  We report on finite-sized-induced transitions to synchrony in a population of
phase oscillators coupled via a nonlinear mean field, which microscopically is
equivalent to a hypernetwork organization of interactions. Using a
self-consistent approach and direct numerical simulations, we argue that a
transition to synchrony occurs only for finite-size ensembles, and disappears
in the thermodynamic limit. For all considered setups, that include purely
deterministic oscillators with or without heterogeneity in natural oscillatory
frequencies, and an ensemble of noise-driven identical oscillators, we
establish scaling relations describing the order parameter as a function of the
coupling constant and the system size.
