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Sur un problème de la théorie des relations

Starting from well-quasi-orders (wqos), we motivate step by step the introduction of the complicated notion of better-quasi-order (bqo). We then discuss the equivalence between the two main approaches to defining bqo and state several essential results of bqo theory. After recalling the rôle played by the ideals of a wqo in its bqoness, we give a new presentation of known examples of wqos which fail to be bqo. We also provide new forbidden pattern conditions ensuring that a quasi-order is a better quasi-order. It is the variety of these applications, rather than any depth in the results obtained, that suggests that the theorems may be interesting.

Well-Quasi Orders in Computation, Logic, Language and Reasoning. A Unifying Concept of Proof Theory, Automata Theory, Formal Languages and Descriptive Set Theory

Let κ be a cardinal which is measurable after generically adding ℶκ+ω many Cohen subsets to κ, and let ℚκ = (Q, ≤Q) be the strongly κ-dense linear order of size κ. We prove, for 2 ≤ m < ω, that there is a finite value tm+ such that the set [Q]m of m-tuples from Q can be partitioned into classes 〈Ci: i < tm+}〉 such that for any coloring a class Ci in fewer than κ colors, there is a copy ℚ* of ℚκ such that [ℚ*]m ⋂ Ci is monochromatic. It follows that $ \mathbb{Q}_\kappa \to (\mathbb{Q}_\kappa )_{ < \kappa /t_m^ + }^m $, that is, for any coloring of [ℚκ]m with fewer than κ colors there is a copy Q′ ⊆ Q of ℚκ such that [Q′]m has at most tm+ colors. On the other hand, we show that there are colorings of ℚκ such that if Q′ ⊆ Q is any copy of ℚκ then Ci ⋂ [Q′] ≠ o; for all i < tm+, and hence $ \mathbb{Q}_\kappa \nrightarrow [\mathbb{Q}_\kappa ]_{t_m^ + }^m $.

A partition theorem for a large dense linear order

Let Q_K=(Q, 1 that there is a finite value t_m^+ such that the set of all m-tuples from Q can be divided into t_m^+ many classes, such that whenever any of these classes C is colored with fewer than K many colors, there is a copy Q* of Q_K such that all m-tuples from Q* in C receive the same color. As a consequence we obtain that whenever we color the m-tuples of Q with fewer than K many colors, there is a copy of Q_K all m-tuples from which are colored in at most t_m^+ colors. In other words, the partition relation Q_K -->(Q_K)^m_{ 2 we have t_m^+ > t_m, the m-th tangent number. The paper also contains similar partition results about K-Rado graphs. A consequence of our work and some earlier results of Hajnal and Komjath is that a theorem of Shelah known to follow from a large cardinal assumption in a generic extension, does not follow from any large cardinal assumption on its own.

Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove --- again in ZFC --- that for a large class of cardinals there is no universal linear order (e.g. in every $\aleph_1<\l<2^{\aleph_0}$). In fact, what we show is that if there is a universal linear order at a regular $\l$ and its existence is not a result of a trivial cardinal arithmetical reason, then $\l$ ``resembles'' $\aleph_1$ --- a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the non existence of a universal linear order, we show the non-existence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).

Non-existence of Universal Orders in Many Cardinals

A linear ordering is scattered if it does not contain a copy of the rationals. Hausdorff characterised the class of scattered linear orderings as the least family of linear orderings that includes the class $ \mathcal B$ of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in $ \mathcal B$. More generally, we say that a partial ordering is $ \kappa $-scattered if it does not contain a copy of any $ \kappa $-dense linear ordering. We prove analogues of Hausdorff's result for $ \kappa $-scattered linear orderings, and for $ \kappa $-scattered partial orderings satisfying the finite antichain condition. We also study the $ \mathbb{Q}_\kappa $-scattered partial orderings, where $ \mathbb{Q}_\kappa $ is the saturated linear ordering of cardinality $ \kappa $, and a partial ordering is $ \mathbb{Q}_\kappa $-scattered when it embeds no copy of $ \mathbb{Q}_\kappa $. We classify the $ \mathbb{Q}_\kappa $-scattered partial orderings with the finite antichain condition relative to the $ \mathbb{Q}_\kappa $-scattered linear orderings. We show that in general the property of being a $ \mathbb{Q}_\kappa $-scattered linear ordering is not absolute, and argue that this makes a classification theorem for such orderings hard to achieve without extra set-theoretic assumptions.

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A scattering of orders

This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered” posets that we call κ-scattered. These posets cannot embed any order such that for every two subsets of size < κ, one being strictly less than the other, there is an element in between. If a linear order has this property and has size κ it is unique and called ℚ(κ). Partial orders such that for every a < b the set {x: a < x < b} has size ≥ κ are called weakly κ-dense, and posets that do not have a weakly κ-dense subset are called strongly κ-scattered. We prove that κℌ* includes all strongly κ-scattered FAC posets and is included in the class of all FAC κ-scattered posets. For κ = ℵ0 the notions of scattered and strongly scattered coincide and our hierarchy is exactly aug(ℌ) from the Abraham-Bonnet theorem.

A poset hierarchy

In this paper it is shown that the univerality spectrum of well-founded posets is exactly the same as the spectrum of the class of well-orders. A universality result for a restricted class of well-founded posets under rank and order preserving embeddings is also proved. This is done using a club guessing method generalised by Kojman which demonstrates a surjective homomorphism with subsets of the reals of bounded size ordered by inclusion.

Universality results for well-founded posets

When no single universal model for a set of structures exists at a given cardinal, then one may ask in which models of set theory does there exist a small family which embeds the rest. We show that for ?+-graphs (? regular) omitting cliques of some finite or uncountable cardinality, it is consistent that there are small universal families and 2? > ?+. In particular, we get such a result for triangle-free graphs.

Katherine Thompson

Papers

A scattering of orders

A poset hierarchy

Universality results for well-founded posets

Small universal families for graphs omitting cliques without GCH