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arXiv · 2311.08556

Colouring versus density in integers and Hales-Jewett cubes

Christian Reiher·Vojtěch Rödl·Marcelo Sales
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Abstract

We construct for every integer $k\geq 3$ 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 $|Z|\geq μ|Y|$ 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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Christian Reiher, Vojtěch Rödl, Marcelo Sales. 2024-10-07. Colouring versus density in integers and Hales-Jewett cubes. https://arxiv.org/abs/2311.08556

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