  We prove a scale-invariant, semi-global existence result and a trapped
surface formation result in the context of coupled Einstein-Yang-Mills theory,
without symmetry assumptions. More precisely, we prove a scale-invariant
semi-global existence theorem from past null-infinity and show that the
focusing of the gravitational and/or chromoelectric-chromomagnetic waves could
lead to the formation of a trapped surface. Adopting the signature for decay
rates approach introduced in \cite{A19}, we develop a novel gauge (and scale)
invariant hierarchy of non-linear estimates for the Yang-Mills curvature which,
together with the estimates for the gravitational degrees of freedom, yields
the desired semi-global existence result. Once semi-global existence has been
established, the formation of a trapped surface follows from a standard ODE
argument.
