The authors prove two theorems about totally imperfect subsets of \(2^\omega\). The first theorem deals with the notion of dual Ramsey set introduced by \textit{T. J. Carlson} and \textit{S. G. Simpson} [Adv. Math. 53, 265--290 (1984; Zbl 0564.05005)]. Here it is proved that every strongly meager subset of \(2^{\omega \times \omega}\) is dual Ramsey null. In the second part of the paper the authors introduce the following definition: a \(\sigma\)-ideal \(\mathcal I\) of subsets of \(2^\omega\) has property (\ddag) if every \(X \subseteq 2^{\omega}\) such that for every \(f \in \omega^{\uparrow \omega}\) the set \(\{g \in \omega^{\uparrow \omega} : f \nprec g\} \cap X \in \mathcal I\) belongs to \(\mathcal I\). Then they prove that each of the following three \(\sigma\)-ideals has property (\ddag): perfectly meager sets, universally meager sets, and perfectly meager sets in the transitive sense. As a corollary of the last of these results it is proved, assuming CH, the existence of a perfectly meager set in the transitive sense which is not completely Ramsey null and of a perfectly meager set in the transitive sense which is not an \(l_0\) set (the latter notion is related to Laver forcing).