  Collisions were suggested to potentially play a role in the formation of
massive stars in present day clusters, and have likely been relevant during the
formation of massive stars and intermediate mass black holes within the first
star clusters. In the early Universe, the first stellar clusters were
particularly dense, as fragmentation typically only occurred at densities above
$10^9$cm$^{-3}$, and the radii of the protostars were enhanced due to the
larger accretion rates, suggesting a potentially more relevant role of stellar
collisions. We present here a detailed parameter study to assess how the number
of collisions as well as the mass growth of the most massive object depends on
the properties of the cluster, and we characterize the time evolution with
three effective parameters, the time when most collisions occur, the duration
of the collisions period, as well as the normalization required to obtain the
total number of collisions. We apply our results to typical Population III
(Pop.III) clusters of about $1000$M$_\odot$, finding that a moderate
enhancement of the mass of the most massive star by a factor of a few can be
expected. For more massive Pop.III clusters as expected in the first atomic
cooling halos, we expect a more significant enhancement by a factor of $15-32$.
We therefore conclude that collisions in massive Pop.III clusters were likely
relevant to form the first intermediate mass black holes.
