Question 123
Given some collection of subsets $A_1, A_2, \dots$ of the set $A = \{1, 2, \dots, n\}$ we say that a subset $B \subset A$ is *sparse* if no $A_i \subset B$. Suppose that for any collection $A_1, A_2, \dots$, with $A_i \subset A$, $|A_i| = 3$, and $|A_i \cap A_j| \le 1$ ($i \neq j$), any sparse set containing 29 elements can be expanded to 30 element set such that after this expansion the new set remains sparse. Find the smallest possible value of $n$.