The Topological Pigeonhole Principle for Ordinals

Hilton, Jacob

doi:10.1017/jsl.2015.45

Cited by 5 publications

(13 citation statements)

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“…This is presented in Theorem 3.2. (The corresponding result for P top , building on [Bau86, Theorem 2.3], is the main theorem of [Hil16].) We also prove a simple lower bound for Ramsey numbers in terms of pigeonhole numbers, which remains our best lower bound with the exception of a couple of special cases.…”

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confidence: 83%

“…. , k}, there is a closed copy of ω βi + 1 in color i if and only if i = r, and no closed copy of any ordinal larger than ω βr + 1 in color r. To obtain such a coloring, simply extend the coloring given in [Hil16,Theorem 4.5] by setting c r (ω β1#β2#···#β k ) = r. This equation allows P cl (α 1 , α 2 , . .…”

Section: The Closed Pigeonhole Principle For Ordinalsmentioning

confidence: 99%

“…Analogously to Ramsey numbers, we define the pigeonhole number P (α i ) i∈κ to be the least ordinal β such that β → (α i ) 1 i∈κ , and we define the topological pigeonhole number P top (α i ) i∈κ and the closed pigeonhole number P cl (α i ) i∈κ in a similar fashion. An algorithm for finding the topological pigeonhole numbers is given in [Hil16], and we will describe the modifications required to obtain the closed pigeonhole numbers (see Section 3). Repeatedly we make use of the classical fact that the indecomposable ordinals, the powers of ω, satisfy P (ω α ) k = ω α for all finite k.…”

Section: Introductionmentioning

confidence: 99%

“…For a general introduction to topological Ramsey theory, see [Wei90]. In a sense, [Bau86] is the direct precursor of this paper and [Hil16]. Both authors were working independently on this topic when the talk on which [Cai14] is based was given.…”

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confidence: 99%

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Topological Ramsey numbers and countable ordinals

Caicedo¹,

Hilton²

2017

Foundations of Mathematics

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, 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.

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confidence: 83%

Section: The Closed Pigeonhole Principle For Ordinalsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Topological Ramsey numbers and countable ordinals

Caicedo¹,

Hilton²

2017

Foundations of Mathematics

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“…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].…”

Section: Introductionmentioning

confidence: 99%