Special Seminar: 5-chromatic unit-distance graphs in the plane: initial discovery and subsequent progress

Seminar | May 2 | 4:10-5 p.m. | 3 Evans Hall

 Aubrey de Grey, SENS Research Foundation

 Department of Mathematics

Earlier this year I made the first improvement since 1950 to the bounds of the Hadwiger-Nelson problem, which is to determine the chromatic number of the plane (CNP); the lower bound was previously 4, since there are 4-chromatic unit-distance (UD) graphs in the plane. The improvement to CNP ≥ 5 was achieved by identifying, though not actually defining precisely, a numerical function of UD graphs that must be high in some 4-chromatic cases and low in others, and using examples of the two classes as building-blocks. One of the classes required computer search to identify, and the smallest 5-chromatic case that I found has 1581 vertices. Interest has immediately been high in identifying simpler cases, and a Polymath project is underway for this purpose. I shall describe the initial discovery and the project's recent progress.