Vern Paulsen, University of Houston
The chromatic number of a graph has a description as the classical value of a three-person game. If instead one plays a quantum version of this game, then this yields a smaller value--the quantum chromatic number of the graph. However, using the Algebraic Quantum Field Theory (AQFT) model could yield a larger set of quantum correlations, and a different value for the quantum chromatic number. Whether or not these different sets of quantum correlation matrices are equal or not is related to whether or not conjectures of Connes and Tsirelson have positive answers. Thus, we introduce and study several possible quantum chromatic numbers and study their properties. Proving that these quantum chromatic numbers are all equal would give a partial affirmation of these conjectures, any inequalities would yield a counterexample to one or more conjectures.