The <Ramsey number> $R(k,\ell)$ is the least positive <integer> $N$ such that every red-blue coloring of the edges of $K_N$ has a red $K_k$ or a blue $K_\ell$. Equivalently, every $N$-vertex <graph> has a <clique> of size $k$ or an <independent set> of size $\ell$.

For $N=k$, either there is a blue edge or every edge is red, producing the red $K_k$. Conversely an entirely red $K_{k-1}$ has neither a red $K_k$ nor a blue edge. Therefore
$$
\boxed{R(k,2)=k.}
$$

Solution

Past exam of the short options course of the University of Oxford / 2014 / 6 / b / i / Solution

 Solution (source code)