Counting paths in digraphs

Say a digraph is k-free if it has no directed cycles of length at most k, for k ∈ Z⁺. Thomassé conjectured that the number of induced 3-vertex directed paths in a simple 2-free digraph on n vertices is at most (n - 1) n (n + 1) / 15. We present an unpublished result of Bondy proving that there are at most 2 n³ / 25 such paths, and prove that for the class of circular interval digraphs, an upper bound of n³ / 16 holds. We also study the problem of bounding the number of (non-induced) 4-vertex paths in 3-free digraphs. We show an upper bound of 4 n⁴ / 75 using Bondy's result for Thomassé's conjecture.