High-diversity assemblages are very common in nature, still the factors allowing for the maintenance of biodiversity are 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 a stable and uninvadable (SU) subset of species play a crucial role. Here we analyze a schematic model of community dynamics, mapping the community SUs question to the geometric problem of finding maximal cliques of the corresponding graph and providing numeric and analytic results for the case of random, Erdos-Renyi networks. 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 a long-standing wisdom. Our results rule out the recently suggested ”glass transition” in this system. Instead we show that symmetric systems relax rapidly to an SU, while in the asymmetric case the relaxation time diverges with N, suggesting an excitable dynamics under noise.