  We give an effective uniform bound on the multigraded regularity of a
subscheme of a smooth projective toric variety X with a given multigraded
Hilbert polynomial. To establish this bound, we introduce a new combinatorial
tool, called a Stanley filtration, for studying monomial ideals in the
homogeneous coordinate ring of X. As a special case, we obtain a new proof of
Gotzmann's regularity theorem. We also discuss applications of this bound to
the construction of multigraded Hilbert schemes.
