Corpus ID: 235421986

# 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
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
• O. Spinas
• 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
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
• O. Spinas
• 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
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
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 thatExpand
Definable discrete sets with large continuum
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 $\mathbf{L}$ byExpand
Combinatorial Cardinal Characteristics of the Continuum
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 manyExpand
The Integers and Topology
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, andExpand