The cardinal characteristic for relative γ-sets

For X a separable metric space define p ( X ) to be the smallest cardinality of a subset Z of X which is not a relative γ-set in X, i.e., there exists an ω-cover of X with no γ-subcover of Z. We give a characterization of p ( 2 ω ) and p ( ω ω ) in terms of definable free filters on ω which is related to the pseudo-intersection number p . We show that for every uncountable standard analytic space X that either p ( X ) = p ( 2 ω ) or p ( X ) = p ( ω ω ) . We show that the following statements are each relatively consistent with ZFC: (a) p = p ( ω ω ) < p ( 2 ω ) and (b) p < p ( ω ω ) = p ( 2 ω )