$F\subseteq2^{[n]}$ and it has no disjoint elements in it. Show that exists an $F'\supseteq F$, such that $F'$ still has no disjoint elements and $|F'|=2^{n-1}$!
I tried to construct $F'$ of $F$ fixing one element $a\in [n]$ and taking all subsets of $[n]$ that includes $a$, but clearly if I don't have a special element like that, then my reasoning is not working.
Can you please give me a hint on how to do it?
Thanks!