Paths with many shortcuts in tournaments

A shortcut of a directed path v₁v₂⋯v_n is an edge v_iv_j with j>i+1. If j=i+2 the shortcut is a hop. If all hops are present, the path is called hop complete so the path and its hops form a square of a path. We prove that every tournament with n≥4 vertices has a Hamiltonian path with at least (4n−10)∕7 hops, and has a hop complete path of order at least n^0.295. A spanning binary tree of a tournament is a spanning shortcut tree if for every vertex of the tree, all its left descendants are in-neighbors and all its right descendants are out-neighbors. It is well-known that every tournament contains a spanning shortcut tree. The number of shortcuts of a shortcut tree is the number of shortcuts of its unique induced Hamiltonian path. Let t(n) denote the largest integer such that every tournament with n vertices has a spanning shortcut tree with at least t(n) shortcuts. We almost determine the asymptotic growth of t(n) as it is proved that [Formula presented].