Orthonormal representations, vector chromatic number, and extension complexity

We construct a bipartite generalization of Alon and Szegedy's nearly orthogonal vectors, thereby obtaining strong bounds for several extremal problems involving the Lovász theta function, vector chromatic number, minimum semidefinite rank, nonnegative rank, and extension complexity of polytopes. In particular, we answer a question from our previous work together with Letzter and Sudakov, while also addressing a question of Hrubeš and of Kwan, Sauermann, and Zhao. Along the way, we derive a couple of general lower bounds for the vector chromatic number which may be of independent interest.