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

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 Turán numbers $T^{\downarrow }(n,m)$ with $m\ge n\mathrm{/}4$. 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.