A "Challenging Question" of Bj\"orner from 1976: Every Infinite Geometric Lattice of Finite Rank Has a Matching

Abstract

It is proven that every geometric lattice of finite rank greater than 1 has a matching between the points and hyperplanes. This answers a question of P\'olya Prize-winner Anders Bj\"orner from the 1981 Banff Conference on Ordered Sets, which he raised as a "challenging question" in 1976.