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.