A tight constraint on some varying speed of light theories

We argue that if the velocity of electromagnetic waves in vacuum, \cem, is different from the limiting velocity for massive particles, \cst, then, a proton moving with velocity $\text{cem}<v<\text{cst}$ would radiate Cherenkov radiation. Because we have seen protons with energy $\sim {10}^{19}$ eV coming from a distance of order more than 1 Mpc, we can put a constraint on $\Delta := (\cst - \cem)/\cst$:$\Delta < 10^{-27}$.