Let $\mathcal{C}$ be an abstract simplicial complex on some finite set $\Omega$. I say that a subset $\Lambda\subset\Omega$ is minimally non-simplicial if it is not a simplex, but all of its subsets are. Write $h(\mathcal{C})$ for the size of the largest minimally non-simplicial subset of $\Omega$.

I need to calculate $h(\mathcal{C})$ for a particular family of abstract simplicial complexes. I'm wondering if this parameter can be seen somehow using algebraic topology -- via some (co)homology calculation or suchlike. Also, if this parameter has been studied somewhere in the literature, then I would be very interested to know about that.

I apologise if my question is vague/ ill-posed/ inappropriate -- I am not at all an expert on abstract simplicial complexes. For those who are interested, this family of complexes arose when studying some properties of permutation groups.