A simplicial complex is said to satisfy complementarity if exactly one of each complementary pair of nonempty vertex-sets constitutes a simplex of the complex. In this article we show that if there exists a n-vertex d-dimensional pseudo-manifold M with complementarity and either n≤d+6 or d≤ 6 then d = 0, 2, 4 or 6 with n = 3d/2 + 3. We also show that if M is a d-dimensional pseudo-manifold with complementarity and the number of vertices in M is ≤ d+5 then M is either a set of three points or the unique 6-vertex real projective plane or the unique 9-vertex complex projective plane.
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