Monochromatic simplices of any volume

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