set $S$ of [*${\alpha}$-small*]{} objects such that $S{{}^{\perp}}={\{0\}}$; see Definition \[dwg\](\[idpg\])). Then for any smashing weight structure $w$ on ${\underline{C}}$ there exists a cardinal ${\alpha}'$ such that for any regular ${\beta}\ge {\alpha}'$ the weight structure $w$ is strongly ${\beta}$-well generated in the following sense: the couple $({\underline{C}}_{w\le 0}\cap {\operatorname{Obj}}{\underline{C}}^{\beta}, {\underline{C}}_{w\ge 0}\cap {\operatorname{Obj}}{\underline{C}}^{\beta})$ is a weight structure on the triangulated subcategory ${\underline{C}}^{\beta}$ of ${\underline{C}}$ consisting of ${\beta}$-compact objects (see Definition \[dbecomp\](\[idcomp\])), the class ${\mathcal{P}}={\underline{C}}_{w\le 0}\cap {\operatorname{Obj}}{\underline{C}}^{\beta}$ is essentially small and perfect, and $w=(L,R)$ for $L$ and $R$ being the classes described in Theorem \[textw\] (if these conditions are fulfilled for $w$ and some ${\mathcal{P}}$ then we also say that $w$ is perfectly generated by ${\mathcal{P}}$).

A significant part of the “easier” results of the current paper is stated in terms of [*torsion pairs*]{} (as defined in [@aiya]; cf. Remark \[rgen\](4) below); these essentially generalize both weight structures and $t$-structures.[^14]  This certainly makes the corresponding results more general; note also that (the main subject of) [@bkw] yields an interesting family of examples of torsion pairs that do not come either from weight structures or from $t$-structures. Probably the most interesting result about general torsion pairs proved in this paper is the classification 