## Lecture 10: Matching Polytopes

The lower bound for the matching polytope is actually the following: if you can write $M$ 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 $K_n$), then $M = 2^{\Omega(n)}$.

Since Edmonds’ formulation of the polytope (based on odd-sized cuts) uses $M = 2^{O(n)}$ constraints, we know the pretty much the correct answer. (This is due to this recent paper of Thomas Rothvoss.)