Colouring versus density in integers and Hales–Jewett cubes

IF 1 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2024-10-10 DOI:10.1112/jlms.12987

Christian Reiher, Vojtěch Rödl, Marcelo Sales

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引用次数: 0

Abstract

We construct for every integer $k ⩾ 3$ and every real $μ \in (0, \frac{k - 1}{k})$ a set of integers $X = X (k, μ)$ which, when coloured with finitely many colours, contains a monochromatic $k$ -term arithmetic progression, whilst every finite $Y \subseteq X$ has a subset $Z \subseteq Y$ of size $| Z | ⩾ μ | Y |$ that is free of arithmetic progressions of length $k$ . This answers a question of Erdős, Nešetřil and the second author. Moreover, we obtain an analogous multidimensional statement and a Hales–Jewett version of this result.

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整数和黑尔斯-祖耶特立方体中的着色与密度关系

We construct for every integer k ⩾ 3 $k\geqslant 3$ and every real μ ∈ ( 0 , k − 1 k ) $\mu \in (0, \frac{k-1}{k})$ a set of integers X = X ( k , μ ) $X=X(k, \mu)$ which, when coloured with finitely many colours, contains a monochromatic k $k$ -term arithmetic progression, whilst every finite Y ⊆ X $Y\subseteq X$ has a subset Z ⊆ Y $Z\subseteq Y$ of size | Z | ⩾ μ | Y | $|Z|\geqslant \mu |Y|$ that is free of arithmetic progressions of length k $k$ .这回答了厄尔多斯、奈舍特日尔和第二位作者的一个问题。此外，我们还得到了一个类似的多维声明以及这一结果的黑尔斯-杰伊特版本。

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来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.