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Now we are writing a paper on minimal covers and minimal vertex-covers in hypergraphs and would like to know if there are any standard names for the following two (dual) properties of a hypergraph $(V,E)$, depending on an integer parameter $n\in\mathbb N$:

Property $1_n$: For any $n$-element family of edges $F\subset E$ the intersection $\bigcap F$ is finite.

Property $2_n$: For any $n$-element subset $F\subset V$ the family of edges $\{e\in E:F\subset e\}$ is finite.

Do you have any idea (or better information) how these properties can (or should) be called?

Remark. The Property $1_1$ says that each edge of the hypergraph is finite and the Property $2_1$ says that the family $E$ of edges is point-finite in $V$. This suggests to call Property $1_n$ $n$-point-finiteness and Property $2_n$ $n$-edge-finiteness. But maybe there is a better (or standard) terminology?

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  • $\begingroup$ These look like variations of compactness conditions on lattices. I recommend (in addition to asking here) talking to order theorists and researchers in general algebra for something similar. My guess is these have been considered before, but likely under something like complexes. Gerhard "Hypergraph Known By Many Names" Paseman, 2018.08.21. $\endgroup$
    – Gerhard Paseman
    Commented Aug 21, 2018 at 21:13

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The problems you are discussing in your paper could just as well (or better) be stated in terms of a bipartite graph, i.e., the vertex-edge incidence graph of your hypergraph. In terms of the bipartite graph, the properties might be called $K_{\aleph_0,n}$-free and $K_{n,\aleph_0}$-free, or $K_{\omega,n}$-free and $K_{n,\omega}$-free, depending on which notation you prefer for infinite cardinal numbers.

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