  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 this
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 resolution
of its homogeneous ideal is known, what can be said about possible resolutions
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
that this generic resolution is not always achieved, by incorporating an
example of Eisenbud and Popescu. However, we show that it {\em is} achieved for
sets of points in projective two space: given any finite set of points in
projective two space for which the minimal free resolution is known, there must
exist a subset having the predicted resolution.
