Colouring versus density in integers and Hales-Jewett cubes
Christian ReiherVojt\v{e}ch R\"odlMarcelo Sales
Christian ReiherVojt\v{e}ch R\"odlMarcelo Sales
Nov 2023
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摘要原文
We construct for every integer $k\geq 3$ and every real $\mu\in(0, \frac{k-1}{k})$ a set of integers $X=X(k, \mu)$ 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 $|Z|\geq \mu |Y|$ that is free of arithmetic progressions of length $k$. This answers a question of Erd\H{o}s, Ne\v{s}et\v{r}il, and the second author. Moreover, we obtain an analogous multidimensional statement and a Hales-Jewett version of this result.
