  We consider intermediate Ricci curvatures $Ric_k$ on a closed Riemannian
manifold $M^n$. These interpolate between the Ricci curvature when $k=n-1$ and
the sectional curvature when $k=1$. By establishing a surgery result for
Riemannian metrics with $Ric_k>0$, we show that Gromov's upper Betti number
bound for sectional curvature bounded below fails to hold for $Ric_k>0$ when
$\lfloor n/2 \rfloor+2 \le k \le n-1$. This was previously known only in the
case of positive Ricci curvature.
