A linear Ramsey theorem

Abstract

We state and prove a theorem in exterior algebra which is an analogue of Ramsey's theorem in combinatorics, with vector spaces and alternating multilinear maps taking the roles played by sets and colorings. Our result can also be formulated as a theorem about the geometry of Grassmannians. The infinite-dimensional version of the theorem admits a quantum mechanical interpretation, and implies an interesting fact about operators on Hilbert space.