Strong saturation of a uniform hypergraph ($e\notin E(H)⟹\exists T\supseteq e:T^{(r)}\setminus \{e\}\subseteq E(H)$)

An $r$-uniform hypergraph is strongly saturated with respect to a complete uniform hypergraph of order $r+t$ when every missing edge immediately completes such a clique. Existing cliques are allowed. All $r$-sets meeting a fixed $t$-set give an example: the witness for a missing edge is its union with that fixed set. It has $\left(\genfrac{}{}{0ex}{}{n}{r}\right)-\left(\genfrac{}{}{0ex}{}{n-t}{r}\right)$ edges, and the exterior-power quotient bound for strong hypergraph saturation proves optimality.

Take $U$ to be the kernel of a $t$-row Vandermonde matrix on distinct coordinates. For each $(r+t)$-set $T$, $U$ intersects its coordinate subspace in dimension $r$. Wedge a basis of that intersection: its coefficient on every $r$-subset of $T$ is nonzero, because projection onto that subset is an isomorphism by independence of the complementary $t$ columns. Modulo the exterior power of $U$, this gives a dependence involving every clique edge with nonzero coefficient. Strong saturation expresses every missing-edge vector in the span of present ones. Since all coordinate-edge vectors span the quotient, edge count is at least its dimension.