Fix a number $n$, and define $\gamma(n)$ to be the number of simplicial complexes on $n$ unlabeled vertices up to homotopy equivalence. It is unlikely that an explicit formula exists, but what is known about the growth of $\gamma(n)$ as $n$ increases?


This seems to be a fairly basic generalization of "how many non-isomorphic graphs on $n$ unlabeled vertices?" but while this problem even has an [OEIS entry](http://oeis.org/A000088), I can't find any decent references or calculations for $\gamma$.


**Note**: I do not mean to ask about the [Dedekind number](http://en.wikipedia.org/wiki/Dedekind_number) which simply counts all possible simplicial complexes on $n$ vertices without regard to homotopy equivalence.



How many simplicial complexes on n vertices up to homotopy equivalence?
Andrew Newman just posted a preprint to the arXiv showing that the answer is doubly exponential in $n$.


In particular, he showed that the number of homotopy types of simplicial complexes on $n$ vertices is at least
$$\exp \left( \exp \left( 0.004n \right) \right),$$
for all large enough $n$.


This matches the upper bound from Dedekind numbers, up to the constant $0.004$.


Newman's result depends on showing that this many different torsion subgroups are possible for homology in dimension $d$, where $d \approx \delta n$ for some small constant $\delta > 0$. The existence proof is partly constructive and partly depends on the probabilistic method.


<https://arxiv.org/abs/1804.06787>


