 Hypergraph extension property

An $r$-uniform hypergraph has the extension property when, for every finite vertex set $S$ and every subfamily $\mathcal{A}\subseteq \left(\genfrac{}{}{0ex}{}{S}{r-1}\right)$, there is a vertex $v\notin S$ such that $F\cup \{v\}$ is a hyperedge exactly for $F\in \mathcal{A}$, among all $F\in \left(\genfrac{}{}{0ex}{}{S}{r-1}\right)$. Every prescribed pattern has infinitely many witnesses: enlarge $S$ by any finite set of witnesses already found and prescribe the original pattern on its old $(r-1)$-subsets, with arbitrary choices on the new ones. For $r=2$, this says that any finite adjacency and nonadjacency prescription has a new realizing vertex.