Resolutions of subsets of finite sets of points in projective space

Given a finite set, X, of points in projective space for which the Hilbert function is known, a standard result says that there exists a subset of thi s finite set whose Hilbert function is "as big as possible" inside X. Given a finite set of points in projective space for which the minimal free reso lution of its homogeneous ideal is known, what can be said about possible r esolutions of ideals of subsets of this finite set? We first give a maximal rank type description of the most generic possible resolution of a subset. Then we show, via two very different kinds of counterexamples, that this g eneric resolution is not always achieved. However, we show that it is achie ved for sets of points in projective two space: given any finite set of poi nts in projective two space for which the minimal free resolution is known, there must exist a subset having the predicted resolution.