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Consider any network of $n$ identical Kuramoto oscillators in which eachoscillator is coupled bidirectionally with unit strength to at least $\mu(n-1)$ other oscillators. There is a critical value of the connectivity,$\mu_c$, such that whenever $\mu>\mu_c$, the system is guaranteed to convergeto the all-in-phase synchronous state for almost all initial conditions, butwhen $\mu<\mu_c$, there are networks with other stable states. The precisevalue of the critical connectivity remains unknown, but it has been conjecturedto be $\mu_c=0.75$. In 2020, Lu and Steinerberger proved that $\mu_c\leq0.7889$, and Yoneda, Tatsukawa, and Teramae proved in 2021 that $\mu_c >0.6838$. In this paper, we prove that $\mu_c\leq 0.75$ and explain why this isthe best upper bound that one can obtain by a purely linear stability analysis.

Sufficiently dense Kuramoto networks are globally synchronizing