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.