# Journal of Operator Theory

Volume 84, Issue 1, Summer 2020  pp. 49-65.

An infinite quantum Ramsey theorem

Authors:  Matthew Kennedy (1), Taras Kolomatski (2), Daniel Spivak (3)
Author institution:(1) Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
(2) Mathematics Department, Stony Brook University, Stony Brook, New York, 11794-3651, U.S.A.
(3) Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

Summary: 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 $\mathcal V$ is such a subspace, then provided there is no obvious obstruction, there is an infinite rank projection $P$ with the property that the compression $P \mathcal V P$ is either maximal or minimal in a certain natural sense.

DOI: http://dx.doi.org/10.7900/jot.2018dec11.2259
Keywords: Ramsey theory, graph theory, operator systems, operator theory, quantum information theory, quantum error correction