Erdős–Szekeres upper bound for Ramsey numbers

**Step 1 (pigeonhole split of the neighborhood of $v$).** Let $N = R(s - 1, t) + R(s, t - 1)$ and fix a vertex $v \in V(K_N)$. Partition the remaining $N - 1$ vertices into $V_R$ (red edge to $v$) and $V_B$ (blue edge to $v$). Since $|V_R| + |V_B| = R(s - 1, t) + R(s, t - 1) - 1$, either $|V_R| \ge R(s - 1, t)$ or $|V_B| \ge R(s, t - 1)$.

Step 2 (inductive conclusion). If $|V_R| \ge R(s - 1, t)$, the subgraph on $V_R$ contains a blue $K_t$ or a red $K_{s-1}$ that joins $v$ to form a red $K_s$. Symmetrically for $V_B$.

Step 3 (binomial bound by induction). Base cases $R(2, t) = t$ and $R(s, 2) = s$ match $\binom{t}{1}$ and $\binom{s}{s-1}$. For $s, t \ge 3$, Pascal's identity gives $R(s, t) \le \binom{s + t - 3}{s - 2} + \binom{s + t - 3}{s - 1} = \binom{s + t - 2}{s - 1}$.