Exact unprovability results for compound well-quasi-ordered combinatorial classes

In this paper we prove general exact unprovability results that show how a threshold between provability and unprovability of a finite well-quasi-orderedness assertion of a combinatorial class is transformed by the sequence-construction, multiset-construction, cycle-construction and labeled-tree-construction. Provability proofs use the asymptotic pigeonhole principle, unprovability proofs use Weiermann-style compression techniques and results from analytic combinatorics.