How to show that if G has an induced subgraph which is a complete graph on n vertices, then the chromatic number is at least $\chi(G)\ge n$.
Consider coloring such an induced subgraph, say $H$. Clearly, since $H = K_n$, you will needed $n$ colors to color $H$. However, $\chi(G) \geq \chi(H)$, because $G$ includes some other vertices and edges built on top of $H$. Hence, $\chi(G) \geq n$.