  We show that if a sample of galaxy clusters is complete above some mass
threshold, then hierarchical clustering theories for structure formation
predict its autocorrelation function to be determined purely by the cluster
abundance and by the spectrum of linear density fluctuations. Thus if the shape
of the initial fluctuation spectrum is known, its amplitude $\sigma_8$ can be
estimated directly from the correlation length of a cluster sample in a way
which is independent of the value of $\Omega_0$. If the cluster mass
corresponding to the sample threshold is also known, it provides an independent
estimate of the quantity $\sigma_8\Omega_0^{0.6}$. Thus cluster data should
allow both $\sigma_8$ and $\Omega_0$ to be determined observationally. We
explore these questions using N-body simulations together with a simple but
accurate analytical model based on extensions of Press-Schechter theory.
Applying our results to currently available data we find that if the linear
fluctuation spectrum has a shape similar to that suggested by the APM galaxy
survey, then a correlation length $r_0$ in excess of $20\mpch$ for Abell
clusters would require $\sigma_8>1$, while $r_0<15\mpch$ would require
$\sigma_8<0.5$. With conventional estimates of the relevant mass threshold
these imply $\Omega_0\la 0.3$ and $\Omega_0\ga 1$ respectively.
