= Finite product closure of well-quasi-orderings
A finite <Cartesian product> of <well-quasi-orderings> is well-quasi-ordered in the componentwise order. Every infinite sequence in one well-quasi-order has an infinite nondecreasing subsequence: colour pairs by whether they are ordered, apply <Ramsey's theorem>, and exclude an all-bad subsequence. Restrict successively to such a subsequence in each coordinate. The same argument, or the infinite pigeonhole principle, proves closure under finite disjoint unions with no cross-component comparisons.
Back to article page
