An infinite quantum Ramsey theorem

TL;DR

This paper establishes an infinite Ramsey theorem for noncommutative graphs represented as operator subspaces on infinite-dimensional Hilbert spaces, revealing structural dichotomies under certain conditions.

Contribution

It introduces a novel infinite Ramsey theorem for noncommutative graphs in infinite dimensions, expanding the understanding of their structural properties.

Findings

Existence of an infinite rank projection P with specific compression properties

Compression PVP is either maximal or minimal under certain conditions

Provides a new perspective on the structure of noncommutative graphs in infinite dimensions

Abstract

We prove an infinite Ramsey theorem for noncommutative graphs realized as unital self-adjoint subspaces of linear operators acting on an infinite dimensional Hilbert space. Specifically, we prove that if V is such a subspace, then provided there is no obvious obstruction, there is an infinite rank projection P with the property that the compression PVP is either maximal or minimal in a certain natural sense.