Louis van Gaal has compiled a list of 20 soccer teams, ordered by how good he thinks they are, but strictly refuses to share the ranking. Alex Ferguson knows the names of the 20 teams, but does not know the ranking. Ferguson may mention three arbitrary teams $A,B,C$ to Van Gaal, and Van Gaal will then choose one of the following two options:
Van Gaal may announce to Ferguson which he thinks is the weakest team of the three ("I think that team $X$ is the weakest team among these three teams").
Van Gaal may announce to Ferguson which he thinks is the strongest team of the three ("I think that team $X$ is the strongest team among these three teams").
Alex Ferguson may do this as many times as he likes.
Problem: Determine the largest integer $n$ such that Ferguson can guarantee to find a sequence $T_1, T_2, \ldots, T_n$ of $n$ teams with the property that he knows that Van Gaal thinks that $T_i$ is better than $T_{i+1}$ for $i=1,\ldots,n$.