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, _{μ c}, such that whenever μ > _{μ c}, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when μ < _{μ c}, there are networks with other stable states. The precise value of the critical connectivity remains unknown, but it has been conjectured to be _{μ c} = 0.75 . In 2020, Lu and Steinerberger proved that _{μ c} ≤ 0.7889 , and Yoneda, Tatsukawa, and Teramae proved in 2021 that _{μ c} ><!-- > --> 0.6838 . This paper proves that _{μ c} ≤ 0.75 and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.