On well-quasi-ordering transfinite sequences

. Let Q be a well-quasi-ordered set, i.e. a set on which a reflexive and transitive relation ≤ is defined and such that, for every infinite sequence q 1 , q 2 ,… of elements of Q , there exist i and j such that i j and q i ≤ q i . A restricted transfinite sequence on Q is a function from a well-ordered set onto a finite subset of Q . If f, g are restricted transfinite sequences on Q with domains A, B respectively and there exists a one-to-one order-preserving mapping μ of A into B such that f (α) ≤ h (μ(α)) for every α ∈ A , we write f ≤ g . It is proved that this rule well-quasi-orders the set of restricted transfinite sequences on Q . The proof uses the following subsidiary theorem, which is a generalization of a classical theorem of Ramsey (4). Let P be the set of positive integers, and A ( I ) denote the set of ascending finite sequences of elements of a subset I of P . If s, t ∈ A ( P ), write s ≺ t if, for some m , the terms of s are the first m terms of t . Let T 1 ,…, T n be disjoint subsets of A ( P ) whose union T does not include two distinct sequences s, t such that s ≺ t . Then there exists an infinite subset I of P such that T ∩ A ( I )is contained in a single T j .