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A UNIFIED APPROACH TO HINDMAN, RAMSEY, AND VAN DER WAERDEN SPACES

Part of: Fairly general properties Set theory Generalities Real functions Connections with other structures, applications Convergence and divergence of infinite limiting processes Extremal combinatorics

Published online by Cambridge University Press:  12 February 2024

RAFAŁ FILIPÓW Open the ORCID record for RAFAŁ FILIPÓW [Opens in a new window] ,
KRZYSZTOF KOWITZ Open the ORCID record for KRZYSZTOF KOWITZ [Opens in a new window]  and
ADAM KWELA Open the ORCID record for ADAM KWELA [Opens in a new window]
RAFAŁ FILIPÓW
Affiliation:
INSTITUTE OF MATHEMATICS FACULTY OF MATHEMATICS, PHYSICS AND INFORMATICS UNIVERSITY OF GDAŃSK UL. WITA STWOSZA 57, 80-308 GDAŃSK POLAND E-mail: Rafal.Filipow@ug.edu.pl URL: http://mat.ug.edu.pl/~rfilipow
KRZYSZTOF KOWITZ
Affiliation:
INSTITUTE OF MATHEMATICS FACULTY OF MATHEMATICS, PHYSICS AND INFORMATICS UNIVERSITY OF GDAŃSK UL. WITA STWOSZA 57, 80-308 GDAŃSK POLAND E-mail: Krzysztof.Kowitz@ug.edu.pl
ADAM KWELA*
Affiliation:
INSTITUTE OF MATHEMATICS FACULTY OF MATHEMATICS, PHYSICS AND INFORMATICS UNIVERSITY OF GDAŃSK UL. WITA STWOSZA 57, 80-308 GDAŃSK POLAND URL: https://mat.ug.edu.pl/~akwela
*
E-mail: Adam.kwela@ug.edu.pl
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Abstract

For many years, there have been conducting research (e.g., by Bergelson, Furstenberg, Kojman, Kubiś, Shelah, Szeptycki, Weiss) into sequentially compact spaces that are, in a sense, topological counterparts of some combinatorial theorems, for instance, Ramsey’s theorem for coloring graphs, Hindman’s finite sums theorem, and van der Waerden’s arithmetical progressions theorem. These spaces are defined with the aid of different kinds of convergences: IP-convergence, R-convergence, and ordinary convergence.

The first aim of this paper is to present a unified approach to these various types of convergences and spaces. Then, using this unified approach, we prove some general theorems about existence of the considered spaces and show that all results obtained so far in this subject can be derived from our theorems.

The second aim of this paper is to obtain new results about the specific types of these spaces. For instance, we construct a Hausdorff Hindman space that is not an $\mathcal {I}_{1/n}$-space and a Hausdorff differentially compact space that is not Hindman. Moreover, we compare Ramsey spaces with other types of spaces. For instance, we construct a Ramsey space that is not Hindman and a Hindman space that is not Ramsey.

The last aim of this paper is to provide a characterization that shows when there exists a space of one considered type that is not of the other kind. This characterization is expressed in purely combinatorial manner with the aid of the so-called Katětov order that has been extensively examined for many years so far.

This paper may interest the general audience of mathematicians as the results we obtain are on the intersection of topology, combinatorics, set theory, and number theory.

Keywords

Ramsey’s theoremHindman’s theoremvan der Waerden’s theoremsequentially compact spacecompact spaceBolzano–Weierstrass theoremRamsey spaceHindman spacedifferentially compact spaceidealfilteralmost disjoint familiesKatětov order

MSC classification

Primary: 54A20: Convergence in general topology (sequences, filters, limits, convergence spaces, etc.) 05D10: Ramsey theory 03E75: Applications of set theory
Secondary: 03E02: Partition relations 54D30: Compactness 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) 26A03: Foundations 40A35: Ideal and statistical convergence 40A05: Convergence and divergence of series and sequences 03E17: Cardinal characteristics of the continuum 03E15: Descriptive set theory 03E05: Other combinatorial set theory 03E35: Consistency and independence results
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 53
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© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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