Multilinear polynomials and Frankl-Ray-Chaudhuri-Wilson type intersection theorems

We give a very simple new proof of the celebrated intersection theorem of D. K. Ray-Chaudhuri and R. M. Wilson. The new proof yields a generalization to nonuniform set systems. Let N(n, s, r) = ( n s ) + ( n s− 1 ) + · · ·+ ( n s− r + 1 ) . Generalized Ray-Chaudhuri – Wilson Theorem. Let K = {k1, . . . , kr}, L = {l1, . . . , ls}, and assume ki > s − r for all i. Let F be a family of subsets of an n-element set. Suppose that |F | ∈ K for each F ∈ F ; and |E∩F | ∈ L for each pair of distinct sets E,F ∈ F . Then |F| ≤ N(n, s, r). The proof easily generalizes to equicardinal geometric semilattices. As a particular case we obtain the q-analogue (subspace version) of this result, thus extending a result of P. Frankl and R. L. Graham. – A modular version of the Ray-Chaudhuri – Wilson Theorem was found by P. Frankl and R. M. Wilson. We generalize this result to nonuniform set systems: Generalized Frankl – Wilson Theorem. Let p be a prime and K,L two disjoint subsets of {0, 1, . . . , p − 1}. Let |K| = r, |L| = s, and assume r(s − r + 1) ≤ p − 1 and n ≥ s + kr, where kr is the maximal element of K. Let F be a family of subsets of an n-element set. Suppose that |F | ∈ K + pZ for each F ∈ F ; and |E ∩ F | ∈ L + pZ for each pair of distinct sets E,F ∈ F (where pZ denotes the set of multiples of p). Then |F| ≤ N(n, s, r). Our proofs operate on spaces of multilinear polynomials and borrow ideas from a paper by A. Blokhuis on 2-distance sets.