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The link at a fixed vertex $v$ of an $r$-uniform hypergraph has vertex set $V(H)\setminus \{v\}$ and the displayed $(r-1)$-element hyperedges. For a universal countable uniform hypergraph, its link has the hypergraph extension property: include $v$ in the finite set of vertices and prescribe only those $r$-edges containing $v$, completing the remaining prescription arbitrarily. Thus the link is a universal countable $(r-1)$-uniform hypergraph. In particular the link at a fixed vertex in the three-uniform case is the Rado graph. This operation retains a specified vertex; deleting the maximum from every edge is a different operation.

The printed assertion is false when maximal means largest in the natural-number labelling. Both the original PDF and the TeX have this wording. The specified operation gives the complete graph, not the Rado graph.

Fix any two distinct vertices $u,v\in \mathbb{N}$, and choose $M\ge \max \{u,v\}$. Apply the hypergraph extension property of $G_{\mathrm{univ}}^{(3)}$ to $S=\{1,\dots ,M\}$, prescribing that the pair $\{u,v\}$ extend to a hyperedge; prescribe all other pairs in $S$ arbitrarily. The resulting witness $w\notin S$ has $w>M$, so $\{u,v,w\}$ is a hyperedge whose largest vertex is $w$. Deleting that vertex produces the edge $\{u,v\}$ in $G_0$.

Since $u,v$ were arbitrary,The Rado graph has nonedges: its hypergraph extension property supplies a new vertex nonadjacent to any specified existing vertex. Therefore $K_{\mathbb{N}}$ and $G_{\mathrm{univ}}^{(2)}$ cannot be isomorphic graphs. This argument holds for every representative of the universal three-uniform structure labelled by $\mathbb{N}$, so changing its labelling does not repair the assertion. Equivalently, in an independently sampled representative each pair has infinitely many larger vertices that complete it to a triple, almost surely.

A valid related construction is to fix a vertex $a$ and take its link hypergraph: make $\{u,v\}$ an edge exactly when $\{a,u,v\}$ is a hyperedge, on the remaining vertex set. This is indeed the Rado graph. To prove it, take any disjoint finite sets $U,W$ of vertices other than $a$, put $S=\{a\}\cup U\cup W$, and prescribe triples through the pairs $\{a,u\}$ for $u\in U$ and no triples through $\{a,w\}$ for $w\in W$. Complete all other pair prescriptions arbitrarily. The hypergraph extension property produces a new vertex adjacent in the link to all of $U$ and none of $W$. Uniqueness from part (i), for $r=2$, now proves the claimed hypergraph isomorphism for this corrected operation.

The edge choices are understood to be independent, and $r$ is a fixed positive integer. We prove the hypergraph extension property and then show directly that it determines a unique countable hypergraph isomorphism type.

For a finite $S\subseteq \mathbb{N}$, prescribe a family $\mathcal{A}\subseteq \left(\genfrac{}{}{0ex}{}{S}{r-1}\right)$. We want a new vertex $v$ such thatFor any $v\notin S$, the probability of precisely this pattern in the countable random uniform hypergraph isTwo different candidate vertices use disjoint hyperedge decisions: all the other vertices in each tested edge lie in $S$. The successes are therefore independent random variables with the same positive probability $q$. The probability that the first $N$ candidates all fail is $(1-q{)}^N\to 0$. The same argument after discarding any finite initial set of candidates shows that infinitely many witnesses exist almost surely.

There are only countably many finite subsets of $\mathbb{N}$, and finitely many prescriptions for each. A countable union of null events is null. Consequently the random uniform hypergraph satisfies every such extension prescription simultaneously almost surely.

Now let $H$ and $K$ be two countably infinite $r$-uniform hypergraphs with the hypergraph extension property. Enumerate their vertex sets and construct increasing finite partial hypergraph isomorphisms by the back-and-forth method. At a forward step, take the least unused vertex $u$ of $H$. Its incident hyperedges with each $(r-1)$-subset of the current domain specify a pattern on the matched vertices of $K$. The hypergraph extension property supplies an unused vertex $v$ of $K$ with exactly this pattern. Adding $u\mapsto v$ preserves and reflects every hyperedge in the enlarged domain. At the next step, take the least unused vertex of $K$ and perform the identical construction with $H$ and $K$ interchanged.

The union of the finite maps is defined on every vertex and has every vertex in its range. It is a bijection, and each finite $r$-set appears at some finite stage, so all hyperedges and nonedges are preserved. Thus it is a hypergraph isomorphism. Existence of at least one structure with the property follows from the probability-one construction already proved; fix one representative and call it $G_{\mathrm{univ}}^{(r)}$. ThereforeThe hypergraph isomorphism type does not depend on $p$. In the case $r=2$ it is the Rado graph. For $r=1$, the argument says that the infinitely many selected and unselected singleton vertices determine the same one-uniform structure for every $p\in (0,1)$.