On Ramsey number R(4, 3, 3) and triangle-free edge-chromatic graphs in three colors Original Research Article

By applying a look-ahead algorithm, we show that there are, up to isomorphism, exactly two colorings of the edges of K15 with three colors which do not contain a monochromatic triangle. Based on this result we derive a new upper bound for the three color Ramsey number, R(4, 3, 3) ⩽ 32. The best previously known bounds were 30⩽R(4,3,3)⩽33.