  We construct open sets of degenerate unfoldings of heterodimensional cycles
of any co-index $c>0$ and homoclinic tangencies of arbitrary codimension $c>0$.
These sets are known to be the support of unexpected phenomena in families of
diffeomorphisms, such as the Kolmogorov typical co-existence of infinitely many
attractors. As a prerequisite we also construct robust homoclinic tangencies of
large codimension which cannot be inside a strong partially hyperbolic set.
