  We use direct N-body simulations of gas embedded star clusters to study the
importance of stellar collisions for the formation and mass accretion history
of high-mass stars. Our clusters start in virial equilibrium as a mix of gas
and proto-stars. Proto-stars then accrete matter using different mass accretion
rates and the amount of gas is reduced in the same way as the mass of stars
increases. During the simulations we check for stellar collisions and we
investigate the role of these collisions for the build-up of high-mass stars
and the formation of runaway stars. We find that a significant number of
collisions only occur in clusters with initial half-mass radii r_h < 0.1 pc.
After emerging from their parental gas clouds, such clusters end up too compact
compared to observed young, massive open clusters. In addition, collisions lead
mainly to the formation of a single runaway star instead of the formation of
many high mass stars with a broad mass spectrum. We therefore conclude that
massive stars form mainly by gas accretion, with stellar collisions only
playing a minor role if any at all. Collisions of stars in the pre-main
sequence phase might however contribute to the formation of the most massive
stars in the densest star clusters and possibly to the formation of
intermediate-mass black holes with masses up to a few 100 Msun.
