Choose $A$ uniformly from $\mathcal A$, and let $X_i=\mathbf1_{\{i\in A\}}$. The <characteristic vector of a set> determines $A$, so
$$
H(X_1,ldots,X_n)=\log|\mathcal A|.
$$
For $F\in\mathcal F$, the projected vector $X_F$ determines the intersection $A\cap F$ and therefore takes values in the <trace of a set family> $T_F(\mathcal A)$. The maximum-entropy bound on a finite set gives
$$
H(X_F)\leq\log|T_F(\mathcal A)|.
$$
Every coordinate belongs to at least $t$ members of $\mathcal F$, so <Shearer's inequality> gives
$$
t\log|\mathcal A|
\leq\sum_{F\in\mathcal F}\log|T_F(\mathcal A)|.
$$
Exponentiation proves the <Shearer trace inequality>
$$
\boxed{|\mathcal A|\leq
\left(\prod_{F\in\mathcal F}|T_F(\mathcal A)|\right)^{1/t}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 161 / 2 / iii / Solution

{title2=$T_F(\mathcal A)$}

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


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 Incoming links: Trace of a set family