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Let's recall a Frankl's conjecture.

Consider a finite family of finite sets $\mathcal{F}$, such for every pair of sets $A\in \mathcal{F}$ and $B\in\mathcal{F}$, we have $A\cup B\in\mathcal{F}$(the family $\mathcal{F}$ is said to be union closed). The conjecture says that there is an element $a\in \bigcup_{A\in\mathcal{F}}A$, so that $a\in A$ for at least $\frac{|\mathcal{F}|}{2}$ sets $A$.

Now, let's take a set $U=\{1,2,...,n\}$ and an union closed family $\mathcal{F}'\subseteq\mathcal{P}(U)$, where $\{U,\emptyset\}\cup\mathcal{P}_{n-1}(U)\subseteq\mathcal{F}'$.

Is it known whether this case is solved?

I would like to see a proof of this specific case.

I was trying to figure it out myself by considering an induction in respect to $|U|$, but i failed.

Thank you for any help or hint.

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  • $\begingroup$ What do you mean by $\mathcal{P}_{n-1}(U)$? $\endgroup$
    – JoshuaZ
    Commented Sep 14, 2021 at 19:24
  • $\begingroup$ This is a set of all subsets $A\subset U$, so that $|A|=n-1$ $\endgroup$
    – mkultra
    Commented Sep 14, 2021 at 20:24
  • $\begingroup$ i doubt this case is known $\endgroup$
    – mathworker21
    Commented Sep 15, 2021 at 6:15
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    $\begingroup$ For $n\ge2$ there are $n+2$ sets in this family, $n$ of them contains $1$, and $n\ge (n+2)/2$. So this case is trivial. $\endgroup$
    – juan
    Commented Sep 15, 2021 at 7:00

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