  We prove a flatness result for entire nonlocal minimal graphs having some
partial derivatives bounded from either above or below. This result generalizes
fractional versions of classical theorems due to Bernstein and Moser. Our
arguments rely on a general splitting result for blow-downs of nonlocal minimal
graphs.
  Employing similar ideas, we establish that entire nonlocal minimal graphs
bounded on one side by a cone are affine.
  Moreover, we show that entire graphs having constant nonlocal mean curvature
are minimal, thus extending a celebrated result of Chern on classical CMC
graphs.
