  A class of Keller-Segel-Stokes systems generalizing the prototype \[
  \left\{ \begin{array}{rcl}
  n_t + u\cdot\nabla n &=& \Delta n - \nabla \cdot \Big(n(n+1)^{-\alpha}\nabla
c\Big),
  c_t + u\cdot\nabla c &=& \Delta c-c+n,
  u_t +\nabla P &=& \Delta u + n \nabla \phi + f(x,t), \qquad \nabla\cdot u =0,
  \end{array} \right.
  \qquad \qquad (\star) \] is considered in a bounded domain $\Omega\subset
R^3$, where $\phi$ and $f$ are given sufficiently smooth functions such that
$f$ is bounded in $\Omega\times (0,\infty)$.
  It is shown that under the condition that \[
  \alpha>\frac{1}{3}, \] for all sufficiently regular initial data a
corresponding Neumann-Neumann-Dirichlet initial-boundary value problem
possesses a global bounded classical solution. This extends previous findings
asserting a similar conclusion only under the stronger assumption
$\alpha>\frac{1}{2}$.
  In view of known results on the existence of exploding solutions when
$\alpha<\frac{1}{3}$, this indicates that with regard to the occurrence of
blow-up the criticality of the decay rate $\frac{1}{3}$, as previously found
for the fluid-free counterpart of ($\star$), remains essentially unaffected by
fluid interaction of the type considered here.
