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. 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. We give a necessary and sufficient condition that a graded Stanley-Reisner ring is realizable when there is no pair of generators x, y such that I x I = I y I = 2 n and xy # 0.