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First, give the definition.

  1. A facet is any simplex in a complex that is not a face of any larger simplex. (aka maximal face)
  2. A simplex $\tau$ is called a free face if it is the face of only one facet in a simplicial complex.

Here is an example. Suppose we have a simplicial complex {{1,2,3}, {3,4}}. {1,2,3} and {3,4} are all facets and {4}, {1,2}, {2,3}, {1,3} are free faces.

Here is my question: is there any research about the maximal number of free faces given an arbitrary simplicial complex which has n 0-simplex, which can be regarded vertices? Is there any relationship between the number of free faces and of the facets for an arbitrary simplicial complex?

Sooner
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  • Your definition of free face is not the usual one. Normally a free face is required to be a codimension one face of a facet (and not contained in any other facets), so ${1}$ and ${2}$ are not free faces in your example. – Eric Wofsey Dec 17 '18 at 15:46
  • Thanks for your mention! Do you have any idea that given n 0-simplex, how to construct a simplicial complex which has the maximum number of free faces? – Sooner Dec 18 '18 at 02:11

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