Multiplicative form of the Ramsey Diagonal Conjecture

Prove that for every positive integers \(s_1,s_2,t_1,t_2\) satisfying \(s_1t_1\le s_2t_2\), the Ramsey numbers obey \(R(s_1,t_1)\le R(s_2,t_2)\).

Background

The multiplicative form compares Ramsey numbers according to the product of their two parameters rather than their sum. The paper presents the weak multiplicative conjecture from the introduction as a weaker version of this statement.

The multiplicative form would imply the ordinary Ramsey Diagonal Conjecture, as the paper explains, and is therefore a stronger unresolved conjectural principle.

References

We note that Conjecture~\ref{conj:weak.mult.RDC} can be stated as a weak version of the following conjecture.

\begin{conjecture}[Multiplicative form of RDC] \label{conj:mult.RDC} For every $s_1 s_2 t_2 t_1$ such that $s_1t_1 s_2t_2$ we have that \begin{equation*} R(s_1, t_1) R(s_2, t_2). \end{equation*} \end{conjecture}

— On the maximum ratio between chromatic number and clique number  (2512.16062 - Araujo et al., 18 Dec 2025) in Conjecture 4, Appendix, Section Ramsey Diagonal Conjecture