Monochromatic simplices of any volume

We give a very short proof of the following result of Graham from 1980: For any finite coloring of RdRd, d≥2d≥2, and for any α>0α>0, there is a monochromatic (d+1)(d+1)-tuple that spans a simplex of volume αα. Our proof also yields new estimates on the number A=A(r)A=A(r) defined as the minimum positive value AA such that, in any rr-coloring of the grid points Z2Z2 of the plane, there is a monochromatic triangle of area exactly AA.