Nik Weaver - Washington University in St. Louis In finite dimensions, an operator system is a linear subspace of the n x n complex matrices which contains the identity matrix and is stable under Hermitian transpose. Recently these objects have come to be understood as a "quantum" analog of finite simple graphs. This point of view arose in quantum error correction and has a good theoretical basis. I will discuss some of the many interesting and basic questions which come out of this idea, including a quantum Ramsey theorem, a quantum Turan problem, and the notion of quantum chromatic number. The last of these can be seen as a kind of paving problem.