Colouring finite products
type: Item
instance of: scholarly article
title: Colouring finite products
zbMATH Open document ID: 1499.03037
published in: Periodica Mathematica Hungarica
publication date: 2022-02-22T00:00:00Z
full work available at URL: Todorcevic - 2021 - Colouring finite products.pdf
review text: Let \(X_1,\dots,X_n\) be a finite sequence of countable infinite sets. Then, there is a colouring of the product \(\prod_{i=1}^n X_i\) with \(n!\) colours such that for each family \(\{ Y_i :i<n\}\) of infinite subsets \(Y_i \subseteq X_i\), each of the \(n!\) colours shows up in the subproduct \(\prod_{i=1}^n Y_i\).\par Here, the author investigates this phenomenon in case when some of the sets \(X_i\) are uncountable. He shows:\par Theorem 1.1: Suppose \(X_1,\dots,X_n\) are infinite sets with at least one of them uncountable. Then for each finite colouring of \(\prod_{i=1}^n X_i\) there exists infinite sets \(Y_i \subseteq X_i\) such that \(\prod_{i=1}^n Y_i\) uses no more that \((n-1)!\) colours.\par Theorem 1.2: Let \(\Omega\) denote the set of countable ordinals. Then for each \(n<\omega\) there is a finite colouring of \(\Omega^n\) which has at least \((n-1)!\) colours on any subproduct \(\prod_{i=1}^n Y_i\) with \(Y_i \subseteq \Omega\) infinite.\par The first theorem can be seen as an interpretation of a result of R. Laver. For the proof of the two theorems, the author uses \(\rho\)-functions which he studied in detail in \textit{S. Todorcevic} [Walks on ordinals and their characteristics. Basel: Birkhäuser (2007; Zbl 1148.03004)].
reviewed by: Martin Weese
Mathematics Subject Classification ID: 03E02
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 05D10
zbMATH DE Number: 7479497
zbMATH Keywords: colourings
zbMATH Keywords: products
zbMATH Keywords: \(\rho\)-functions
author: Stevo Todorčević
MaRDI profile type: Publication
OpenAlex ID: W3171510180
cites work: Q5635446
cites work: Walks on ordinals and their characteristics
cites work: Introduction to Ramsey Spaces (AM-174)
cites work: Combinatorial set theory
DOI: 10.1007/S10998-021-00389-8
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description: scientific article; zbMATH DE number 7479497