The Dedekind number (https://en.wikipedia.org/wiki/Dedekind_number) $M(n)$ is supposed to be the number of abstract simplicial complexes with $n$ elements.
I don't quite understand why $M(0)=2$ and why $M(1)=3$ for that matter.
If $n=0$, then there are no elements, so isn't there just one abstract simplicial complex, namely the emptyset $\emptyset$? Unless they are considering $\{\emptyset\}$, which violates the definition in (https://en.wikipedia.org/wiki/Abstract_simplicial_complex) since the faces are supposed to be non-empty?
Similarly, for $n=1$, the only abstract simplcial complexes I can see are $\{v_1\}$ and $\emptyset$.
Hope it makes sense (I may have made a serious mistake here). Thanks for any help.
Ok, I think I got it, but from another Wikipedia site:
"The number of abstract simplicial complexes on up to $n$ elements is one less than the $n$th Dedekind number."
https://en.wikipedia.org/wiki/Abstract_simplicial_complex