= Universal countable uniform hypergraph
{title2=$G^{(r)}_{\mathrm{univ}}$}
The universal countable uniform hypergraph is the unique, up to <hypergraph isomorphism>, countably infinite $r$-<uniform hypergraph> with the <hypergraph extension property>. To prove uniqueness, enumerate two such structures and construct finite partial <hypergraph isomorphisms>. Alternately add the least unused vertex of the first enumeration to the domain and the least unused vertex of the second to the range. Prescribe its incidences with every $(r-1)$-subset of the already matched vertices and apply the extension property in the other structure. The resulting union is a <bijection> preserving every <hyperedge> and nonedge. Existence follows from the <countable random uniform hypergraph> construction. A construction with only the forward steps embeds every countable $r$-<uniform hypergraph> as an induced subhypergraph, explaining the word universal. Its isomorphism type is independent of $p\in(0,1)$.
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