= Low-degree evaluation rank on a uniform layer
{title2=$\operatorname{rank}_{\mathbb F_p}M\leq\binom ns$}
Let $0\leq s\leq k\leq n$ and form the integer <matrix> with rows indexed by $S\subseteq[n]$, $|S|\leq s$, columns indexed by $k$-subsets $T$, and entry $M_{S,T}=\mathbf1_{S\subseteq T}$. Its <matrix rank> over any <finite field> $\mathbb F_p$ is at most $\binom ns$. Over the <rational numbers>, the row of a $j$-subset $S$ is $\binom{k-j}{s-j}^{-1}$ times the sum of the rows of all $s$-subsets containing $S$. Thus the rows of degree exactly $s$ span, giving the bound over the <rational numbers>. Every larger square minor therefore has integer <determinant> zero, and remains zero modulo $p$. The <matrix rank> cannot increase under this reduction. This argument avoids dividing by an integer that might vanish in the <finite field>, and gives the sharp uniform-layer step in the <Frankl-Wilson theorem>.
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