What are some important (discrete) theorems involving (abstract) simplicial complexes

Question

I am interested in looking for some (preferably discrete) theorems involving simplicial complexes.

To elaborate, by discrete I mean the opposite of continuous, which are those theorems relating to continuous maps/topological spaces.

Most books I read introduce simplicial complexes as a means to do simplicial homology etc, hence they don't have many theorems devoted to simplicial complexes themselves.

Thanks. Pointing to a reference will be great too.

Answer 1

There are many theorems "devoted to (abstract) simplicial complexes". Here are a few examples:

1. The Kruskal–Katona theorem, characterising $f$-vectors of simplicial complexes.

2. The Alexander duality for simplicial complexes.

3. Stokes Theorem for simplicial complexes.

4. de Rham's Theorem for Simplicial Complexes.

5. Erdős-Ko-Rado Theorem for Simplicial Complexes.