Rigid Borel sets and better quasi-order theory

Total Representations

“…Nevertheless, some important information on these structures is already available. For any A ⊆ P (N ), let A k denote the set of k-partitions ν ∈ k N such that ν −1 (i) ∈ A for each i < k. In [EMS87] it was shown that the structure ((∆ 1 1 (N )) k ; ≤) is a well preorder, i.e. it has neither infinite descending chains nor infinite antichains.…”

Thin Maximal Antichains in the Turing Degrees

“…Our proof combines Corollary 2.3 and the argument in [3] which is based on [5]. Assuming V = L, we define a function F on ω 1 × α<ω 1 P(α × ω) as follows: For each α < ω 1 and antichain X ⊆ α × ω with α < ω 1 , we define F (α, X) to be the real z such that there exists a lexicographically least triple (β, E, e 0 ) (where the ordering on the second coordinate is < L ) satisfying the following properties:…”

Well-quasi-ordering infinite graphs with forbidden finite planar minor

“…The following theorem is essentially due to Nash-Williams [13] although it was first enunciated explicitly by Laver [9]. The proof we present here is due to [1] (see [21] for a different pi oof).…”

“…t(/) = (7,P, (Plin : (t, t') G E(T))), g = (Dg', g', g'), 2) It is asked in [1] whether the QO-category of infinite trees (with homeomorphic embeddings as morphisms) satisfies ( 10.1 iii). It follows from (7.1) that it is well-behaved, but the method used probably cannot give more.…”