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

The inequality $$R(k_1,\dots ,k_r)\le 2-r+\sum _{i=1}^{r}R(k_1,\dots ,k_{i-1},k_i-1,k_{i+1},\dots ,k_r)$$ 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 $R(3,3,6)\le 91$.