= Intersection polynomial
{title2=$p_A(x)=\prod_{\ell\in L}(\sum_{a\in A}x_a-\ell)$}
For a finite <set> $A\subseteq[n]$ and a finite list $L$ of allowed intersection sizes, the displayed <polynomial> satisfies $p_A(\chi_B)=\prod_{\ell\in L}(|A\cap B|-\ell)$ at <characteristic vectors of sets>. For an $r$-<uniform set family> whose pairwise intersection sizes lie in $L\subseteq\{0,\ldots,r-1\}$, these <polynomials> vanish at all other family members and have nonzero values at their own members. Their restrictions are therefore <linearly independent>. <Multilinear reduction on the Boolean cube> and <homogenisation on a uniform layer> then prove the <Ray-Chaudhuri–Wilson theorem> bound.
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