  We give a simple classification of the independent $n$-point interaction
vertices for bosonic higher-spin gauge fields in $d$-dimensional Minkowski
space-times. We first give a characterisation of such vertices for large
dimensions, $d \geq 2n - 1$, where one does not have to consider Schouten
identities due to over-antisymmetrisation of space-time indices. When the
dimension is lowered, such identities have to be considered, but their
appearance only leads to equivalences of large-$d$ vertices and does not lead
to new types of vertices. We consider the case of low dimensions, $d<n$, in
detail, where the large number of Schouten identities leads to strong
restrictions on independent vertices. We also comment on the generalisation of
our results to the intermediate case $n \leq d \leq 2n - 2$. In all cases, the
independent vertices are expressed in terms of elementary manifestly
gauge-invariant quantities, suggesting that no deformations of the gauge
transformations are induced.
