  We study the effects of repulsive interactions on the critical density for
the Bose-Einstein transition in a homogeneous dilute gas of bosons. First, we
point out that the simple mean field approximation produces no change in the
critical density, or critical temperature, and discuss the inadequacies of
various contradictory results in the literature. Then, both within the
frameworks of Ursell operators and of Green's functions, we derive
self-consistent equations that include the dynamic correlations in the system
and predict the change of the critical density. We argue that the dominant
contribution to this change can be obtained within classical field theory and
show that the lowest order correction introduced by interactions is linear in
the scattering length, $a$, with a positive coefficient. Finally, we calculate
this coefficient within various approximations, and compare with various recent
numerical estimates.
