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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?

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