Let $mathcal{F}$ be a family of subsets of some finite set $X$, where $left| Xright|geq 5$, with the property that for every $a, b, cin X$ there exists $Finmathcal{F}$ with $Fsubseteq Xbackslash{a, b, c}$.
Must there always exist three sets $F, G, Hinmathcal{F}$, at least two of which are inclusion-maximal in $mathcal{F}$, and three points $f, g, hin X$ satisfying $fin Fbackslash{(Gcup H)}$, $gin Gbackslash{(Fcup H)}$ and $hin Hbackslash{(Fcup G)}$?
By “inclusion-maximal in $mathcal{F}$” I mean an $Minmathcal{F}$ such that for $Ninmathcal{F}$ with $Msubseteq N$ we necessarily have $M=N$.