Colouring versus density in integers and Hales–Jewett cubes

Abstract

We construct for every integer $k⩾3$ 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 $\left|Z\right|⩾\mu \left|Y\right|$ 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.