contains a subgraph isomorphic to $K_n$?

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You are trying to compare the chromatic number to the so called clique number. There is a whole literature about this problem. Google should provide relevant information. – Mariano Suárez-Alvarez May 13 '13 at 22:25

Consider the cycle on $5$ vertices.
Yes, $\chi(G)=3$ here but no $K_3$. – coffeemath Jan 29 '13 at 11:56
If $G$ is a triangle-free graph with chromatic number $n$, its Mycelskian $M(G)$ is a triangle free graph with chromatic number $n+1$.