  High-diversity assemblages are very common in nature, and yet the factors
allowing for the maintenance of biodiversity remain obscure. The competitive
exclusion principle and May's complexity-diversity puzzle both suggest that a
community can support only a small number of species, turning the spotlight at
the dynamics of local patches or islands, where stable and uninvadable (SU)
subsets of species play a crucial role. Here we map the community SUs question
to the geometric problem of finding maximal cliques of the corresponding graph.
We solve for the number of SUs as a function of the species richness in the
regional pool, $N$, showing that this growth is subexponential, contrary to
long-standing wisdom. We show that symmetric systems relax rapidly to an SU,
where the system stays until a regime shift takes place. In asymmetric systems
the relaxation time grows much faster with $N$, suggesting an excitable
dynamics under noise.
