  We extend a result of B\`{e}s, Martin, Peris and Shkarin by stating: $B_w$ is
$\mathscr{F}$-weighted backward shift if and only if $(B_w,\dots, B_w^r)$ is
$d$-$\mathscr{F}$, for any $r\in \mathbb{N}$, where $\mathscr{F}$ runs along
some filters containing strictly the family of cofinite sets, which are
frequently used in Ramsey theory. On the other hand, we point out that this
phenomenon does not occur beyond the weighted shift frame by showing a mixing
linear operator $T$ on a Hilbert space such that the tuple $(T, T^2)$ is not
$d$-syndetic. We also investigate the relationship between reiteratively
hypercyclic operators and $d$-$\mathscr{F}$ tuples, for filters $\mathscr{F}$
contained in the family of syndetic sets. Finally, we examine conditions to
impose in order to get reiterative hypercyclicity from syndeticity in the
weighted shift frame.
