  The Frankl conjecture, also known as the union-closed sets conjecture, states
that in any finite non-empty union-closed family, there exists an element in at
least half of the sets. From an optimization point of view, one could instead
prove that $2a$ is an upper bound to the number of sets in a union-closed
family on a ground set of $n$ elements where each element is in at most $a$
sets for all $a,n\in \mathbb{N}^+$. Similarly, one could prove that the minimum
number of sets containing the most frequent element in a (non-empty)
union-closed family with $m$ sets and $n$ elements is at least $\frac{m}{2}$
for any $m,n\in \mathbb{N}^+$. Formulating these problems as integer programs,
we observe that the optimal values we computed do not vary with $n$. We
formalize these observations as conjectures, and show that they are not
equivalent to the Frankl conjecture while still having wide-reaching
implications if proven true. Finally, we prove special cases of the new
conjectures and discuss possible approaches to solve them completely.
