Monochromatic simplices of any volume.
Adrian Dumitrescu, Minghui Jiang · 2008
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. 1