Generalizations of the geometric de Bruijn Erdős theorem

Pierre Aboulker, École Normale Supérieure, Paris. Part of the Algebra, Logic, and Algorithms Seminar Series.

A classic Theorem of de Bruijn and Erdős states that every noncollinear set of n points in the plane determines at least n distinct lines.  The line L(u,v) determined by two points u, v in the plane consists of all points p such that  u, v, and p are collinear.  With this definition of line L(uv) in an arbitrary metric space (V, dist), Chen and Chvátal conjectured that every metric space on n points, where n is at least 2, has at least n distinct lines or a line that consists of all n points.  The talk will survey results on and around this conjecture.