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

https://arxiv.org/abs/1804.06787

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

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

https://arxiv.org/abs/1804.06787

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

https://arxiv.org/abs/1804.06787