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 $\mu (n-1)$otheroscillators\mathrm{.}Thereisacriticalvalueoftheconnectivity,$\mu_c$,suchthatwhenever$\mu>\mu_c$, the system is guaranteed to converge to the all-in-phase synchronous state for almost all initial conditions, but when $\mu <{\mu }_c$, there are networks with other stable states. The precise value of the critical connectivity remains unknown, but it has been conjectured to be ${\mu }_c=0.75$. In 2020, Lu and Steinerberger proved that $\mu_c\leq 0.7889$,andYoneda,Tatsukawa,andTeramaeprovedin2021that$\mu_c > 0.6838$\mathrm{.}Inthispaper,weprovethat$\mu_c\leq 0.75$ and explain why this is the best upper bound that one can obtain by a purely linear stability analysis.