Introduction
Van der Waerden ideal
Questions
Some cardinal invariants
related to analytic quotients
Jana Flašková
Department of Mathematics
University of West Bohemia in Pilsen
Infinite and finite sets – June 2011, Budapest
References
Introduction
Van der Waerden ideal
Questions
Cardinal invariants of the continuum
Most of the classical cardinal invariants of the continuum are
associated to the quotient algebra P(ω)/ Fin.
References
Introduction
Van der Waerden ideal
Questions
References
Cardinal invariants of the continuum
Most of the classical cardinal invariants of the continuum are
associated to the quotient algebra P(ω)/ Fin.
The pseudointersection number p is the minimal cardinality of a
centered family F ⊆ [ω]ω with no pseudointersection i.e.
¬((∃B ∈ [ω]ω )(∀F ∈ F)B ⊆∗ F )
Introduction
Van der Waerden ideal
Questions
References
Cardinal invariants of the continuum
Most of the classical cardinal invariants of the continuum are
associated to the quotient algebra P(ω)/ Fin.
The pseudointersection number p is the minimal cardinality of a
centered family F ⊆ [ω]ω with no pseudointersection i.e.
¬((∃B ∈ [ω]ω )(∀F ∈ F)B ⊆∗ F )
The almost disjointness number a is the minimal cardinality of
an infinite maximal almost disjoint family A ⊆ [ω]ω .
Introduction
Van der Waerden ideal
Questions
References
Cardinal invariants of the continuum
Most of the classical cardinal invariants of the continuum are
associated to the quotient algebra P(ω)/ Fin.
The pseudointersection number p is the minimal cardinality of a
centered family F ⊆ [ω]ω with no pseudointersection i.e.
¬((∃B ∈ [ω]ω )(∀F ∈ F)B ⊆∗ F )
The almost disjointness number a is the minimal cardinality of
an infinite maximal almost disjoint family A ⊆ [ω]ω .
Fact. ℵ1 ≤ p ≤ a ≤ c
Introduction
Van der Waerden ideal
Questions
Cardinal invariants of analytic quotients
All ideals are tall, analytic and contain Fin.
Given an ideal I on ω we write B ⊆I A if B \ A ∈ I.
Note ⊆Fin = ⊆∗ .
References
Introduction
Van der Waerden ideal
Questions
Cardinal invariants of analytic quotients
All ideals are tall, analytic and contain Fin.
Given an ideal I on ω we write B ⊆I A if B \ A ∈ I.
Note ⊆Fin = ⊆∗ .
A family F ⊆ I + is (I + -)centered if
k ∈ ω, Fi ∈ F, i = 1, . . . , k .
Tk
i=1 Fi
∈ I + for every
References
Introduction
Van der Waerden ideal
Questions
References
Cardinal invariants of analytic quotients
All ideals are tall, analytic and contain Fin.
Given an ideal I on ω we write B ⊆I A if B \ A ∈ I.
Note ⊆Fin = ⊆∗ .
A family F ⊆ I + is (I + -)centered if
k ∈ ω, Fi ∈ F, i = 1, . . . , k .
Tk
i=1 Fi
∈ I + for every
p(I) is the minimal cardinality of an I + -centered family F ⊆ I +
with no pseudointersection in I + i.e.
¬((∃B ∈ I + )(∀F ∈ F)B ⊆I F )
Introduction
Van der Waerden ideal
Questions
Cardinal invariants of analytic quotients
A family A ⊆ I + is I-almost disjoint (I-AD in short)
if A ∩ B ∈ I for every A 6= B, A, B ∈ A.
References
Introduction
Van der Waerden ideal
Questions
Cardinal invariants of analytic quotients
A family A ⊆ I + is I-almost disjoint (I-AD in short)
if A ∩ B ∈ I for every A 6= B, A, B ∈ A.
a(I) is the minimal cardinality of an infinite I-MAD family
A ⊆ I + i.e. (∀B ∈ I + )(∃A ∈ A)B ∩ A ∈ I +
References
Introduction
Van der Waerden ideal
Questions
Cardinal invariants of analytic quotients
A family A ⊆ I + is I-almost disjoint (I-AD in short)
if A ∩ B ∈ I for every A 6= B, A, B ∈ A.
a(I) is the minimal cardinality of an infinite I-MAD family
A ⊆ I + i.e. (∀B ∈ I + )(∃A ∈ A)B ∩ A ∈ I +
Observation. p(I) ≤ a(I) ≤ c
References
Introduction
Van der Waerden ideal
Questions
Cardinal invariants of analytic quotients
A family A ⊆ I + is I-almost disjoint (I-AD in short)
if A ∩ B ∈ I for every A 6= B, A, B ∈ A.
a(I) is the minimal cardinality of an infinite I-MAD family
A ⊆ I + i.e. (∀B ∈ I + )(∃A ∈ A)B ∩ A ∈ I +
Observation. p(I) ≤ a(I) ≤ c
In general, p(I) and a(I) need not be uncountable
References
Introduction
Van der Waerden ideal
Questions
Some known results
Theorem (Farkas, Soukup).
a(Zµ ) = ℵ0 for any tall density zero ideal Zµ .
a(Ih ) ≥ ℵ1 for any tall summable ideal Ih .
References
Introduction
Van der Waerden ideal
Questions
Some known results
Theorem (Farkas, Soukup).
a(Zµ ) = ℵ0 for any tall density zero ideal Zµ .
a(Ih ) ≥ ℵ1 for any tall summable ideal Ih .
Definition (Farkas, Soukup).
Let ā(I) be the cardinality of an uncountable I-MAD family.
References
Introduction
Van der Waerden ideal
Questions
Some known results
Theorem (Farkas, Soukup).
a(Zµ ) = ℵ0 for any tall density zero ideal Zµ .
a(Ih ) ≥ ℵ1 for any tall summable ideal Ih .
Definition (Farkas, Soukup).
Let ā(I) be the cardinality of an uncountable I-MAD family.
Theorem (Farkas, Soukup).
ā(I) ≥ b for any tall analytic P-ideal I.
References
Introduction
Van der Waerden ideal
Questions
Some known results
Theorem (Brendle).
p(I) ≥ p for any tall Fσ ideal.
References
Introduction
Van der Waerden ideal
Questions
Some known results
Theorem (Brendle).
p(I) ≥ p for any tall Fσ ideal.
The idea of the proof.
p(I) can be increased by a σ-centered forcing.
References
Introduction
Van der Waerden ideal
Questions
Some known results
Theorem (Brendle).
p(I) ≥ p for any tall Fσ ideal.
The idea of the proof.
p(I) can be increased by a σ-centered forcing.
Combining results of Farkas, Soukup and Brendle
we obtain for Fσ P-ideals:
p ≤ p(I) ≤ b ≤ a(I)
References
Introduction
Van der Waerden ideal
Questions
References
AP-sets and van der Waerden theorem
Definition.
A set A ⊆ N is called an AP-set if it contains arbitrary long
arithmetic progressions.
Van der Waerden Theorem.
If an AP-set is partitioned into finitely many pieces then at least
one of them is again an AP-set.
Introduction
Van der Waerden ideal
Questions
References
AP-sets and van der Waerden theorem
Definition.
A set A ⊆ N is called an AP-set if it contains arbitrary long
arithmetic progressions.
Van der Waerden Theorem.
If an AP-set is partitioned into finitely many pieces then at least
one of them is again an AP-set.
Sets which are not AP-sets form a proper ideal on N
— van der Waerden ideal denoted by W
Introduction
Van der Waerden ideal
Questions
References
Van der Waerden ideal W
The van der Waerden ideal W is
• a tall ideal — because every infinite A ⊆ N contains an infinite
subset with no arithmetic progressions of length 3
Introduction
Van der Waerden ideal
Questions
References
Van der Waerden ideal W
The van der Waerden ideal W is
• a tall ideal — because every infinite A ⊆ N contains an infinite
subset with no arithmetic progressions of length 3
• Fσ -ideal — because W =
S
n∈N Wn
where
Wn = {A ⊆ N : A contains no a. p. of length n}
Introduction
Van der Waerden ideal
Questions
References
Van der Waerden ideal W
The van der Waerden ideal W is
• a tall ideal — because every infinite A ⊆ N contains an infinite
subset with no arithmetic progressions of length 3
• Fσ -ideal — because W =
S
n∈N Wn
where
Wn = {A ⊆ N : A contains no a. p. of length n}
• not a P-ideal — consider for example the sets
Ak = {2n + k : n ∈ ω} for k ∈ ω
Introduction
Van der Waerden ideal
Questions
Van der Waerden ideal W
Szemerédi Theorem.
W⊆Z
where Z = {A ⊆ N : lim sup
n→∞
|A ∩ n|
= 0}
n
References
Introduction
Van der Waerden ideal
Questions
References
Van der Waerden ideal W
Szemerédi Theorem.
W⊆Z
where Z = {A ⊆ N : lim sup
n→∞
|A ∩ n|
= 0}
n
Erdős Conjecture.
W ⊆ I1/n
where I1/n = {A ⊆ N :
X1
a∈A
a
< ∞}
Introduction
Van der Waerden ideal
Questions
Pseudointersection number of P(ω)/W
Since W is an Fσ ideal, we have p ≤ p(W).
References
Introduction
Van der Waerden ideal
Questions
Pseudointersection number of P(ω)/W
Since W is an Fσ ideal, we have p ≤ p(W).
Proposition 1.
p(W) ≤ p
References
Introduction
Van der Waerden ideal
Questions
Pseudointersection number of P(ω)/W
Since W is an Fσ ideal, we have p ≤ p(W).
Proposition 1.
p(W) ≤ p
The idea of the proof.
There exists a regular embedding of P(ω)/ Fin into P(ω)/W.
References
Introduction
Van der Waerden ideal
Questions
Pseudointersection number of P(ω)/W
Since W is an Fσ ideal, we have p ≤ p(W).
Proposition 1.
p(W) ≤ p
The idea of the proof.
There exists a regular embedding of P(ω)/ Fin into P(ω)/W.
Corollary 2.
p(W) = p
References
Introduction
Van der Waerden ideal
Questions
Almost disjointness number of P(ω)/W
Proposition 3.
a(W) ≤ a
References
Introduction
Van der Waerden ideal
Questions
Almost disjointness number of P(ω)/W
Proposition 3.
a(W) ≤ a
The idea of the proof.
There exists a regular embedding of P(ω)/ Fin into P(ω)/W.
References
Introduction
Van der Waerden ideal
Questions
Almost disjointness number of P(ω)/W
Proposition 3.
a(W) ≤ a
The idea of the proof.
There exists a regular embedding of P(ω)/ Fin into P(ω)/W.
Define f : [ω]ω → W + by
f (A) =
[
[2n , 2n+1 )
n∈A
References
Introduction
Van der Waerden ideal
Questions
More about the almost disjointness number
Theorem (Farkas, Soukup).
a(I) ≥ b for any tall Fσ P-ideal.
References
Introduction
Van der Waerden ideal
Questions
More about the almost disjointness number
Theorem (Farkas, Soukup).
a(I) ≥ b for any tall Fσ P-ideal.
Question A.
Is a(W) ≥ b?
References
Introduction
Van der Waerden ideal
Questions
More about the almost disjointness number
Theorem (Farkas, Soukup).
a(I) ≥ b for any tall Fσ P-ideal.
Question A.
Is a(W) ≥ b?
We have seen ā(Z) ≤ a and a(W) ≤ a hold.
References
Introduction
Van der Waerden ideal
Questions
More about the almost disjointness number
Theorem (Farkas, Soukup).
a(I) ≥ b for any tall Fσ P-ideal.
Question A.
Is a(W) ≥ b?
We have seen ā(Z) ≤ a and a(W) ≤ a hold.
Question B.
Is a(I1/n ) ≤ a?
References
Introduction
Van der Waerden ideal
Questions
More about the pseudointersection number
We observed that p(W) ≤ p and thus p(W) = p hold.
References
Introduction
Van der Waerden ideal
Questions
More about the pseudointersection number
We observed that p(W) ≤ p and thus p(W) = p hold.
Question C.
Is p(I1/n ) = p?
References
Introduction
Van der Waerden ideal
Questions
More about the pseudointersection number
We observed that p(W) ≤ p and thus p(W) = p hold.
Question C.
Is p(I1/n ) = p?
Question D.
Is there an Fσ ideal I such that p(I) > p is consistent?
References
Introduction
Van der Waerden ideal
Questions
References
B. Farkas, L. Soukup, More on cardinal invariants of analytic
P-ideals,
Comment. Math. Univ. Carolin. 50 (2), 281 – 295, 2009.
J. Brendle, Cardinal invariants of analytic quotients,
presentation in ESI workshop, 2009.
References