Complete uniform hypergraph ($K_r^{\ell }$)

The complete $\ell$-uniform hypergraph on $r$ vertices contains every $\ell$-subset of its vertex set. A hypergraph clique here means a restriction to a vertex subset of this form, rather than merely a clique in the pairwise shadow. For $\ell =2$, it is an ordinary complete graph.

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.

The three-uniform tetrahedron is the hypergraph on four vertices with all four possible triple hyperedges. It is the complete uniform hypergraph $K_4^3$, and differs from its geometric realization as a tetrahedron. Its edge pattern encodes four-term arithmetic progressions.