Abstract. We will characterize-under appropriate axiomatic assumptions-when a linear order is minimal with respect to not being a countable union of scattered suborders. We show that, assuming PFA + , the only linear orders which are minimal with respect to not being σ-scattered are either Countryman types or real types. We also outline a plausible approach to demonstrating the relative consistency of: There are no minimal non-σ-scattered linear orders. In the process of establishing these results, we will prove combinatorial characterizations of when a given linear order is σ-scattered and when it contains either a real or Aronszajn type.

We show that for any infinite cardinal κ, every (κ+1)-strategically closed poset is κ +-strategically closed if and only if κ holds. This extends previous results of Velleman, et.al.

We investigate club guessing sequences and filters. We prove that assuming V = L, there exists a strong club guessing sequence on μ if and only if μ is not ineffable for every uncountable regular cardinal μ. We also prove that for every uncountable regular cardinal μ, relative to the existence of a Woodin cardinal above μ, it is consistent that every tail club guessing ideal on μ is precipitous.

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We shall show the consistency of CH+⌝(+) and CH+(+)+there are no club guessing sequences on ω1. We shall also prove that ◊+ does not imply the existence of a strong club guessing sequence on ω1.