This question initially arose out of a question in asymptotic matroid theory. The matroid question has since been answered in a different way, but the extremal set theory question remains unanswered and may be of interest in its own right. 



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> **Question.** Let $\mathcal{F}$ be a set family such that for all distinct sets $A,B,C \in \mathcal{F}$ both $A \cap B \not \subseteq C$ and $C \not \subseteq A \cup B$. Let $f(n)$ be the maximum size of such a set family on ground set $[n]$. What are the best known upper and lower bounds on $f(n)$?
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Here is a simple argument that shows at least that $f(n)$ goes to infinity. Just draw a Venn diagram and make sure that all the cells contain a point. 


I am hoping the extremal combinatorics people already know the answer. For example, there is this nice answer of Sergey Norin to this [MO question](https://mathoverflow.net/questions/59939/identifying-poisoned-wines) on poisoned wines which describes the related concept of *strongly union-free* families (which just means that all pairwise unions are distinct). 


My condition implies that $\mathcal{F}$ is both strongly union-free *and* strongly interection-free, but the converse does not hold. For example, the set family consisting of $\{1,2\}, \{2,3\}$, and $\{3,4\}$ is both strongly union-free and strongly interection-free, but does not satisfy my condition. 



Set family $\mathcal{F}$ such that for all $A,B,C \in \mathcal{F}$ both $A \cap B \not \subseteq C$ and $C \not \subseteq A \cup B $
$f(n)$ grows exponentially in $n$ as may be seen by a simple probabilistic argument: choose $M$ subsets $A\_1,\dots,A\_M$ at random ($M$ to be specified later) independently. Then for given indices $i,j,k$ the probability that $A\_i\cap A\_j\subset A\_k$ is $(7/8)^n$ (there should be no element which does not belong to $A\_k$, but belongs to $A\_i$ and to $A\_j$), and the probability that $A\_k\subset A\_i\cup A\_j$ is the same. So, if $2M(M-1)(M-2)(7/8)^n<1$, with positive probability there is no such an event. This may be further improved by applying Lovasz Local Lemma, since these events are not much dependent (each event depend on $O(M^2)$ other events only).


