this paper, a tree T = (V,) is a partial ordering with a minimum element, where V is finite, and the ancestors of any x V are linearly ordered under . The minimum element of T is called the root of T, and is written r(T). A tree is said to be trivial if and only if it has exactly one vertex, which must be its root. V = V(T) represents the set of all vertices of the tree T = (V,). In a tree T, if x < y and for no z is x < z < y, then we say that y is a child of x and x is the parent of y. Every vertex has at most one parent. However, vertices may have zero or more children. We write p(x,T) for the parent of x in T. We use Ch(T) = V(T)\{r(T)} for the set of all children of T. We write T 1 T 2 if and only if i) r(T 1 ) = r(T 2 ); ii) for all x Ch(T 1 ), p(x,T 1 ) = p(x,T 2 ). This is a partial ordering on trees. Note that if T 1 T 2

Abstract. Let κ a regular uncountable cardinal and λ a cardinal> κ, and suppose λ <κ is less than the covering number for category cov(Mκ,κ). Then (a) I + κ κ,λ−→(I + κ,λ, ω + 1)2, (b) I + κ κ,λ−→[I + κ,λ]2 κ + if κ is a limit cardinal, and (c) I + κ κ,λ−→(I + κ,λ)2 if κ is weakly compact. 0.