A proof of Frankl's conjecture on cross-union families

Abstract

The families F0, …, Fs of k-element subsets of [n]:= {1, 2, …, n} are called cross-union if there is no choice of (Math Presents) such that (Math Presents). A natural generalization of the celebrated Erdős–Ko–Rado theorem, due to Frankl and Tokushige, states that for (Math Presents) the geometric mean of |Fi | is at most (Math Presents). Frankl conjectured that the same should hold for the arithmetic mean under some mild con-ditions. We prove Frankl’s conjecture in a strong form by showing that the unique (up to isomorphism) maximizer for the arithmetic mean of cross-union families is the natural one (Math Presents).