Ramsey Diagonal Conjecture

Prove that whenever positive integers \(s_1,s_2,t_1,t_2\) satisfy \(s_1+t_1\le s_2+t_2\), the Ramsey numbers obey \(R(s_1,t_1)\le R(s_2,t_2)\).

Background

The Ramsey Diagonal Conjecture asserts that, for a fixed or bounded total parameter, Ramsey numbers do not increase as the pair of parameters moves away from the diagonal. The paper describes this conjecture as widely believed and very difficult to prove.

The appendix notes that even the special comparison $R(t-1,t+1)\le R(t,t)$ has seen no relevant progress, providing explicit context for the conjecture’s unresolved status.