Amazing: Erdős’ Unit Distance Problem was Disproved! It was achieved by AI!

Paul Erdős’s, in his 1946 paper published in the American Mathematical Monthly, posed two general questions about the distribution of distances determined by a finite set of points in a metric space. 1. Unit Distance Problem: At most how many times can the same distance (say, distance 1) occur among a set of n points? 2. Distinct Distances Problem: What is the minimum number of distinct distances determined by a set of n points? Erdős conjectured that in the plane the number of unit distances determined by n points is at most , for a positive constant c, but the best known upper bound, due to Spencer, Szemeredi, and Trotter is only . As for the Distinct Distances Problem, the order of magnitude of the conjectured minimum is . In 2010 a sensational paper of Guth and Katz presents a proof of an almost tight lower bound of the order of Janos Pach wrote about it in this 2010 post. See also this post from 2008. I have just learned that an internal model of OpenAI have very recently…