Topological Bounds on the Dimension of Orthogonal Representations of Graphs

An orthogonal representation of a graph is an assignment of nonzero real vectors to its vertices such that distinct non-adjacent vertices are assigned to orthogonal vectors. We prove general lower bounds on the dimension of orthogonal representations of graphs using the Borsuk-Ulam theorem from algebraic topology. Our bounds strengthen the Kneser conjecture, proved by Lov\'asz in 1978, and some of its extensions due to B\'ar\'any, Schrijver, Dol'nikov, and Kriz. As applications, we determine the integrality gap of fractional upper bounds on the Shannon capacity of graphs and the quantum one-round communication complexity of certain promise equality problems.