DMS Graduate Student Seminar

Speaker: Professor Pete Johnson 

Title: Euclidean Coloring Problems:  The Origin Story

Abstract:  The Four-Colour Conjecture begat not only graph theory but also "coloring problems."  For instance, around 1916 Issai Schur proved that if r > 1 and you color [N] = {1,,,,,N} with r colors, then for all N sufficiently large (depending on r) you cannot avoid the existence of integers a, b, and c in [N], all of the same color, such that a + b = c.  This is a coloring result that is only distantly related to graph theory results. In 1950 an 18-year-old student, Edward Nelson, asked a classmate:  What do you think is the smallest number of colors with which the points of the plane are colorable so that any two points (Euclidean) distance 1 from each other are colored differently? This question is still open, as are a great number of other questions that have germinated in certain human minds exposed to Nelson's question.