On the Size of Higman-Haines Sets

In fact, this statement is a corollary to a more general theorem on well-partiallyordered sets. Here a partially ordered set is called well-partially-ordered, if every non-empty subset has at least one, but no more than a finite number of minimal elements (finite basis property). For instance, the set A∗, where A is a finite alphabet, under the scattered subword relation ≤, i.e., v ≤ w if and only if v = v1 . . . vk and w = w1v1 . . . wkvkwk+1, for some integer k, where vi and wj are in A∗, for 1 ≤ i ≤ k and 1 ≤ j ≤ k+1, is a well-partially-ordered set. Interestingly, the concept of well-partially-orders has been frequently rediscovered, for example, see [4, 5, 6, 7, 8]. Moreover, although Higman’s result appears to be only of theoretical interest, it has some nice applications in formal language theory; see, e.g., [2, 3, 7]. It seems that one of the first applications has been given by Haines in [4, Theorem 3], where it is shown that the set of all scattered subwords, i.e., the Higman-Haines sets