  The study of intersection problems on families of sets is one of the most
important topics in extremal combinatorics. As we all know, the extremal
problems involving certain intersection constraints are equivalent to that with
the union properties by taking complement of sets. A family of sets is called
$s$-union if the union of any two sets in this family has size at most $s$.
Katona [Acta Math. Hungar. 15 (1964)] provided the maximum size of an $s$-union
family of sets of $[n]$, and he also determined the extremal families up to
isomorphism. Recently, Frankl [J. Combin. Theory Ser. B 122 (2017) 869--876]
sharpened this result by establishing the maximum size of an $s$-union family
that is not a subfamily of the so-called Katona family. In this paper, we
determine the maximum size of an $s$-union family that is neither contained in
the Katona family nor in the Frankl family. Moreover, we characterize all
extremal families achieving the upper bounds.
