  We consider dark sector particles (DSPs) that obtain sizeable interactions
with Standard Model fermions from a new mediator. While these particles can
avoid observation in direct detection experiments, they are strongly
constrained by LHC measurements. We demonstrate that there is an important
complementarity between searches for DSP production and searches for the
mediator itself, in particular bounds on (broad) dijet resonances. This
observation is crucial not only in the case where the DSP is all of the dark
matter but whenever - precisely due to its sizeable interactions with the
visible sector - the DSP annihilates away so efficiently that it only forms a
dark matter subcomponent. To highlight the different roles of DSP direct
detection and LHC monojet and dijet searches, as well as perturbativity
constraints, we first analyse the exemplary case of an axial-vector mediator
and then generalise our results. We find important implications for the
interpretation of LHC dark matter searches in terms of simplified models.
