A GEOMETRIC HALL-TYPE THEOREM

Abstract

We introduce a geometric generalization of Hall's marriage theorem. For any family F = {X-1,...,X-m} of finite sets in R-d, we give conditions under which it is possible to choose a point x(i# is an element of X-i for every 1 <= i <= m in such a way that the points {x#1#,..., x#m)} subset of R-d are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxell's celebrated generalization of Hall's theorem.