Here is a lovely piece of news: the following paper by James Davies was posted on the arXive a few weeks ago. The paper uses spectral methods to settle an old question, posed in 1994, by Moshe Rosenfeld.
(See this 2009 post over here and this 2008 paper by Moshe. Rosenfeld problem first appeared in print in a 1994 paper by Paul Erdos and Alexander Soifer also wrote in several places about this problem.)
Every finite colouring of the plane contains a monochromatic pair of points at an odd (integral) distance from each other.
The paper proves more general result and discover a new class of triangle free graphs with large chromatic number. Congratulations, James.
Challenge: what is the smallest odd distance between a monochromatic pair in the Hoffman-Soifer coloring?