  The families $\mathcal F_0,\ldots,\mathcal F_s$ of $k$-element subsets of
$[n]:=\{1,2,\ldots,n\}$ are called cross-union if there is no choice of $F_0\in
\mathcal F_0, \ldots, F_s\in \mathcal F_s$ such that $F_0\cup\ldots\cup
F_s=[n]$. A natural generalization of the celebrated Erd\H{o}s--Ko--Rado
theorem, due to Frankl and Tokushige, states that for $n\le (s+1)k$ the
geometric mean of $\lvert \mathcal F_i\rvert$ is at most $\binom{n-1}{k}$.
Frankl conjectured that the same should hold for the arithmetic mean under some
mild conditions. 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 $\mathcal F_0=\ldots=\mathcal F_s={[n-1]\choose
k}$.
