The quantum Ramsey numbers $QR(2,k)$

TL;DR

This paper determines quantum Ramsey numbers and Turán numbers for operator systems, establishing exact values and confirming conjectures, thereby advancing the understanding of quantum analogues of classical combinatorial problems.

Contribution

The paper explicitly calculates quantum Ramsey numbers $QR(2,k)$ and lower quantum Turán numbers $T^ ightarrow(n,m)$, confirming Weaver's conjecture and providing new results on anticliques in low-dimensional quantum graphs.

Findings

$QR(2,2) = 4$

Confirmed Weaver's conjecture $T^ ightarrow(4,1) = 4$

New results on anticliques in low-dimensional quantum graphs

Abstract

Operator systems of matrices can be viewed as quantum analogues of finite graphs. This analogy suggests many natural combinatorial questions in linear algebra. We determine the quantum Ramsey numbers $QR(2,k)$ and the lower quantum Tur\'an numbers $T^\downarrow(n, m)$ with $m \geq n/4$. In particular, we conclude that $QR(2,2) = 4$ and confirm Weaver's conjecture that $T^\downarrow(4, 1) = 4$. We also obtain a new result for the existence of anticliques in quantum graphs of low dimension.