Optimizing the CGMS upper bound on Ramsey numbers
Parth Gupta, Ndiame Ndiaye, Sergey Norin, Louis Wei
TL;DR
This work provides a streamlined inductive framework for upper-bounding Ramsey numbers, replacing the prior book algorithm with a simpler density-based approach. It establishes an explicit off-diagonal bound R(k,ℓ) ≤ 4(k+ℓ)((√5+1)(k+2ℓ)/(4ℓ))^ℓ((k+2ℓ)/k)^{k/2} for ℓ ≤ k and a diagonal bound R(k,k) ≤ (3.7992...)^{k+o(k)}, with further refinement via higher-moment analysis of red-edge excess yielding a bound R(k,ℓ) ≤ e^{G(ℓ/k)k+o(k)} binom(k+ℓ,ℓ) where G(λ) = (-0.25λ+0.03λ^2+0.08λ^3)e^{−λ}. The results extend to multicolor Ramsey numbers, introducing Θ(ℓ⃗) to capture the color-allocation factor, and discuss technical challenges in fully generalizing the book-algorithm optimization to the multicolor setting. Collectively, the paper sharpens exponential upper bounds for Ramsey numbers and clarifies how parameter choices drive the improvements in a density-based inductive framework.
Abstract
In a recent breakthrough Campos, Griffiths, Morris and Sahasrabudhe obtained the first exponential improvement of the upper bound on the diagonal Ramsey numbers since 1935. We shorten their proof, replacing the underlying book algorithm with a simple inductive statement. This modification allows us - to give a very short proof of an improved upper bound on the off-diagonal Ramsey numbers, which extends to the multicolor setting, and - to clarify the dependence of the bounds on underlying parameters and optimize these parameters, obtaining, in particular, an upper bound $$R(k,k) \leq (3.8)^{k+o(k)}$$ on the diagonal Ramsey numbers.