Given a finite set E of n symbols a family S of subsets of E (called vertices) form an abstract polytope if (1) Each vertex is a subset of m symbols of E. (2) Every subset of m + 1 symbols of E contains either zero or two vertices (called adjacent). (3) Every pair of vertices V sup 0 and V sup * can be joined by a path V sup 0 = V sub 1, ..., V sub k = V sup * such that V sub i, V sub (i + 1) are adjacent and (V sub i) contained in (V sup 0) joined to (V sup *) i = 1, ..., k-1. It is shown that if two vertices of a given abstract polytope contain the same symbol (say x) then there exists a path such that every vertex along the path contains x. (Author)
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