A constructive two-parameter Ramsey increment

Abstract

We prove the inequality $R(k+1,s+1) \ge R(k,s) + 2k + 2s$ for classical Ramsey numbers, valid for all $5 \le k \le s$, and formalize the proof in Lean 4. As a consequence, we obtain the new lower bound $R(12,12) \ge 1641$.