The lower bound for the matching polytope is actually the following: if you can write linear inequalities to define the perfect matching polytope on the complete graph (i.e., so that the resulting polytope is equal to the convex hull of perfect matchings in ), then .

Since Edmonds’ formulation of the polytope (based on odd-sized cuts) uses constraints, we know the pretty much the correct answer. (This is due to this recent paper of Thomas Rothvoss.)

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