Intersection polynomial ($p_A(x)={\prod }_{\ell \in L}({\sum }_{a\in A}{x}_a-\ell )$)

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 $L\subseteq \{0,\dots ,r-1\}$, 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.