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1. Let ''V'' be a finite set of [[cardinality]] {{math|''n'' + 1}}. The '''combinatorial ''n''-simplex''' with vertex-set ''V'' is an ASC whose faces are all nonempty subsets of ''V'' (i.e., it is the [[power set]] of ''V''). If {{math|''V'' {{=}} ''S'' {{=}} {0, 1, ..., ''n''},}} then this ASC is called the '''standard combinatorial ''n''-simplex'''.
2. Let ''G'' be an undirected graph. The '''[[clique complex]]''' '''of ''G''''' is an ASC whose faces are all [[Clique (graph theory)|cliques]] (complete subgraphs) of ''G''. The '''independence complex of ''G''''' is an ASC whose faces are all [[Independent set (graph theory)|independent sets]] of ''G'' (it is the clique complex of the [[complement graph]] of G). Clique complexes are the prototypical example of [[flag complex]]es. A '''flag complex''' is a complex ''K'' with the property that every set,
3. Let ''H'' be a [[hypergraph]]. A [[Matching in hypergraphs|matching]] in ''H'' is a set of edges of ''H'', in which every two edges are [[Disjoint sets|disjoint]]. The '''matching complex of ''H''''' is an ASC whose faces are all [[Matching in hypergraphs|matchings]] in ''H''. It is the [[independence complex]] of the [[Line graph of a hypergraph|line graph]] of ''H''.
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