Trace of a set family ($T_F(\mathcal{A})$)

For a set family $\mathcal{A}\subseteq \mathcal{P}([n])$ and $F\subseteq [n]$, its trace on $F$ is

If every ground element belongs to at least $t$ members of a family $\mathcal{F}\subseteq \mathcal{P}([n])$, thenApply Shearer's inequality to the characteristic vector of a uniformly random member of $\mathcal{A}$.