  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.
