• Corpus ID: 246441973

On the bounding, splitting, and distributivity numbers

@inproceedings{Dow2022OnTB,
  title={On the bounding, splitting, and distributivity numbers},
  author={Alan Dow and Saharon Shelah},
  year={2022}
}
The cardinal invariants h, b, s of Ppωq are known to satisfy that ω1 ď h ď mintb, su. We prove that all inequalities can be strict. We also introduce a new upper bound for h and show that it can be less than s. The key method is to utilize finite support matrix iterations of ccc posets following [4]. 

References

SHOWING 1-10 OF 27 REFERENCES
On the cofinality of the splitting number
Mad families, splitting families and large continuum
TLDR
Using a finite support iteration of ccc posets, if μ is a measurable cardinal and μ < κ < λ, then using similar techniques the authors obtain the consistency of .
Adjoining Dominating Functions
Abstract If dominating functions in ω ω are adjoined repeatedly over a model of GCH via a finite-support c.c.c. iteration, then in the resulting generic extension there are no long towers, every
Pseudo P-points and splitting number
TLDR
A model in which the splitting number is large and every ultrafilter has a small subset with no pseudo-intersection is constructed.
Ultrafilters with small generating sets
It is consistent, relative to ZFC, that the minimum number of subsets ofω generating a non-principal ultrafilter is strictly smaller than the dominating number. In fact, these two numbers can be any
Matrix iterations and Cichon’s diagram
TLDR
Using matrix iterations of ccc posets, it is proved that it is consistent with ZFC to assign, at the same time, several arbitrary regular values on the left hand side of Cichon’s diagram.
Souslin forcing
Abstract We define the notion of Souslin forcing, and we prove that some properties are preserved under iteration. We define a weaker form of Martin's axiom, namely , and using the results on Souslin
COHERENT SYSTEMS OF FINITE SUPPORT ITERATIONS
TLDR
A forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets is introduced, which are referred to as 3D-coherent systems.
...
...