Question 89
A $k$-set is a set with exactly $k$ elements. For a 6-set $A$ and any collection $\mathcal{F}$ of 4-sets, we say that $A$ is $\mathcal{F}$-good if there are exactly three elements $B_1, B_2, B_3$ in $\mathcal{F}$ that are subsets of $A$, and they furthermore satisfy
$$
(A \setminus B_1) \cup (A \setminus B_2) \cup (A \setminus B_3) = A.
$$
Find all $n \ge 6$ so that there exists a collection $\mathcal{F}$ of 4-subsets of $\{1, 2, \dots, n\}$ such that every 6-set $A \subset \{1, 2, \dots, n\}$ is $\mathcal{F}$-good.
$$
(A \setminus B_1) \cup (A \setminus B_2) \cup (A \setminus B_3) = A.
$$
Find all $n \ge 6$ so that there exists a collection $\mathcal{F}$ of 4-subsets of $\{1, 2, \dots, n\}$ such that every 6-set $A \subset \{1, 2, \dots, n\}$ is $\mathcal{F}$-good.