The notion of a [Formula: see text]-generic sequence of P-points is introduced in this paper. It is proved assuming the Continuum Hypothesis (CH) that for each [Formula: see text], any [Formula: see text]-generic sequence of P-points can be extended to an [Formula: see text]-generic sequence. This shows that the CH implies that there is a chain of P-points of length [Formula: see text] with respect to both Rudin–Keisler and Tukey reducibility. These results answer an old question of Andreas Blass.
We analyze the forcing notion P of finite matrices whose rows consists of isomorphic countable elementary submodels of a given structure of the form H θ . We show that forcing with this poset adds a Kurepa tree T . Moreover, if Pc is a suborder of P containing only continuous matrices, then the Kurepa tree T is almost Souslin, i.e. the level set of any antichain in T is not stationary in ω1.
We investigate the poset P(X)∪{∅}, ⊂ , where P(X) is the set of isomorphic suborders of a countable ultrahomogeneous partial order X. For X different from (resp. equal to) a countable antichain the order types of maximal chains in P(X) ∪ {∅}, ⊂ are characterized as the order types of compact (resp. compact and nowhere dense) sets of reals having the minimum non-isolated.
For a countable ultrahomogeneous graph G = G, ρ let P(G) denote the collection of sets A ⊂ G such that A, ρ ∩ [A] 2 ∼ = G. The order types of maximal chains in the poset P(G) ∪ {∅}, ⊂ are characterized as:(I) the order types of compact sets of reals having the minimum nonisolated, if G is the Rado graph or the Henson graph H n , for some n ≥ 3;(II) the order types of compact nowhere dense sets of reals having the minimum non-isolated, if G is the union of µ disjoint complete graphs of size ν, where µν = ω. 2010 MSC: 05C63, 05C80, 05C60, 06A05, 06A06, 03C50, 03C15.
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