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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).

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  • $\begingroup$ What does complete mean? $\endgroup$
    – Wlod AA
    Dec 14, 2020 at 20:01
  • $\begingroup$ Does k-uniform mean that each simplex is a face of a k-simplex? $\endgroup$
    – Wlod AA
    Dec 14, 2020 at 20:03
  • $\begingroup$ $k$-uniform -- every hyperedge has $k$ vertices. Complete -- every $(k-1)$-face appears in some facet; alternatively every set of vertices of size $(k-1)$ appears as a subset of some hyperedge (of size $k$). $\endgroup$
    – Craig
    Dec 14, 2020 at 20:21
  • $\begingroup$ Note, the $k=3$ case is possibly related to this recent work: arxiv.org/abs/2010.07191, which is discussed on Gil Kalai's blog. $\endgroup$
    – Craig
    Dec 14, 2020 at 21:44

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