  We use a self-consistent modeling of x-ray cluster properties to constrain
cosmological scenarios of structure formation in the case of open cosmological
models. We first show that an unbiased open model can reproduce present day
observations, provided that the density parameter is in the range 0.15-0.4.
Although this estimate is derived in a rather different way, it is very close
to dynamical measurements. We also obtain constraints on the shape of the
initial fluctuation power spectrum: we find that the local spectral index n
falls in the range -1.1 < n < -1.6 on scales between 10 and 30h^{-1}Mpc. We
focus on differences between open models and the Einstein--de Sitter model, and
we exhibit a test which depends only on the cosmological parameter and is
insensitive to the shape of the initial fluctuations power spectrum: we show
that, contrary to what is generally admitted, the redshift distribution of
X-ray clusters selected on the basis of their apparent temperature is almost
independent of the power spectrum and that it is only sensitive to the value of
density of the universe. Since the relation between the virial mass and the
X-ray temperature seems rather robust, we expect very little theoretical
uncertainty in the modeling. This allows us to show that the combined knowledge
of the redshift distribution of X-ray flux selected clusters and that of the
luminosity-temperature correlation at high redshifts offers a new and efficient
way to measure the mean density of the universe. This appears to be a clean
test not suffering of the problems attached to local density estimators. We
have tentatively applied our test to the redshift distribution of clusters
detected within the Einstein medium sensitivity X-ray survey.
