CAPS 2014 Problem 6

Let $n \ge 6$ be an integer and $F$ be the system of the $3$ -element subsets of the set $\{1, 2,...,n \}$ satisfying the following condition: for every $1 \le i < j \le n$ there is at least $\lfloor \frac{1}{3} n \rfloor -1$ subsets $A\in F$ such that $i, j \in A$ . Prove that for some integer $m \ge 1$ exist the mutually disjoint subsets $A_1, A_2 , ... , A_m \in F$ also, that $|A_1\cup A_2 \cup ... \cup A_m |\ge n-5$ (Poland)

PS. just in case my translation does not make sense, I leave the original in Slovak, in case someone understands something else

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