K. Kunen proved that it is relatively consistent with Martin´s Axiom that every $(\omega_1,\mathfrak c)$-gap and every $(\mathfrak c,\mathfrak c)$-gap can be separated in $\wp(\omega)/Fin$. What about $(\kappa,\mathfrak c)$-gaps, with $\omega_1<\kappa<\mathfrak c$?
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2$\begingroup$ In Kunen's model, there are actually no linear pre-gaps of type $(\omega_1, \mathfrak{c})$. This implies there are no linear pre-gaps of type $(\lambda, \mathfrak{c})$ for all regular $\lambda \ge \omega_1$ and so vacuously, all gaps of type $(\lambda, \mathfrack{c})$ ($cf(\lambda) \ge \omega_1$) are separated. $\endgroup$– Not MikeCommented Jul 12, 2018 at 16:21
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$\begingroup$ @NotMike, do you know what happens if the gap is not ordered? $\endgroup$– Claudia CorreaCommented Jul 12, 2018 at 17:20
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1$\begingroup$ Not really.The flexibility of the structure of non-linear gaps in models of $\mathsf{MA}$ is not particularly well understood. In the case of Kunen's model, this structure depends how the $\diamondsuit$-sequence behaves when it fails to yield a satisfactory object. $\endgroup$– Not MikeCommented Jul 12, 2018 at 17:47
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