Looking for minimal graded Betti numbers

We consider O-sequences that occur for arithmetically Cohen-Macaulay (ACM) schemes X of codimension three in P n. These are Hilbert functions ' of Artinian algebras that are quotients of the coordinate ring of X by a linear system of parameters. Using suitable decompositions of ', we determine the minimal number of generators possible in some degree c for the dening ideal of any such ACM scheme having the given O-sequence. We apply this result to construct Artinian Gorenstein O-sequences ' of codimension 3 such that the poset of all graded Betti sequences of the Artinian Gorenstein algebras with Hilbert function ' admits more than one minimal element. Finally, for all 3- codimensional complete intersection O-sequences we obtain conditions under which the corresponding poset of graded Betti sequences has more than one minimal element.