A Partition Theorem for Scattered Order Types

If φ is a scattered order type, μ a cardinal, then there exists a scattered order type ψ such that ψ → [φ] μ,א0 holds. In this note we prove a Ramsey type statement on scattered order types. A trivial fact on ordinals implies the following statement. If μ is an infinite cardinal, then μ → (μ+)1μ. It is less trivial but still easy to show that if φ is an order type, μ a cardinal then there is some order type ψ that ψ → (φ)1μ holds. One can say that these results show that the classes of ordinals and order types are both Ramsey classes in the natural sense; given a target element and a cardinal for the number of colors, there is another element of the class, which, when colored with the required number of colors, always has a monocolored copy of the target. One can wonder which other classes have similar Ramsey properties. A natural, and well investigated, class in between is the class of scattered order types. For this class, the Ramsey property fails for the following well known and simple reason. There is some scattered order type ψ that for every scattered φ one has φ 6→ [ψ]1ω. See Lemma 1. In this paper we show that this is the most in the negative direction, that is, for every scattered order type φ and cardinal μ there exists a scattered order type ψ such that ψ → [φ]1μ,ω holds. Notation. We use the standard axiomatic set theory notation. If φ, ψ are order types, then φ ≤ ψ denotes that there is an order preserving embedding of φ into ψ, that is, every ordered set of order type ψ has a subset of order type φ. If φ is an order type, then φ denotes the reverse order type, that is, if φ is the order type of (S,<), then φ is the order type of (S,>). ω is the ordinal of the set of natural numbers, (N, <). η is the order type of the set of rational numbers, (Q, <). If φ, ψ are order types, μ is a cardinal, φ → (ψ)1μ denotes the following statement. If (S,<) is an ordered set of order type φ and f : S → μ then for some i < μ the subset f(i) contains a subset of order type ψ. That is, if a Research partially supported by Hungarian National Research Grant T 032455. This research was supported by the Israel Science Foundation. Publication 796.