= Exterior-power quotient bound for strong hypergraph saturation
{title2=$|E(H)|\geq\dim(\bigwedge^r\mathbb R^n/\bigwedge^rU)=\binom nr-\binom{n-t}{r}$}
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.
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