Let $T$ be a tournament with $n$ vertices (i.e., between every pair of vertices there exists an edge in exactly one direction.) For any $k$, the vertices $A\_1,A\_2,...,A\_k$ form a **transitive path** if there exists an edge from $A\_i$ to $A\_j$ for all $i<j$. The **number of transitive paths** in $T$ is the sum of the number of transitive paths of each size.


What is the minimum possible number of transitive paths in a tournament $T$ with $n$ vertices?


Any approximations, references, etc. would be appreciated.



Minimum number of transitive paths in tournament
As pointed out in the comments, the established name for what you call a transitive path is "transitive subtournament". Call a transitive subtournament "maximal" if it isn't included in a larger transitive subtournament. [This paper](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2899028&fileId=S0013091500009639) contains exponential upper and lower bounds on the maximum number $M\_n$ of maximal transitive subtournaments that a tournament on $n$ vertices can have. Namely, there is a constant $1.4757 \le \theta \le 1.717$ such that $$M\_n = (\theta+o(1))^n.$$


