Undergraduate Math Club Seminar

Throw n points randomly onto a plane, and you should expect to have on the order of n^2 distinct distances between pairs of those points. Place those points carefully, and you can make that number much smaller. The Erdős distinct distances problem asks just how small you can make it as n grows large. We will use some elementary graph theory to find a lower bound for the answer, and a result of Landau to find an upper bound. We'll then look at what kind of improvements to our bounds have been obtained over the years, ending with the very recent result of Katz and Guth.