Known facets of the Travelling Salesman Problem polytope
For asymptotic bounds, Fiorini, Massar, Pokutta, Tiwari, and de Wolf recently showed exponential lower bounds on the number of facets of any polytope that projects to the TSP polytope (the TSP polytope, being the convex hull of feasible TSP solutions). This is stronger than what you ask for, and implies that even adding extra variables will not make the TSP polytope efficiently representable. 

Their paper is follow up to the classical 1988 paper by Yannakakis, who showed the same result but only for polytopes that satisfy a certain symmetry condition.