  This paper is mainly concerned with sets which do not contain four-term
arithmetic progressions, but are still very rich in three term arithmetic
progressions, in the sense that all sufficiently large subsets contain at least
one such progression. We prove that there exists a positive constant $c$ and a
set $A \subset \mathbb F_q^n$ which does not contain a four-term arithmetic
progression, with the property that for every subset $A' \subset A$ with $|A'|
\geq |A|^{1-c}$, $A'$ contains a nontrivial three term arithmetic progression.
We derive this from a more general quantitative Roth-type theorem in random
subsets of $\mathbb{F}_{q}^{n}$, which improves a result of
Kohayakawa-Luczak-R\"odl/Tao-Vu. We also discuss a similar phenomenon over the
integers, where we show that for all $\epsilon >0$, and all sufficiently large
$N \in \mathbb N$, there exists a four-term progression-free set $A$ of size
$N$ with the property that for every subset $A' \subset A$ with $|A'| \gg
\frac{1}{(\log N)^{1-\epsilon}} \cdot N$ contains a nontrivial three term
arithmetic progression. Finally, we include another application of our methods,
showing that for sets in $\mathbb{F}_{q}^{n}$ or $\mathbb{Z}$ the property of
"having nontrivial three-term progressions in all large subsets" is almost
entirely uncorrelated with the property of "having large additive energy".
