The exact minimum transitive-triple packing conjecture for tournaments

Let $T_n$ range over all tournaments on $n$ vertices. Let $P(T_n)$ be the maximum size of a set of edge-disjoint transitive triples in $T_n$, and define

$$f(n)=\min_{T_n}P(T_n).$$

Exact transitive-triple packing conjecture.

$$f(n)=\left\lceil \frac{n(n-1)}{6}-\frac{n}{3}\right\rceil.$$

The paper notes that this is an upper bound obtained by an explicit construction and conjectures that it is attained for every $n$. The broader problem is to determine the minimum number of edge-disjoint transitive triples guaranteed in an arbitrary tournament; the paper proves a lower bound exceeding $0.13n^2(1+o(1))$, leaving the exact formula open.