Let $f(2,3,4)$} denote the smallest integer $n$ such that there exists a l$K_4$}-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, was $f(2,3,4)$ < 3 \cdot 10^9.
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 this, we will
generalize an idea of Goodman's, giving a necessary and sufficient condition for a graph
$G$ to yield a monochromatic triangle for every edge coloring.