Abstract
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.
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Some of this work was conducted at Ohio State University and funded in part by the National Science Foundation grants NSF-DMS #1547357 and #60041693, and the rest was conducted at Technische Universität Berlin funded by Deutsche Forschungsgemeinshaft (DFG, German Research Foundation) Graduiertenkolleg “Facets of Complexity” (GRK 2434)
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Newman, A. A Lower Bound on the Number of Homotopy Types of Simplicial Complexes on N Vertices. Combinatorica 42 (Suppl 2), 1439–1450 (2022). https://doi.org/10.1007/s00493-022-4877-6
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DOI: https://doi.org/10.1007/s00493-022-4877-6