Proof that MAX CLIQUE is NP-Hard
There is probably a proof in Karp's original paper. Here is a simple reduction from SAT. Given an instance $\varphi$ of SAT with $m$ clauses, construct an instance of MAX-CLIQUE as follows. For each clause $C$ and each literal $\ell$ appearing in $C$, there is a vertex $(C,\ell)$. Two vertices $(C_1,\ell_1),(C_2,\ell_2)$ are connected if $C_1 \neq C_2$ and $\ell_1 \neq \lnot \ell_2$. This has a clique of size $m$ iff $\varphi$ is satisfiable (exercise).