Improved lower bound on monotone circuit complexity of perfect matching?
Eva Tardos proved that the gap is truly exponential by showing that there is a monotone boolean function that has poly size circuits but requires exponential size monotone circuits. Nothing better than super-polynomial is known for matching.

Raz has a result that monotone circuits for matching have linear depth. (Thanks Klauck, for pointing the typo.) 

AFAIK, we know nothing better.

Ref: (1) http://www.springerlink.com/index/P25X5838624J0352.pdf

(2) http://www.wisdom.weizmann.ac.il/~ranraz/publications/Pmatching.ps