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A Lower Bound on the Number of Homotopy Types of Simplicial Complexes on N Vertices
Combinatorica ( IF 1.0 ) Pub Date : 2022-09-21 , DOI: 10.1007/s00493-022-4877-6
Andrew Newman

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:

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\(2^{2^{n}}\)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。

更新日期:2022-09-22
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