Unavoidable subtrees

Let T k be a family of all k -vertex trees. For T ⊆ T k 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 ⁄ → 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 ⊆ T k , | T | = q } = 2 Θ ( k log q − 1 k ) , where log q − 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.