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?



Inequality in Frankl's conjecture
The 2018 paper [A lower bound for the minimal counter-example to Frankl’s conjecture by Ankush Hore](https://ajc.maths.uq.edu.au/pdf/72/ajc_v72_p350.pdf) improved the bound to:
$$\mid\mathcal{A}\mid \geq 4q+1$$


