= Maximal-vertex deletion in a countable universal hypergraph
Label the vertices of a <universal countable uniform hypergraph> by the <natural numbers> and remove the largest vertex of each <hyperedge>. The resulting $(r-1)$-<uniform hypergraph> is complete. For any prescribed $(r-1)$-set $F$, apply the <hypergraph extension property> to a finite initial segment containing $F$, prescribing that $F$ extend to an edge. The resulting vertex is larger than every member of $F$, so this deletion produces $F$. This supplies a useful example of a projection that destroys the nonedge information of a random structure.
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