### Model-theoretic properties of ultrafilters built by independent families of functions

by Malliaris and Shelah. [MiSh:997]

J Symbolic Logic, 2014

In this paper, the second of two, we continue our
investigations of model-theoretic properties of ultrafilters:
mu ({D}), the minimum size of a product of an
unbounded sequence of natural numbers modulo {D} ;
mathrm {lcf}(aleph_0, {D}) the lower cofinality
(coinitiality) of aleph_0 modulo {D} ;flexibility, discussed
extensively in the paper; realization of symmetric
cuts; and goodness. We work in ZFC except where noted.
Our main results are as follows. First, we prove that any
ultrafilter {D} which is lambda-flexible
(thus: lambda^+-o.k.) must have mu ({D}) =
2^lambda . Thus, a fortiori, {D} will saturate any
stable theory. This is the strongest possible statement about
the saturation power of flexibility alone, in light of our
proof in the companion paper (I) that consistently, flexibility
does not imply saturation of the random graph. In the remainder
of the paper, we focus on the method of constructing
ultrafilters via families of independent functions.
Our second result is a constraint, that is, a tool for
building ultrafilters which are not flexible. Specifically, we
prove that if, at any point in a construction by independent
functions the cardinality of the range of the remaining
independent family is strictly smaller than the index set, then
essentially no subsequent ultrafilter can be flexible. This is
a useful point of leverage since any ultrafilter which is not
flexible will fail to saturate any non-low theory.
The third and fourth results are ultrafilter constructions.
Third, assuming the existence of a measurable cardinal
kappa (to obtain a kappa-complete ultrafilter), we prove
that on any lambda >= kappa^+ there is a regular ultrafilter
which is flexible but not good. This gives a second proof, of
independent interest, of a question from Dow [1985], complementing
the proof in the companion paper (I). Fourth, assuming the existence
of a weakly compact cardinal kappa, we prove that for aleph_0
< theta = textrm {cf}(theta) < kappa
<= lambda there is a regular ultrafilter {D}
on lambda such that mathrm {lcf}(aleph_0, {D})
= theta but ({N}, <)^lambda / {D} has no
(kappa, kappa)-cuts. This appears counter to model-theoretic
intuition, since it shows some families of cuts in linear order
can be realized without saturating the minimum unstable theory.
We give several extensions of this last result, and show how
to eliminate the large cardinal hypothesis in the case of
asymmetric cuts.

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