New Upper Bounds for the Classical Ramsey Numbers $R(4,4,4)$, $R(3,4,5)$ and $R(3,3,6)$

TL;DR

This paper establishes new upper bounds for certain three-color classical Ramsey numbers, improving the known limits using the well-known inequality and computational methods.

Contribution

It provides the first improved upper bounds for $R(4,4,4)$, $R(3,4,5)$, and $R(3,3,6)$ beyond previously known results, using advanced combinatorial techniques.

Findings

$R(4,4,4) \,\leq 229$

$R(3,4,5) \,\leq 157$

$R(3,3,6) \,\leq 91$

Abstract

The inequality \[ R(k_1,\ldots,k_r)\le 2-r+\sum_{i=1}^r R(k_1,\ldots,k_{i-1},k_i-1,k_{i+1},\ldots,k_r) \] is well known, and it is strict whenever the right-hand side and at least one of the terms in the sum are even. Except for two known cases, the best upper bounds for classical Ramsey numbers with at least three colors have so far been obtained from this inequality. In this paper we present new bounds such as $R(4,4,4)\le 229$, $R(3,4,5)\le 157$ and $R(3,3,6)\le 91$.