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

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.