Clique

Definition (Clique)

The clique number, $\omega (G)$ of a Graph $G$ is the largest $n$ so that Complete Graph $K_n$ is a subgraph of $G$.

Corollary (Chromatic Number Lower Bound)

For any graph $G$, chromatic number $\chi (G)\ge \omega (G)$ by Lemma (Complete Chromatic Number).

This bound is unfortunately far from tight.