  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.
