Unavoidable subtrees

Let Tk be a family of all k-vertex trees. For T ⊆ Tk and a tree T , we write T → T if T contains at least one of the trees from T as a subtree, we write T 6→ T otherwise. Let ex(T ) be the smallest integer n, if such exists, such that for any tree T on at least n vertices T → T . It is shown that min{ex(T ) : T ⊆ Tk, |T | = q} = 2 log q−1 , where logq−1 is the q− 1 times iterated logarithm. In addition, the bounds on ex(T ) for families T with a given number of spiders are given.