  In the gravitational collapse of matter beyond spherical symmetry,
gravitational waves are necessarily present. On the other hand, gravitational
waves can collapse to a black hole even without matter. One might therefore
wonder how the interaction and competition between the matter fields and
gravitational waves affects critical phenomena at the threshold of black hole
formation. As a toy model for this, we study the threshold of black-hole
formation in 4+1 dimensions, where we add a massless minimally coupled scalar
matter field to the gravitational wave ansatz of Biz\'on, Chmaj and Schmidt (in
a nutshell, Bianchi~IX on $S^3\times\text{radius}\times\text{time}$). In order
to find a stable discretisation of the equation governing the gravitational
waves in 4+1 physical dimensions, which has the same principal part as the
spherical wave equation in 9+1 dimensions, we first revisit the problem of
critical spherical scalar field collapse in $n+2$ dimensions with large $n$.
Returning to the main problem, we find numerically that weak gravitational wave
perturbations of the scalar field critical solution decay, while weak scalar
perturbations of the gravitational wave critical solution also decay. A
dynamical systems picture then suggests the existence of a codimension-two
attractor. We find numerical evidence for this attractor by evolving mixed
initial data and fine-tuning both an overall amplitude and the relative
strength of the two fields.
