Cliques, stable sets, the clique number $\omega(G)$ and stability number $\alpha(G)$

Let $G$ be a finite simple graph. A set $X\subseteq V(G)$ is a clique when every two distinct vertices of $X$ are adjacent, equivalently when $G[X]$ is complete. It is a stable set, or independent set, when no two distinct vertices of $X$ are adjacent, equivalently when $G[X]$ is edgeless (Subgraphs, induced subgraphs and spanning subgraphs, Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices).

The clique number and stability number are

$\omega(G):=\max\{|X|:X\text{ is a clique in }G\},\qquad \alpha(G):=\max\{|X|:X\text{ is a stable set in }G\}.$

Both maxima exist because the families are nonempty, containing $\varnothing$, and lie inside the finite power set of $V(G)$ (The cardinality $\lvert A\rvert$ of a finite set, $\lvert\mathcal{P}(A)\rvert = 2^{\lvert A\rvert}$ for finite $A$, A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$, and equality holds if and only if $B = A$, The well-ordering principle). In particular, $\omega(K_0)=\alpha(K_0)=0$.