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?
 A: 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 $\tau$.
A: A warm-up exercise. Let's consider the homogeneous case of all free-face simplexes $\ s^d\ $ being of the same dimension $\ d.\ $ Then perhaps we want every $\ s^d\ $ being a face of its unique simplex of dimension $\ d+1,\ $ and we want the pair intersections of $\ (d+1)$-dimensional simplexes having dimension always $\ < d,\ $ i.e. having intersections that have at the most $\ d\ $ vertices.

Let's start simple with $\ d=1\ $ (case $\ d=0\ $ is trivial). The first interesting series consists of finite projection planes that have $3$-point straight lines. Let $\ P^d\ $ be an $\ n$-dimensional space as this. With $\ P^d,\ $ we associate the $2$-dimensional complex $\ S(P^d),\ $ where the 2-dimensional simplexes will be all straight lines of $\ P^d.\ $ Of course all subsets $\ s\subseteq P^d\ $ such that $\ |s|\le2\ $ are simplexes too. Then the number of vertices is
$$ V\ =\ 2^{d+1}-1 $$
while the number of free-face simplexex (they are 1-simplexex) is:
$$ F\ :=\ \binom V2. $$
we may also consider finite affine spaces $\ A^d\ $ of dimension $\ d,\ $% such that every straight line has 3 points.
Let the respective 2-dimensional simplicial complex $\ Q\ $ have straight lines as its $2$-dimensional simplexes. Then the number of vertices will be
$$ W\ :=\ 3^d $$
and the number of free-face simplexes will be
$$ G\ :=\ \binom W2. $$
That much for the starters.
A: In the previous answer, I've provided examples of finite simplicial complexes that have an absolutely maximal family of face-free 1-simplexes. Now, let me present the case of free-face 2-simplexes.


Let $\ B\ $ be a finite Boolean algebra, say $\ B:=K^d,\ $ where $\ K\ $ is a $2$-element field. Thus $\ B\ $ admits $K$-vector Abelian addition; and
$$ |B|\ =\ 2^d. $$
Consider $3$-dimensional simplicial complex such that $\ s\subseteq B\ $ is a simplex whenever $\ |s|<4;\ $ and on the top of it, $\ 4$-set $\ \{a\ b\ c\ d\}\subseteq B\ $ is
a simplex $\ \Leftarrow:\Rightarrow\ a+b+c+d=0.$
Then,
Theorem: The set of all face-free $\ 2$-simplexes of $\ B\ $ is equal to:
$$ F\ :=\ \binom B3 $$
hence
$$ |F|\ :=\ \binom {|B|}3 $$
is the cardinality of this set of the free-face $2$ simplexes.
Thank you.
