On the Folkman Number f(2, 3, 4)

Let f(2, 3, 4) denote the smallest integer n such that there exists a K4-free graph of order n for which any 2-coloring of its edges yields at least one monochromatic triangle. It is well-known that such a number must exist. For a long time the best known upper bound, provided by J. Spencer, said that f(2, 3, 4) < 3 · 10. Recently, L. Lu announced that f(2, 3, 4) < 10 000. In this note, we will give a computer assisted proof showing that f(2, 3, 4) < 1000. To prove it we will generalize the idea of Goodman giving a necessary and sufficient condition for a graph G to yield a monochromatic triangle for every edge coloring.