We continue investigations of reasonable ultrafilters on uncountable cardinals defined in Shelah [8]. We introduce a general scheme of generating a filter on λ from filters on smaller sets and we investigate the combinatorics of objects obtained this way.

We show: There are pairs of universes V1V2 and there is a notion of forcing PV1 such that the change mentioned in the title occurs when going from V1[G] to V2[G] for a P-generic filter G over V2. We use forcing iterations with partial memories. Moreover, we implement highly transitive automorphism groups into the forcing orders.

Suppose that λ=λ <λ ≥ℵ0, and we are considering a theory T. We give a criterion on T which is sufficient for the consistent existence of λ++ universal models of T of size λ+ for models of T of size ≤λ+, and is meaningful when 2λ +>λ++. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories. Having in mind (...) possible applications in analysis, we further observe that for such λ, for any fixed μ>λ+ regular with μ=μλ+, it is consistent that 2λ=μ and there is no normed vector space over ℚ of size <μ which is universal for normed vector spaces over ℚ of dimension λ+ under the notion of embedding h which specifies (a,b) such that ||h(x)/||x∈(a,b) for all x. (shrink)

A strong coloring on a cardinal$\kappa $is a function$f:[\kappa ]^2\to \kappa $such that for every$A\subseteq \kappa $of full size$\kappa $, every color$\unicode{x3b3} <\kappa $is attained by$f\restriction [A]^2$. The symbol$$ \begin{align*} \kappa\nrightarrow[\kappa]^2_{\kappa} \end{align*} $$asserts the existence of a strong coloring on$\kappa $.We introduce the symbol$$ \begin{align*} \kappa\nrightarrow_p[\kappa]^2_{\kappa} \end{align*} $$which asserts the existence of a coloring$f:[\kappa ]^2\to \kappa $which isstrong over a partition$p:[\kappa ]^2\to \theta $. A coloringfis strong overpif for every$A\in [\kappa ]^{\kappa }$there is$i<\theta $so that for every color$\unicode{x3b3} <\kappa $is (...) attained by$f\restriction ([A]^2\cap p^{-1}(i))$.We prove that whenever$\kappa \nrightarrow [\kappa ]^2_{\kappa }$holds, also$\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$holds for an arbitrary finite partitionp. Similarly, arbitrary finitep-s can be added to stronger symbols which hold in any model of ZFC. If$\kappa ^{\theta }=\kappa $, then$\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$and stronger symbols, like$\operatorname {Pr}_1(\kappa,\kappa,\kappa,\chi )_p$or$\operatorname {Pr}_0(\kappa,\kappa,\kappa,\aleph _0)_p$, also hold for an arbitrary partitionpto$\theta $parts.The symbols$$ \begin{gather*} \aleph_1\nrightarrow_p[\aleph_1]^2_{\aleph_1},\;\;\; \aleph_1\nrightarrow_p[\aleph_1\circledast \aleph_1]^2_{\aleph_1},\;\;\; \aleph_0\circledast\aleph_1\nrightarrow_p[1\circledast\aleph_1]^2_{\aleph_1}, \\ \operatorname{Pr}_1(\aleph_1,\aleph_1,\aleph_1,\aleph_0)_p,\;\;\;\text{ and } \;\;\; \operatorname{Pr}_0(\aleph_1,\aleph_1,\aleph_1,\aleph_0)_p \end{gather*} $$hold for an arbitrary countable partitionpunder the Continuum Hypothesis and are independent over ZFC$+ \neg $CH. (shrink)