Owings-like theorems for infinitely many colours or finite monochromatic sets

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic Pub Date : 2024-07-10 DOI:10.1016/j.apal.2024.103495

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引用次数: 0

Abstract

Inspired by Owings's problem, we investigate whether, for a given an Abelian group G and cardinal numbers $κ, θ$ , every colouring $c : G ⟶ θ$ yields a subset $X \subseteq G$ with $| X | = κ$ such that $X + X$ is monochromatic. (Owings's problem asks this for $G = Z$ , $θ = 2$ and $κ = ℵ_{0}$ ; this is known to be false for the same G and κ but $θ = 3$ .) We completely settle the question for κ and θ both finite (by obtaining sufficient and necessary conditions for a positive answer) and for κ and θ both infinite (with a negative answer). Also, in the case where θ is infinite but κ is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer.

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无限多色或有限单色集的类欧文定理

受欧文斯问题的启发，我们研究了对于给定的阿贝尔群和心数，每一种着色是否都能产生一个子集，而这个子集又是单色的。(欧文斯的问题是针对、和提出这个问题的；众所周知，对于相同的和，这个问题是假的）。对于和都是有限的（通过得到肯定答案的充分必要条件），以及对于和都是无限的（否定答案），我们完全解决了这个问题。此外，在和是无限的，但和是有限的情况下，我们还得到了一些得到否定答案的充分条件，以及一个得到肯定答案的例子。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Annals of Pure and Applied Logic 数学-数学

CiteScore

1.40

自引率

12.50%

发文量

审稿时长

200 days

期刊介绍： The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.