Intersection-free uniform set family

An intersection-free uniform set family is a uniform set family $\mathcal{F}$ with no three distinct members $A,B,C$ satisfying $A\cap B\subseteq C$. For distinct members of a rank-$r$ family, $|A\cap B|<r$, so using strict containment gives the same restriction. Fixing one member turns the other members' intersections with it into distinct antichain traces, giving the antichain trace bound for intersection-free families.

For an intersection-free uniform set family of rank $r$, fix $X\in \mathcal{F}$. The map $A\mapsto X\cap A$ on $\mathcal{F}\setminus \{X\}$ is injective and its image is an antichain in $\mathcal{P}(X)$: a containment $X\cap A\subseteq X\cap B$ would violate intersection-freeness on the distinct members $X,A,B$. The Sperner theorem then proves the displayed bound.