# Constraints for the number of basis sets vs number of sets for a union-closed sets conjecture counterexample

I searched some literature about constraints on the number of basis sets (or $$\cup$$-irreducible sets, also called generators) $$\vert\mathit{J}(\mathcal{F})\vert$$ with respect to the number of sets $$\vert\mathcal{F}\vert$$ of a possible family $$\mathcal{F}$$ counterexample of the union-closed sets conjecture, but I could not find any.

Also, If no bound as a function of the number of sets is known, I would be interested in any constant bound $$\vert\mathit{J}(\mathcal{F})\vert \ge k$$. What I know so far is that $$\vert\mathit{J}(\mathcal{F})\vert \ge 6$$, because according to this $$\vert\mathcal{F}\vert \ge 53$$ and thus $$\vert\mathit{J}(\mathcal{F})\vert \ge \left\lceil{log_2{\vert\mathcal{F}\vert}}\right\rceil \ge \left\lceil{log_2{53}}\right\rceil = 6$$.

Is there anything like that known?