= Countable random uniform hypergraph
{title2=$G^{(r)}_{\mathbb N,p}$}
Choose each $r$-element subset of the <natural numbers> as a <hyperedge> independently with <probability> $p$, where $0<p<1$. This defines a countable random uniform hypergraph. For any finite pattern in the <hypergraph extension property>, an outside vertex realizes it with probability $q=p^{|\mathcal A|}(1-p)^{|\binom S{r-1}\setminus\mathcal A|}>0$. Distinct outside vertices use disjoint edge decisions, so the probability that the first $N$ candidates all fail is $(1-q)^N\to0$. Taking a countable intersection over all finite patterns proves the extension property <almost surely>. The bounds $0<p<1$ are necessary for all patterns to have positive probability.
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