# Number of free faces given n 0-simplexes

Here is my question: How to construct a simplicial complex with $$n$$ 0-simplex 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?

## 1 Answer

If $$\Delta$$ is the $$(n-1)$$-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 $$(n-1)$$-simplex as the unique maximal coface of $$\tau$$.

• Can you show me how to prove it? We can also split $n$ 0-simplex into several groups and is there any possibility that the total number of free faces is larger than your case? – Sooner Dec 17 '18 at 13:52
• If you glue $n-k$ $k$ -simplices along a common facet, then there are $k(n-k)$ free faces. – Richard Stanley Dec 17 '18 at 14:45
• Hi Stanley, so should this be the maximum number of free faces? Could you please tell me how to prove it? – Sooner Dec 18 '18 at 2:14
• @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. – Richard Stanley Dec 18 '18 at 15:28
• 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 $(n-1)$-simplex $\Delta$ is a free face, giving $2^n-2$ free faces (the minus 2 is to rule out $\tau$ being the emptyset or $\Delta$ itself). Am I interpreting that correctly? – Henry Adams Dec 18 '18 at 19:42