  Suppose that $1<c<9/8$. For any $m\geq 1$, there exist infinitely many $n$
such that $$ \{[n^c],\ [(n+1)^c],\ \ldots,\ [(n+k_0)^c]\} $$ contains at least
$m+1$ primes, if $k_0$ is sufficiently large (only depending on $m$).
