= Injectivity of inclusion between adjacent set layers
{title2=$\operatorname{rank}W_{d,d+1}=\binom nd$}
If $n\geq2d+1$ and the <field characteristic> is zero or greater than $d+1$, the matrix recording inclusion of $d$-sets in $(d+1)$-sets has full row <rank>. Subtracting equations on sets differing in one coordinate reduces the kernel problem to degree $d-1$ on $n-2$ coordinates. <Mathematical induction> then makes all coefficients equal, and the factor $d+1$ forces them to vanish. This gives the auxiliary-polynomial independence used in the uniform <Frankl-Wilson theorem>.
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