Multilinear Polynomials and a Conjecture of Frankl and Füredi

Frankl and Füredi conjectured that given a family F of subsets of [n] such that 1⩽|E∩F|⩽kfor all distinctEandFin F, we must have |F|⩽∑ki=0 (n−1i), (1981, P. Frankl and Z. Füredi,Collog. Math. Soc. János Bolyai37, 305–320). The first proof of this result was given by G. V. Ramanan in (1997,J. Combin. Ser. A79, 53–67). In this note, we present a proof which is a modification of an approach to this problem by H. Snevily (1994,J. Combin. Ser. A.68, 232–238) and like Snevilyʹs, is based on the technique of N. Alon, L. Babai, and H. Suzuki (1991,J. Combin. Theory Ser. A58, 165–180).