For n ∈ ℕ, let h(n) denote the number of simplicial complexes on n vertices up to homotopy equivalence. Here we prove that \(h(n)\geq 2^{2^{0.02n}}\) when n is large enough. Together with the trivial upper bound of \(2^{2^{n}}\) on the number of labeled simplicial complexes on n vertices this proves a conjecture of Kalai that h(n) is doubly exponential in n.

中文翻译：

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N个顶点上单纯复形同伦类型数的下界

对于n ∈ ℕ，令h ( n ) 表示n个顶点上的单纯复形数，直到同伦等价。这里我们证明当n足够大时\(h(n)\geq 2^{2^{0.02n}}\) 。连同\(2^{2^{n}}\)在n个顶点上标记的单纯复形数量的平凡上界，这证明了 Kalai 的猜想，即h ( n ) 在n中是双指数的。