On monochromatic configurations for finite colorings

Answering a question of Gurevich, Graham proved that, given any δ>0, for any finite coloring of the plane, there is a triangle of area δ having all of its three vertices of the same color. Questions were asked about similar results for parallelograms, rhombuses etc. For any coloring of the plane, a trapezoid is called monochromatic if its four vertices have the same color. In this paper, we prove that, for any δ>0 and any finite coloring of the plane, there exist infinitely many monochromatic trapezoids of area δ>0 that are translates of the same trapezoid. We shall have some related results for triangles.