Monochromatic simplices of any volume

We give a very short proof of the following result of Graham from 1980: For any finite coloring of Rd , d 2, and for any > 0, there is a monochromatic (d + 1)-tuple that spans a simplex of volume . 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 Z2 of the plane, there is a monochromatic triangle of area exactly A.

• Type: Conference
• Year: 2008
• Where: CCCG
• Authors: Adrian Dumitrescu, Minghui Jiang
• Keywords: CCCG 2008, Computational Geometry, Finite Coloring, Minimum Positive Value, Short Proof