We construct a bipartite generalization of Alon and Szegedy's nearly orthogonal vectors, thereby obtaining strong bounds for several extremal problems involving the Lov\'asz theta function, vector chromatic number, minimum semidefinite rank, nonnegative rank, and extension complexity of polytopes. In particular, we derive some general lower bounds for the vector chromatic number which may be of independent interest.

翻译：我们构造了Alon和Szegedy近正交向量的二部图推广，从而获得了涉及Lovász theta函数、向量色数、最小半定秩、非负秩以及多面体扩展复杂性的若干极值问题的强界。特别地，我们推导了向量色数的一些一般下界，这些结果可能具有独立的研究价值。