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In the longest plane spanning tree problem, we are given a finite planar point set ▫${\mathcal P}$▫, and our task is to find a plane (i.e., noncrossing) spanning tree for ▫${\mathcal P}$▫ with maximum total Euclidean edge length. Despite more than two decades of research, it remains open whether thi ...

Let ▫$S$▫ be a set of four points chosen independently, uniformly at random from a square. Join every pair of points of ▫$S$▫ with a straight line segment. Color these edges red if they have positive slope and blue, otherwise. We show that the probability that ▫$S$▫ defines a pair of crossing edges ...