Triangulated spheres and colored cliques

In [1] a generalization of Hall's theorem was proved for families of hypergraphs. The proof used Sperner's lemma. In [5] Meshulam proved an extension of this result, using homology and the nerve theorem. In this paper we show how the triangulations method can be used to derive Meshulam's results. As in [1], the proof is based on results on extensions of triangulations from the sphere to the full ball. A typical result of this type is that any triangulation of the (d -1)-dimensional sphere S^d-1 can be extended to a triangulation of the ball B^d, by adding one point at a time, having degree at most 2d to its predecessors.