poset P. 5. Let M be a metric space and δ a real number. The Vietoris–Rips complex is an ASC whose faces are the finite subsets of M with diameter at most δ. It has applications in homology theory, hyperbolic groups, image processing, and mobile ad hoc networking. It is another example of a flag complex. 6. Let be a square-free monomial ideal in a polynomial ring (that is, an ideal generated by products of subsets of variables). Then the exponent vectors of those square-free monomials of that are not in determine an abstract simplicial complex via the map . In fact, there is a bijection between (non-empty) abstract simplicial complexes on vertices and square-free monomial ideals in . If is the square-free ideal corresponding to the simplicial complex then the quotient is known as the Stanley–Reisner ring of . 7. For any open covering C of a topological space, the nerve complex of C is an abstract simplicial complex containing the sub-families of C with a non-empty intersection. Enumeration The number of abstract simplicial complexes on up to n labeled elements (that is on a set S of size n) is one less than the nth Dedekind number. These numbers grow very rapidly, and are known only for ; the Dedekind numbers are (starting with n = 0): 1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787 . This corresponds to the number of non-empty antichains of subsets of an set. The number of abstract simplicial complexes whose vertices are exactly n labeled elements is given by the sequence "1, 2, 9, 114, 6894, 7785062, 2414627396434, 56130437209370320359966" , starting at n = 1. This corresponds to the number of antichain covers of a labeled n-set; there is a clear bijection between antichain covers of an n-set and simplicial complexes on n elements described in terms of their maximal faces. The number of abstract simplicial complexes on exactly n unlabeled elements is given by the sequence "1, 2, 5, 20, 180, 16143" , starting at n = 1. Computational problems The simplicial complex recognition problem is: given a finite ASC, decide whether its geometric realization is homeomorphic to a given geometric object. This problem is undecidable for any d-dimensional manifolds for d ≥ 5. Relation to other concepts An abstract simplicial complex with an additional property called the augmentation property or the exchange property yields a matroid. The following expression