 Solution

The diagonal Ramsey number $R(t)$ is the least positive integer $n$ such that every red-blue colouring of the edges of $K_n$ contains a monochromatic $K_t$.

More generally, let $R(s,t)$ be the least $n$ forcing either a red $K_s$ or a blue $K_t$. We haveAssuming the two smaller numbers exist, colourand choose a vertex $v$. At least $R(s-1,t)$ of its incident edges are red, or at least $R(s,t-1)$ are blue. In the first case, the corresponding neighbourhood contains a red $K_{s-1}$, which extends with $v$ to a red $K_s$, or a blue $K_t$. The second case is symmetric. Thuswhich proves existence by induction.

Pascal's identity then gives the binomial upper bound for a Ramsey numberConsequently, for $t\ge 2$,