For the minimal counter-example to union closed sets conjecture, we have the lower bound $\mid$$\mathcal{A}$$\mid$ $\geq$ $4q-1$ ($\mathcal{A}$ denotes the minimal counter-example family, $q$ denotes the number of elements in $\cup$$\mathcal{A}$). Is there any better lower bound? Is there any research/development happening towards this direction?

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    $\begingroup$ There is quite a bit of information at the Polymath wiki on Frankl's conjecture. $\endgroup$ – Joseph O'Rourke Feb 6 '18 at 13:25
  • $\begingroup$ Yes indeed, thanks. But, under the section 'Partial results', it states the same inequality '$n$ $\leq$ $4m-2$, assuming $\mathcal{A}$ is separating'. So, of not much help in this particular direction. $\endgroup$ – Sisyphus Feb 6 '18 at 13:41
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    $\begingroup$ Which suggests that as of March 2016, when the page was last updated, no better bound was known. $\endgroup$ – Joseph O'Rourke Feb 6 '18 at 17:26

The 2018 paper A lower bound for the minimal counter-example to Frankl’s conjecture by Ankush Hore improved the bound to: $$\mid\mathcal{A}\mid \geq 4q+1$$

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