  Mean curvature flow of clusters of n-dimensional surfaces in R^{n+k} that
meet in triples at equal angles along smooth edges and higher order junctions
on lower dimensional faces is a natural extension of classical mean curvature
flow. We call such a flow a mean curvature flow with triple edges. We show that
if a smooth mean curvature flow with triple edges is weakly close to a static
union of three n-dimensional unit density half-planes, then it is smoothly
close. Extending the regularity result to a class of integral Brakke flows, we
show that this implies smooth short-time existence of the flow starting from an
initial surface cluster that has triple edges, but no higher order junctions.
