Complete bipartite graphs have chromatic number 2, but they may have arbitrarily high colouring number col. These graphs belong to G(T) whenever T contains a path of length 3. Hence for such a T the answer to the posed problem is NO.
In the remaining cases T is a star, and then the answer is YES. Suppose namely that G is k-chromatic without an induced K1,t. If a vertex x has degree d then the graph spanned by the neighbours of x is (k-1)-colorable with at most t-1 independent vertices., hence d is at most (t-1)(k-1). Since col(G)-1 is at most the maximum degree, it follows that (t-1)(k-1)+1 is an upper bound for col(G). This gives the existence of fT when T is a star.
If G(T) instead is defined as the smaller class of graphs not containing T as a subgraph at all, then the answer is YES for all trees T. Let T have t edges and let G be a graph not containing T as a subgraph. Then any subgraph of G has a vertex of degree at most t-1 and hence coloring number col(G) at most t (this is easy to see inductively - it is stated explicitely as Korollar 0.5.4 (Corollary 1.5.4) in [R. Diestel, Graphentheorie, Springer 1996; (English version, Graph Theory, Springer 1997)]. Then for fT(x)=x/t the chromatic number of any non-empty graph G in G(T) is at least fT(col(G)).
Hence, both when "induced" is required and "non-induced" is allowed, the answers to the posed Problem 4.3 are really trivial. We ought to have noticed this before putting the problem into the book.
August 12, 1997. B. Toft.