  We analyse the spatial clustering properties of the RASS1 Bright Sample, an
X-ray flux-limited catalogue of galaxy clusters selected from the southern part
of the $ROSAT$ All-Sky Survey. The two-point correlation function $\xi(r)$ of
the whole sample is well fitted (in an Einstein-de Sitter model) by the
power-law $\xi=(r/r_0)^{-\gamma}$, with $r_0= 21.5^{+3.4}_{-4.4} h^{-1}$ Mpc
and $\gamma=2.11^{+0.53}_{-0.56}$ (95.4 per cent confidence level with one
fitting parameter). We use the RASS1 Bright Sample as a first application of a
theoretical model which aims at predicting the clustering properties of X-ray
clusters in flux-limited surveys for different cosmological scenarios. The
model uses the theoretical and empirical relations between mass, temperature
and X-ray cluster luminosity, and fully accounts for the redshift evolution of
the underlying dark matter clustering and cluster bias factor. The comparison
between observational results and theoretical predictions shows that the
Einstein-de Sitter models display too low a correlation length, while models
with a matter density parameter $\Omega_{\rm 0m}=0.3$ (with or without a
cosmological constant) are successful in reproducing the observed clustering.
The dependence of the correlation length $r_0$ on the X-ray limiting flux and
luminosity of the sample is generally consistent with the predictions of all
our models. Quantitative agreement is however only reached for $\Omega_{\rm 0m}
= 0.3$ models. The model presented here can be reliably applied to future
deeper X-ray cluster surveys: the study of their clustering properties will
provide a useful complementary tool to the traditional cluster abundance
analyses to constrain the cosmological parameters.
