Trees in tournaments

A digraph is said to be n-unavoidable, if every tournament of order n, contains it as a subgraph. Let f(n), be the smallest integer such that every oriented tree of order n, is f(n)-unavoidable. Sumner (see [5]) noted that f(n) ≥ 2 n -2 and conjectured that equality holds. Havet and Thomassé [4] established the upper bound f(n), ≤ 7/2 n. Sumnerʹs conjecture is implied by a stronger one due to Havet and Thomassé: g(k) ≤ k − 1, where g(k), is the smallest integer such that every oriented tree of order n, with k, leaves is (n + g(k))-unavoidable. Häggkvist and Thomason [1] proved that g(k), ≤ 2512k3, and Havet [2] proved g(3) ≤ 5. A tree is constructive, if every path from a node to another node is not directed with first and last block (maximal directed subpath) of length at least two and every path from node to a leaf has first block of length at least two. In this paper, we prove that Havet and Thomasséʹs conjecture is true for constructible trees.