An adaptive upper bound on the Ramsey numbers $R(3,\dots,3)$

TL;DR

This paper introduces an adaptive method to improve upper bounds on Ramsey numbers R(n,3) by leveraging tighter estimates of R(4,3), leading to more precise bounds for all n ≥ 4.

Contribution

It presents a novel adaptive approach that refines upper bounds on R(n,3) based on improved estimates of R(4,3), enhancing previous bounds significantly.

Findings

Improved upper bounds on R(n,3) for all n ≥ 4.

Demonstrates how better estimates of R(4,3) tighten bounds for larger n.

Provides a framework for updating bounds as R(4,3) estimates improve.

Abstract

Since 2002, the best known upper bound on the Ramsey numbers R n (3) = R(3,. .. , 3) is R n (3) $\le$ n!(e -- 1/6) + 1 for all n $\ge$ 4. It is based on the current estimate R 4 (3) $\le$ 62. We show here how any closing-in on R 4 (3) yields an improved upper bound on R n (3) for all n $\ge$ 4. For instance, with our present adaptive bound, the conjectured value R 4 (3) = 51 implies R n (3) $\le$ n!(e -- 5/8) + 1 for all n $\ge$ 4.