$30\le R(3,3,4)\le 31$

Abstract

With the help of computer algorithms, we improve the upper bound on the classical three-color Ramsey number $R(3,3,4)$, and thus we show that the exact value of this number
is $30$ or $31$.

We also present computer enumeration of all $3$-colorings of edges on at least $14$ vertices without monochromatic triangles.