Let $\Delta$ be a simplicial complex, we say that a face $F$ of $\Delta$ is free provided there is a single vertex $v$ not contained in $F$ such that the face determined by the union of $F$ and $v$ is the unique facet (aka maximal face) both containing $F$ and $v$. For instance, the simplicial complex of three vertices given by facets $\{1,2\}$ and $\{3\}$ has two free faces, namely $\{1\}$ and $\{2\}$. My question is the following one: can someone of you provide me an example of a simplicial complex containing a free face (namely, $F$), and such that $F$ has at least three vertices?
In an still ongoing joint work with another colleague, we have defined the notion of free pair; indeed, given a simplicial complex $\Delta$ and a couple of non-empty faces of it (namely, $F_1$ and $F_2$), we say that the pair $(F_1,F_2)$ is free if, on one hand, $F_1\cap F_2=\emptyset$, and, on the other hand, the face of $\Delta$ determined by the union of $F_1$ and $F_2$ is the unique facet containing both $F_1$ and $F_2$. In this way, one can define the set $S$ of all possible free pairs of $\Delta$, and define a partial order on it (in a certain way), so $S$ becomes a poset with this partial order. It turns out that the maximal elements of $S$ are of the form $(F,v)$, where $F$ is a free face and $v$ is as above.
Keeping in mind the previous paragraph, I hope is clear that a response to my question implies that there are examples where $S$ is not only made up by maximal elements.
Thanks in advance for your help.
