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 is f (n)-unavoidable. Sumner (see 7]) noted that f (n) 2n ? 2 and conjetured that equality holds. HH aggkvist and Thomason established the upper bounds f (n) 12n and f (n) (4 + o(1))n. Let g(k) be the smallest integer such that every oriented tree of order n with k leaves is (n+g(k))-unavoidable. HH aggkvist and Thomason 1] proved that g(k) 2 512k 3. Havet and Thomass e conjectured that g(k) k ? 1. We study here the special case where the tree is a merging of paths (the union of disjoint paths emerging from a common origin). We prove that a merging of order n of k paths is (n + 3 2 (k 2 ? 3k) + 5)-unavoidable. In particular, a tree with three leaves is (n + 5)-unavoidable, i.e. g(3) 5. By studying trees with few leaves, we then prove that f (n) 38 5 n ? 6.