Filtering the Heegaard Floer contact invariant

We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set Z(>= 0) U{infinity}. It is zero for overtwisted contact structures, infinity for Stein-fillable contact structures, nondecreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, we give an easily computable obstruction to Stein-fillability on closed contact 3-manifolds with nonvanishing Ozsvath-Szabo contact class.