Minimum Sets Forcing Monochromatic Triangles

Abstract

For a given triangle $T$, consider the problem of finding a finite set $S$ in the plane such that every two-coloring of $S$ results in a monochromatic set congruent to the vertices of $T$. We show that such a set $S$ must have at least seven points. Furthermore, we show by an example that the minimum of seven is achieved.