The notion of a [Formula: see text]-generic sequence of P-points is introduced in this paper. It is proved assuming the Continuum Hypothesis 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.

In this paper we start the analysis of the class, the class of cofinal types of directed sets of cofinality at most ℵ2. We compare elements of using the notion of Tukey reducibility. We isolate some simple cofinal types in, and then proceed to find some of these types which have an immediate successor in the Tukey ordering of.

We show that MAκ implies that each collection of Pc-points of size at most κ which has a Pc-point as an RK upper bound also has a Pc-point as an RK lower bound.

We investigate possible cardinalities of maximal antichains in the poset of copies \,\subseteq \rangle \) of a countable ultrahomogeneous relational structure \. It turns out that if the age of \ has the strong amalgamation property, then, defining a copy of \ to be large iff it has infinite intersection with each orbit of \, the structure \ can be partitioned into countably many large copies, there are almost disjoint families of large copies of size continuum and, hence, there are (...) antichains of size continuum in the poset \\). Finally, we show that the posets of copies of all countable ultrahomogeneous partial orders contain maximal antichains of cardinality continuum and determine which of them contain countable maximal antichains. That holds, in particular, for the generic poset. (shrink)