Interpreting Set Theory in Discrete Mathematics: Boolean Relation Theory

Wqo theory = well quasi ordering theory, is a branch of combinatorics which has proved to be a fertile source of deep metamathematical pheneomena. In wqo theory, one freely uses some overtly set theoretic arguments. They are definitely noticeable, although they come fairly early in the interpretation hierarchy. Way before, say, ZFC. A qo (quasi order) is a reflexive transitive relation (A,£). A wqo (well quasi order) is a qo (A,£) such that for all x 1 ,x 2 ,... from A, there exists i < j such that x i £ x j. Highlights of wqo theory: that certain qo's are wqo's. There are many equivalent definitions of wqo. THEROEM 1.1. Let (A,£) be a qo. The following are equivalent. i. (A,£) is a wqo. ii. Every infinite sequence from A has an infinite subsequence which is increasing (£). iii. For all x 1 ,x 2 ,... OE A there exists n such that every term is ≥ at least one of x 1 ,...,x n. iv. Every infinite subset of A has a two element chain.