Graduate Student Seminar

Speaker: Dr. Pete Johnson

Title: Coloring Euclidean spaces to make translates rainbow

Abstract:  Suppose T is a set of 3 points in the Euclidean plane.  How many colors are needed to color the plane so that every translate of T is rainbow, meaning that the 3 points of the translate are colored with 3 different colors?  The minimum number of colors necessary is either 3 or 4, and that holds not only in the plane, but for 3-point sets in any Euclidean space.  Strangely, the larger number of colors, 4, is necessary only when the 3 points are collinear.

For the corresponding problem for 4-point sets, the minimum number of colors necessary is either 4, 5, 6, or 7, and for non-collinear sets no more than 6 colors are needed.  It has been conjectured, for good reasons, that if no 3 of the 4 points are collinear, then no more than 5 colors are necessary.