Fiorini, Massar, Pokutta, Tiwary and De Wolf (Exponential Lower Bounds for Polytopes in Combinatorial Optimization, Journal of the ACM 62(2):article 17, 2015; PDF, ArXiv) show any linear program that solves travelling salesman needs super-polynomially many constraints.
Suppose $P=NP$ by 'some' method then we can solve the optimal tour explicitly and trivially setup a LP that 'solves' the TSP problem. So $P=NP$ implies that TSP has a poly-size LP formulation. The contrapositive is that TSP has no poly-size LP formulation implies $P\neq NP$. This paper shows TSP needs super-polynomially many constraints.
So why doesn't this show that $P\neq NP$?