  A Stokes wave is a traveling free-surface periodic water wave that is
constant in the direction transverse to the direction of propagation. In 1981
McLean discovered via numerical methods that Stokes waves at infinite depth are
unstable with respect to transverse perturbations of the initial data. Even for
a Stokes wave that has very small amplitude $\varepsilon$, we prove rigorously
that transverse perturbations, after linearization, will lead to exponential
growth in time. To observe this instability, extensive calculations are
required all the way up to order $O(\varepsilon^3)$. All previous rigorous
results of this type were merely two-dimensional, in the sense that they only
treated long-wave perturbations in the longitudinal direction. This is the
first rigorous proof of three-dimensional instabilities of Stokes waves.
