Standard names of two finitary properties of hypergraphs?

Question

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?

Answer 1

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.