Here is my question: How to construct a simplicial complex with $n$ 0simplex which has the maximum number of free faces? Is there any research topic about this? And is there any relationship between the number of facets and of free faces?
If $\Delta$ is the $(n1)$simplex (which has $n$ vertices), then $\Delta$ is the largest simplicial complex that one can build on top of $n$ vertices, and also every proper face $\tau$ of $\Delta$ is a free face (according to the definition at https://en.wikipedia.org/wiki/Collapse_(topology)), with the entire $(n1)$simplex as the unique maximal coface of $\tau$.

$\begingroup$ Can you show me how to prove it? We can also split $n$ 0simplex into several groups and is there any possibility that the total number of free faces is larger than your case? $\endgroup$ – Sooner Dec 17 '18 at 13:52

$\begingroup$ If you glue $nk$ $k$ simplices along a common facet, then there are $k(nk)$ free faces. $\endgroup$ – Richard Stanley Dec 17 '18 at 14:45

$\begingroup$ Hi Stanley, so should this be the maximum number of free faces? Could you please tell me how to prove it? $\endgroup$ – Sooner Dec 18 '18 at 2:14

$\begingroup$ @Sooner, I don't know if this is the maximum (when $k=\lfloor n/2\rfloor$). I just wanted to point out that the answer of Henry Adams is incorrect. $\endgroup$ – Richard Stanley Dec 18 '18 at 15:28

$\begingroup$ Hi all, I'm not sure if I've understood the question correctly or not. If one is looking for only the elementary free faces (those free face pairs $\tau\subseteq\sigma$ where the dimension of $\tau$ is one less than the dimension of $\sigma$), then disregard the idea I propose. However, if one is allowing arbitrary free faces (no dimension requirement), then to my understanding every proper face $\tau$ of the $(n1)$simplex $\Delta$ is a free face, giving $2^n2$ free faces (the minus 2 is to rule out $\tau$ being the emptyset or $\Delta$ itself). Am I interpreting that correctly? $\endgroup$ – Henry Adams Dec 18 '18 at 19:42