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The study of fundamental groups of random two dimensional simplicial complexes calls attention to the small subcomplexes of such objects. Such subcomplexes have fewer triangles than twice the number of their vertices. One gets that this condition on a connected complex (and all of its subcomplexes) implies that it is homotopy equivalent to a wedge of circles, spheres and projective planes and further that it is Gromov hyperbolic with a hyperbolicity constant depending only on the ratio of triangles to vertices. For clique complexes of random graphs there is a similar problem involving complexes with fewer edges than thrice the number of their vertices.