> [!NOTE] Theorem > For all $k\in \mathbb{N},$ $R(k,2)=R(2,k)=k.$ ###### Proof Let $k\in\mathbb{N}.$ By [[Ramsey Numbers on Two Colours is Symmetric]], $R(2,k)=R(k,2).$ We have $R(k,2)\geq k$ since a $(k-1)$-clique does not contain a $k$-clique or an independent set of $2$ vertices. Also $R(k,2)\leq k$: Let $G$ be a graph with $k$ vertices. Either $G$ is a $k$-clique or $G$ has a pair of non-adjacent vertices. In the first case, $G$ contains a $k$-clique and in the second, $G$ contains an independent set of $2$ vertices. This completes the proof of the claim.