  A Riemannian n-manifold M has k-dimensional Uryson width bounded by a
constant c >0 if there exists a continuous map f from M to an k-dimensional
polyhedral space P, such that the pullbacks f^{-1}(p) of all points p in P have
diameters bounded by c. We prove that an n-dimensional Riemannian manifold M
with at least n-k eigenvalues of the Ricci curvature bounded below by a
positive constant (n-1)b has k-dimensional Uryson width bounded by a constant c
>0. The constant c depends only on b.
  In particular, it follows that a Riemannian n-manifold M with scalar
curvature S bounded below by a positive constant n (n-1) s has
(n-1)-dimensional Uryson width bounded by a constant c >0 depending only on s.
This result confirms a conjecture of M. Gromov.
