The topological Baumgartner-Hajnal theorem

Abstract

Two new topological partition relations are proved. These are ω 1 → (top α + 1) 2 k and R → (top α + 1) 2 k for all α < ω 1 and all k < ω. Here the prefix "top" means that the homogeneous set α + 1 is closed in the order topology. In particular, the latter relation says that if the pairs of real numbers are partitioned into a finite number of classes, there is a homogeneous (all pairs in the same class), well-ordered subset of arbitrarily large countable order type which is closed in the usual topology of the reals. These relations confirm conjectures of Richard Laver and William Weiss, respectively. They are a strengthening of the classical Baumgartner-Hajnal theorem.

Topological Ramsey numbers and countable ordinals

“…Since every stationary subset of ω 1 contains topological copies of every α ∈ ω 1 [Fri74], this therefore shows that ω 1 → top (α, ω + 1) 2 for all countable α. In turn, this result was later extended by Schipperus using elementary submodel techniques to show that ω 1 → top (α) 2 k for all α ∈ ω 1 and all finite k [Sch12] (the topological Baumgartner-Hajnal theorem). Meanwhile, both ω 1 → (ω 1 , α) 2 for all α ∈ ω 1 [Tod83] and ω 1 → (ω 1 , ω + 2) 2 [Haj60] are consistent with ZFC, though the topological version of the former remains unchecked.…”

“…Erdős and Rado [ER56] showed that ω 1 → (ω 1 , ω + 1) 2 . Laver noted in [Lav75] (and a proof, using a pressing down argument, is described for example in [Sch12]) that one actually has ω 1 → (Stationary, top ω + 1) 2 , meaning that one can ensure either a 0-homogeneous stationary subset or a 1-homogeneous topological copy of ω + 1. Since every stationary subset of ω 1 contains topological copies of every α ∈ ω 1 [Fri74], this therefore shows that ω 1 → top (α, ω + 1) 2 for all countable α.…”

Infinite closed monochromatic subsets of a metric space

“…The Topological Baumgartner-Hajnal Theorem, proved by Schipperus [3], provides a stronger result for the case k = 2, X = R. The latter states that if the pairs of real numbers are colored with c colors, there is a monochromatic, well-ordered subset of arbitrarily large countable order type which is closed in the usual topology of R. Applying the Topological Baumgartner-Hajnal Theorem to the special case where the order type is ω + 1, provides Theorem 2.1 of this note for k = 2, X = R. For k = 3 however, we show that the result in this note cannot be strengthened in some sense.…”

Stationary and Closed Rainbow subsets

“…In this paper, we are concerned with structural strengthenings of the usual rainbow Ramsey partition relations, loosely motivated by the study of topological partition relations, see [13] for example.…”