Monotonicity of diagonal critical multiplicity

Prove that the two-color diagonal critical multiplicity m_2(s) is monotonically increasing as s increases.

Background

The introduction identifies monotonicity of critical multiplicity as a question attributed to Caicedo. The paper’s first formal conjecture would imply this monotonicity, because it constrains m_2(s) through a product expression based on Ramsey numbers and balanced auxiliary subgraph sizes.

The monotonicity question is not resolved by the bounds proved in the paper; it is presented as an existing open question and as a consequence that would follow from the proposed conjectural characterization.

References

Among other things, \Cref{conjecture} would imply that $m_2(s)$ is monotonically increasing, which has been an open question for some time .

— Bounds on the Critical Multiplicity of Ramsey Numbers with Many Colors  (2501.18869 - Christopherson et al., 31 Jan 2025) in Section ‘Concluding Remarks’, immediately after Conjecture 1