Epimorphisms between linear orders

We study the relation on linear orders induced by order preserving surjections. In particular we show that its restriction to countable orders is a bqo.Comment: 15 pages; in version 2 we corrected some typos and rewrote the paragraphs introducing the results of subsection 3.3 (statements and proofs are unchanged

On antichains of spreading models of Banach spaces

We show that for every separable Banach space $X$, either \spw(X) (the set of all spreading models of $X$ generated by weakly-null sequences in $X$, modulo equivalence) is countable, or \spw(X) contains an antichain of the size of the continuum. This answers a question of S. J. Dilworth, E. Odell and B. Sari.Comment: 14 pages, no figures. Canadian Mathematical Bulletin (to appear

Some interesting problems

A ≤W B. (This refers to Wadge reducible.) Answer: The first question was answered by Hjorth [83] who showed that it is independent. 1.2 A subset A ⊂ ω ω is compactly-Γ iff for every compact K ⊂ ω ω we have that A ∩ K is in Γ. Is it consistent relative to ZFC that compactly-Σ 1 1 implies Σ 1 1? (see Miller-Kunen [111], Becker [11]) 1.3 (Miller [111]) Does ∆ 1 1 = compactly- ∆ 1 1 imply Σ 1 1 = compactly-Σ 1 1? 1.4 (Prikry see [62]) Can L ∩ ω ω be a nontrivial Σ 1 1 set? Can there be a nontrivial perfect set of constructible reals? Answer: No, for first question Velickovic-Woodin [192]. question Groszek-Slaman [71]. See also Gitik [67]

On better-quasi-ordering classes of partial orders

We provide a method of constructing better-quasi-orders by generalising a technique for constructing operator algebras that was developed by Pouzet. We then generalise the notion of $\sigma$-scattered to partial orders, and use our method to prove that the class of $\sigma$-scattered partial orders is better-quasi-ordered under embeddability. This generalises theorems of Laver, Corominas and Thomass\'{e} regarding $\sigma$-scattered linear orders and trees, countable forests and N-free partial orders respectively. In particular, a class of countable partial orders is better-quasi-ordered whenever the class of indecomposable subsets of its members satisfies a natural strengthening of better-quasi-order.Comment: v1: 45 pages, 8 figures; v2: 44 pages, 11 figures, minor corrections, fixed typos, new figures and some notational changes to improve clarity; v3: 45 pages, 12 figures, changed the way the paper is structured to improve clarity and provide examples earlier o