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, I can't find any decent references or calculations for $\gamma$.
Note: I do not mean to ask about the Dedekind number which simply counts all possible simplicial complexes on $n$ vertices without regard to homotopy equivalence.
Best Answer
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