I was thinking about the following problem: Suppose you have a $k$-uniform hyper graph (simplicial complex of dimension $k$) with a complete $(k-1)$-skeleton, and some form of regularity (e.g. every vertex appears in $d$ facets, or every $(k-1)$-simplex appears in $m$ facets). Suppose that the smallest cycle under the $\mathbb{Z}/2\mathbb{Z}$ boundary operator on the $k$-facets is of size D. What can we say about the minimum number of vertices? Of facets?
This is a generalization of Moore graphs in a different direction than Moore geometries. There a cycle is a closed path from vertex-to-facet-to-vertex; here it is a homological cycle (a more geometric interpretation, IMO).