On Fraisse's Order Type Conjecture

ing from the embeddability relation between order types, define a quasi-order to be a reflexive, transitive relation. Throughout this paper, the letters Q and R will range over quasi-ordered sets and classes. Various quasi-ordered spaces will be defined; in each case we will use the symbol < (perhaps with subscripts) to denote the quasi-order under consideration. If q1, q2 E Q, write ql < q2 to mean ql < q2 but q2 ; q,, and write q, q2 to mean q1 _ q2 and q2 < q1. (All results could be done in terms of partial orderings (quasi-orderings where =_ ) instead of quasi-orderings; we elect not to do this since it would mean continually taking equivalence classes.) Whenever a subset Q1 of Q is defined, we assume that Q1 is quasi-ordered as a subordering of Q. We turn now to the definition of well-quasi-ordering, iving two equivalent formulations. Q is well-quasi-ordered (wqo)df (i) for any sequence i<(,, of members of Q, 9i, j < (o: i < j and qj ? qj, equivalently, (ii) every descending sequence of members of Q is finite, and every antichain of members of Q is finite. Thus, in these terms, the first of the two theorems listed in the introduction reads: the class OR is wqo under the embeddability relation. Well-quasi-orderings were first studied by Higman in [5], where the equivalence of the two definitions (immediate from Ramsey's theorem) was observed. If q E Q, let Qq ={r E Q: q S r}. From part (i) in the definition of wqo (which will be the version of wqo used from now on) we have immediately the following Induction principle for well-quasi-orderings: If a proposition F(Q) is true