Kruskal-Friedman Gap Embedding Theorems over Well-Quasi-Orderings

We investigate new extensions of the Kruskal-Friedman theorems concerning well-quasiordering of finite trees with the gap condition. For two labelled trees s and t we say that s is embedded with gap into t if there is an injection from the vertices of s into t which maps each edge in s to a unique path in t with greater-or-equal labels. We show that finite trees are well-quasi-ordered with respect to the gap embedding when the labels are taken from an arbitrary well-quasi-ordering and each tree path can be partitioned into k ∈ N or less comparable sub-paths. This result generalizes both [Křı́89] and [OT87], and is also optimal in the sense that unbounded partiality over tree paths yields a counter example.