Sufficiently dense Kuramoto networks are globally synchronizing.

Consider any network of n identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least μ(n-1) other oscillators. There is a critical value of the connectivity, μc, such that whenever μ>μc, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when μ<μc, there are networks with other stable states. The precise value of the critical connectivity remains unknown, but it has been conjectured to be μc=0.75. In 2020, Lu and Steinerberger proved that μc≤0.7889, and Yoneda, Tatsukawa, and Teramae proved in 2021 that μc>0.6838. This paper proves that μc≤0.75 and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.