# Some combinatorial properties of splitting trees

@inproceedings{Schilhan2021SomeCP, title={Some combinatorial properties of splitting trees}, author={J. Schilhan}, year={2021} }

We show that splitting forcing does not have the weak Sacks property below any condition, answering a question of Laguzzi, Mildenberger and Stuber-Rousselle. We also show how some partition results for splitting trees hold or fail and we determine the value of cardinal invariants after an ω2-length countable support iteration of splitting forcing.

#### References

SHOWING 1-10 OF 17 REFERENCES

Vive la Différence I: Nonisomorphism of Ultrapowers of Countable Models

- Mathematics
- 1992

We show that it is not provable in ZFC that any two countable elementarily equivalent structures have isomorphic ultrapowers relative to some ultrafilter on ω.

Analytic countably splitting families

- Computer Science, Mathematics
- Journal of Symbolic Logic
- 2004

A notion of a splitting tree is defined, by means of which it is proved that every analytic countable splitting family contains a closed countably splitting family. Expand

Splitting squares

- 2007

We show that for every Borel function f:[2ω]n → 2ω there exists a closed countably splitting family A such that f ↾ [A]n omits a perfect set of values in 2ω.

Regularity Properties for Dominating Projective Sets

- Mathematics, Computer Science
- Ann. Pure Appl. Log.
- 1995

It is shown that an inaccessible cardinal is enough to construct a model for projective u -regularity, namely it holds in Solovay's model and that forcing with uniform trees is equivalent to Laver forcing. Expand

Dominating Projective Sets in the Baire Space

- Mathematics, Computer Science
- Ann. Pure Appl. Log.
- 1994

It is shown that every analytic set in the Baire space which is dominating contains the branches of a uniform tree, i.e. a superperfect tree with the property that for every splitnode all the successor splitnodes have the same length, and it is proved that ∑ 1 2 - Kσ -regularity implies ∑ 2 - u - regularity. Expand

Tree forcing and definable maximal independent sets in hypergraphs

- Mathematics
- 2020

We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over $L$, every analytic hypergraph on a Polish space admits a $\mathbf{\Delta}^1_2$… Expand

On a notion of smallness for subsets of the Baire space

- Mathematics
- 1977

Let us call a set A ⊆ ω^ω of functions from ω into ω σ-bounded if there is a countable sequence of functions (α_n: n Є ω)⊆ ω^ω such that every member of A is pointwise dominated by an element of that… Expand

Definable discrete sets with large continuum

- Mathematics
- 2016

Let $\mathcal R$ be a $\Sigma^1_1$ binary relation and call a set $\mathcal R$-discrete iff no two distinct of its elements are $\mathcal R$-related. We show that in the extension of $\mathbf{L}$ by… Expand

Combinatorial Cardinal Characteristics of the Continuum

- Mathematics
- 2010

The combinatorial study of subsets of the set N of natural numbers and of functions from N to N leads to numerous cardinal numbers, uncountable but no larger than the continuum. For example, how many… Expand

The Integers and Topology

- Mathematics
- 1984

Publisher Summary This chapter discusses integers and topology. Role in topology of certain cardinals is associated with ω. This chapter discusses the problems on first countability, convergence, and… Expand