  In this paper, we propose a mild condition, named Condition $(**)$, for
collections of sequence of integers and show that for any measure preserving
system the Pinsker $\sigma$-algebra is a characteristic $\sigma$-algebra for
the averages along a collection satisfying Condition $(**)$. We introduce the
notion of $\Delta$-weakly mixing subsets along a collection of sequences of
integers and show that positive topological entropy implies the existence of
$\Delta$-weakly mixing subsets along a collection of "good" sequences. As a
consequence, we show that positive topological entropy implies multi-variant
Li-Yorke chaos along polynomial times of the shift prime numbers.
