Given a countable transitive model \(\mathcal M\) of ZFC and uncountable cardinals \(\kappa< \mu= 2^{<\mu}\) regular in \(\mathcal M\), the author constructs a generic extension of \(\mathcal M\) which satisfies Martin’s Axiom below \(\kappa\) together with the identities \(\kappa={\mathbf t}={\mathbf b}\), \(\mu= 2^ \omega\) such that the following holds: For all regular \(\lambda< \kappa\) and every \(\subset^*\)-increasing \(\lambda\)-sequence \(a\), there is a \(\subset^*\)-decreasing \(\kappa\)-sequence \(b\) such that \((a,b)\) is a strong \((\lambda,\kappa)\)-gap. This answers one of P. Nyikos’ questions in “On first countable, countably compact spaces. III” [Open problems in topology (J. van Mill and G. M. Reed (eds.)), 127-161 (1990; Zbl 0718.54001)]. The paper concludes with refinements of Hausdorff’s theorem on \((\omega_ 1,\omega_ 1)\) gaps.