A Short Proof that the Extension Complexity of the Correlation Polytope Grows Exponentially

We establish that the extension complexity of the $$n\times n$$n×n correlation polytope is at least $$1.5\,^n$$1.5n by a short proof that is self-contained except for using the fact that every face of a polyhedron is the intersection of all facets it is contained in. The main innovative aspect of the proof is a simple combinatorial argument showing that the rectangle covering number of the unique-disjointness matrix is at least $$1.5^n$$1.5n, and thus the nondeterministic communication complexity of the unique-disjointness predicate is at least $$.58n$$.58n. We thereby slightly improve on the previously best known lower bounds $$1.24^n$$1.24n and $$.31n$$.31n, respectively.