Exterior-power quotient bound for strong hypergraph saturation ($|E(H)|\ge \dim ({\bigwedge }^r{\mathbb{R}}^n/{\bigwedge }^{r}U)=\left(\genfrac{}{}{0ex}{}{n}{r}\right)-\left(\genfrac{}{}{0ex}{}{n-t}{r}\right)$)

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.