Question 118
A family of sets $\mathcal{F}$ is called *perfect* if the following condition holds: For every triple of sets $X_1, X_2, X_3 \in \mathcal{F}$, at least one of the sets
$$
(X_1 \setminus X_2) \cap X_3, \quad (X_2 \setminus X_1) \cap X_3
$$
is empty. Show that if $\mathcal{F}$ is a perfect family consisting of some subsets of a given finite set $U$, then $|\mathcal{F}| \le |U| + 1$.
$$
(X_1 \setminus X_2) \cap X_3, \quad (X_2 \setminus X_1) \cap X_3
$$
is empty. Show that if $\mathcal{F}$ is a perfect family consisting of some subsets of a given finite set $U$, then $|\mathcal{F}| \le |U| + 1$.