  Given a linear equation $\mathcal{L}$, a set $A$ of integers is
$\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to
$\mathcal{L}$. This notion incorporates many central topics in combinatorial
number theory such as sum-free and progression-free sets. In this paper we
initiate the study of (parameterised) complexity questions involving
$\mathcal{L}$-free sets of integers. The main questions we consider involve
deciding whether a finite set of integers $A$ has an $\mathcal{L}$-free subset
of a given size, and counting all such $\mathcal{L}$-free subsets. We also
raise a number of open problems.
