Steenrod problem and some graded Stanley-Reisner rings

``What kind of ring can be represented as the singular cohomology ring of a space?‘’ is a classic problem in algebraic topology, posed by Steenrod. In this paper, we consider this problem when rings are the graded Stanley-Reisner rings, in other words the polynomial rings divided by an ideal generated by square-free monomials. Under some assumption, we give a necessary and sufficiently condition that a graded Stanley-Reisner ring is realizable.