  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. Let $f(n,a)$ be the maximum 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 some $a,n\in \mathbb{N}^+$. Proving that $f(n,a)\leq 2a$ for
all $a, n \in \mathbb{N}^+$ is equivalent to proving the Frankl conjecture. By
considering the linear relaxation of the integer programming formulation that
was proposed in New Conjectures for Union-Closed Families by Pulaj, Raymond and
Theis, we prove that $O(a^2)$ is an upper bound for $f(n,a)$. We also provide
different ways that this result could be strengthened. Additionally, we give a
new proof that $f(n,2^{n-1}-1)=2^n-n$.
