On subschemes of 0-dimensional schemes with given graded Betti numbers

We study the subschemes of a $0$-dimensional scheme $X$ for which is known either the Hilbert function or the graded Betti numbers. In the first case we find which kind of subscheme cannot stay in $X$ and in the codimension $2$ case what subschemes must be in $X.$ In the case of graded Betti numbers we study the case of $2$-codimensional partial intersection schemes or Artinian monomial ideals. More generally we give complete results for almost complete intersections and for other suitable Betti sequences.