= Hypergraph extension property
An $r$-<uniform hypergraph> has the extension property when, for every finite vertex set $S$ and every subfamily $\mathcal A\subseteq\binom S{r-1}$, there is a vertex $v\notin S$ such that $F\cup\{v\}$ is a <hyperedge> exactly for $F\in\mathcal A$, among all $F\in\binom S{r-1}$. 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.
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