Bounds on the Number of 2-Level Polytopes, Cones, and Configurations

Abstract

We prove an upper bound of the form \(2^{O(d^2 \mathop {\mathrm {polylog}}d)}\) on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones, and d-configurations. This in particular answers positively a conjecture of Bohn et al. on 2-level polytopes. We obtain our upper bound by relating affine (resp. linear) equivalence classes of 2-level d-polytopes, d-cones, and d-configurations to faces of the correlation cone. We complement this with a \(2^{\varOmega (d^2)}\) lower bound, by estimating the number of nonequivalent stable set polytopes of bipartite graphs.