Hypergraph link graph ($L_H(v)=\{xy:vxy\in E(H)\}$)

For a three-uniform hypergraph, the link of a vertex $v$ is the ordinary graph on the remaining vertices in which $xy$ is an edge precisely when $vxy$ is a hyperedge. Its edge count is the hypergraph vertex degree of $v$.

If $xyz$ is a hyperedge and the links of $x,y,z$ on four other vertices contain the three perfect matchings of a $K_4$, then those seven vertices contain a Fano plane. There is a second completion: if the link of $v$ contains three disjoint pairs, and the other six vertices contain all transversal triples through those pairs, take four triples of even parity and the three triples containing $v$. This again gives a Fano plane. These finite templates convert forbidden Fano copies into missing cross-part triples.