Two Erdos Problems on Points in the Plane and AI

In a 1946 paper in the American Mathematical Monthly, Paul Erdos posed the Erdos Distinct Distance Problem and the Erdos Unit Distance Problem.--------------------------------------------------------------------THE ERDOS DISTINCT DISTANCE PROBLEMA nice high school math competition problem, if it were not fairly well known, is:Show that for all sets of \(n\) points in the plane, there are at least \(0.5\sqrt{n}\) distinct distances.A bit harder is to show that there exist \(n\) points in the plane where the number of distinct distances is \(O(\frac{n}{(\log n)^{1/2}})\). The points are the grid points of a \(\sqrt{n}\times\sqrt{n}\) integer grid.Let \(g(n)\) be the minimum number such that, for every set of \(n\) points in the plane,there are \(g(n)\) distinct distances. The above results (of Erdos) show that\( \Omega(n^{1/2}) \le g(n) \le O(\frac{n}{(\log n)^{1/2}}) \)Erdos made no conjecture about \(g(n)\) in that 1946 paper. However, later papers attribute to him one of two…