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

Shalom Eliahou (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

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

Domaines: Combinatoire [math.CO]

Shalom Eliahou. An adaptive upper bound on the Ramsey numbers $R(3,\dots,3)$. Integers : Electronic Journal of Combinatorial Number Theory, 2020, 20 (A54), pp.7. ⟨hal-02401197⟩