Topological Ramsey numbers and countable ordinals

Abstract

, on the occasion of his birthday.Abstract. We study the topological version of the partition calculus in the setting of countable ordinals. Let α and β be ordinals and let k be a positive integer. We write β →top (α, k) 2 to mean that, for every red-blue coloring of the collection of 2-sized subsets of β, there is either a red-homogeneous set homeomorphic to α or a blue-homogeneous set of size k. The least such β is the topological Ramsey number R top (α, k).We prove a topological version of the Erdős-Milner theorem, namely that R top (α, k) is countable whenever α is countable. More precisely, we prove that R top (ω ω β , k + 1) ≤ ω ω β·k for all countable ordinals β and finite k. Our proof is modeled on a new easy proof of a weak version of the Erdős-Milner theorem that may be of independent interest. We also provide more careful upper bounds for certain small values of α, proving among other results thatOur computations use a variety of techniques, including a topological pigeonhole principle for ordinals, considerations of a tree ordering based on the Cantor normal form of ordinals, and some ultrafilter arguments.

On the closed Ramsey numbers Rcl(ω + n, 3)

“…Next we define an ordering on ordinals which first appeared in [CH17]. Consider the relation < * on the class of ordinals given by β < * α if and only if α = β + ω θ for some θ > CB(β) for all ordinals α, β.…”

“…These notions were later generalized to topological spaces by Baumgartner in [Bau86]. Recently, Caicedo and Hilton continued this study and provided upper bounds for topological and closed Ramsey numbers for various pairs of countable ordinals in [CH17].…”

Calculating the closed ordinal Ramsey number $R^{cl}(ω\cdot 2,3)^2$

“…Caicedo and Hilton proved recently that ω 2 • 3 ≤ R cl (ω • 2, 3) ≤ ω 3 • 100 [CH17, Theorem 8.1] and provided also the upper bound R cl (ω 2 , k) ≤ ω ω for every positive integer k [CH17, Theorem 7.1]. The lower bound R cl (ω 2 , k) ≥ ω k+1 is a consequence of [CH17,Theorem 3.1].…”

“…Topological partition calculus was considered by Baumgartner in [Bau86]. Baumgartner's work was continued in recent papers on topological (closed) ordinal partition relations by Caicedo, Hilton, and Piña [Pn15], [Hil16], [CH17].…”

Calculating the closed ordinal Ramsey number Rcl(ω · 2, 3)

“…The ordinal partition calculus was introduced by Erdős and Rado in [ER56], and topological partition calculus was considered by Baumgartner in [Bau86]. Baumgartner's work was continued in recent papers on topological (closed) ordinal partition relations by Hilton, Caicedo-Hilton, and Piña, see [Hil16], [CH17], and the sequence of works starting with [Pn15]. See also [OAW19] and the author's [Mer19].…”