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(n, m) with m ≥ n/4. In particular, we conclude that QR(2,2) = 4 and confirm Weaver's conjecture that T(4, 1) = 4. We also obtain a new result for the existence of anticliques in quantum graphs of low dimension.