On properties of families of sets - Uniwersytet Wrocławski

On properties of families of setsLecture 2

Lajos Soukup

Alfréd Rényi Institute of MathematicsHungarian Academy of Sciences

http://www.renyi.hu/∼soukup

7th Young Set Theory Workshop

Recapitulation

Recapitulation

Definition: A family A ⊂ P(X) has property B iff χ(A) = 2,where the chromatic number of A is defined as follows:

χ(A)=min{λ | ∃f : X → λ ∀A ∈ A |f [A]| ≥ 2}.

Recapitulation


χ(A)=min{λ | ∃f : X → λ ∀A ∈ A |f [A]| ≥ 2}.

Theorem (E. W. Miller, 1937)There is an almost disjoint A ⊂

[

ω]ω

with χ(A) = ω.

Recapitulation


χ(A)=min{λ | ∃f : X → λ ∀A ∈ A |f [A]| ≥ 2}.


[

ω]ω

with χ(A) = ω.

Theorem (Gy. Elekes, Gy Hoffman, 1973)For all infinite cardinal κ there is an almost disjoint A ⊂

[

X]ω

withχ(A) ≥ κ.

Recapitulation


χ(A)=min{λ | ∃f : X → λ ∀A ∈ A |f [A]| ≥ 2}.


[

ω]ω

with χ(A) = ω.


[

X]ω

withχ(A) ≥ κ.

Definition: A is n-almost-disjoint iff |A ∩ A′| < n for all A 6= A′ ∈ A

Recapitulation


χ(A)=min{λ | ∃f : X → λ ∀A ∈ A |f [A]| ≥ 2}.


[

ω]ω

with χ(A) = ω.


[

X]ω

withχ(A) ≥ κ.

Definition: A is n-almost-disjoint iff |A ∩ A′| < n for all A 6= A′ ∈ A

Theorem (E. W. Miller, 1937)An n-almost disjoint family of infinite sets has property B.

Covering of the plain.


Theorem (Sierpinski)CH holds iff R2 is the union of countably many functions and theirinverses.



• f−1 = R(90◦)(−f )



• f−1 = R(90◦)(−f )

• R(α) : R2 → R2 the rotation by α degree around the origin.



• f−1 = R(90◦)(−f )


• If CH holds, then R2 is the union of countably many rotations offunctions.



• f−1 = R(90◦)(−f )



• Sierpinski, 1951: Is the converse true?



• f−1 = R(90◦)(−f )




Theorem (Davies, 1963)R2 is the union of countably many rotations of functions.



• f−1 = R(90◦)(−f )




Theorem (Davies, 1963)R2 is the union of countably many rotations of functions.If α0, α1, . . . are pairwise different angles between 0 and π, then thereare function f0, f1 . . . such that

R2 =⋃

n∈ω R(αn)(fn).

Thm. There are function f0, f1 . . . such that R2 =⋃

n∈ω R(αn)(fn).


n∈ω R(αn)(fn).

x0 = R(α0)(x)

x1 = R(α1)(x)


n∈ω R(αn)(fn).

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)


n∈ω R(αn)(fn).

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

• Write E(P)= {e(P, xn) : n ∈ ω}. Let E= {E(P) : P ∈ R2}.


n∈ω R(αn)(fn).

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

Rα0(f0)Rα1(f1)


• Assume R2 =⋃

n∈ω R(αn)(fn).


n∈ω R(αn)(fn).

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

Rα0(f0)Rα1(f1)

e(P, x0) = e(P)


• Assume R2 =⋃

n∈ω R(αn)(fn).

• if P ∈ R(αn)(fn), let e(P) = e(P, xn)


n∈ω R(αn)(fn).

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

Rα0(f0)Rα1(f1)

e(P, x0) = e(P)


• Assume R2 =⋃

n∈ω R(αn)(fn).


• P 6= Q implies e(P) 6= e(Q).


n∈ω R(αn)(fn).

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

Rα0(f0)Rα1(f1)

e(P, x0) = e(P)

Q


• Assume R2 =⋃

n∈ω R(αn)(fn).




n∈ω R(αn)(fn).

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

Rα0(f0)Rα1(f1)

e(P, x0) = e(P)

Q

e(Q, x0) = e(Q)


• Assume R2 =⋃

n∈ω R(αn)(fn).




n∈ω R(αn)(fn).

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

Rα0(f0)Rα1(f1)

e(P, x0) = e(P)

Q

e(Q, x0) = e(Q)

e(Q, x1) = e(Q)


• Assume R2 =⋃

n∈ω R(αn)(fn).




n∈ω R(αn)(fn).

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

Rα0(f0)Rα1(f1)

e(P, x0) = e(P)

Q

e(Q, x0) = e(Q)

e(Q, x1) = e(Q)


• Assume R2 =⋃

n∈ω R(αn)(fn).



• E has a transversal, i.e. an injective choice function.

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

• Write E(P)= {e(P, xn) : n ∈ ω} Let E= {E(P) : P ∈ R2}.

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

e(P, x0) = e(P)


• Assume E has a transversal (an injective choice function) e.

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

e(P, x0) = e(P)

Q

e(Q, x0) = e(Q)



• Let Fn={P : e(P) = e(P, xn)}.

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

e(P, x0) = e(P)

Q

e(Q, x0) = e(Q)

Rα0(f0)



• Let Fn={P : e(P) = e(P, xn)}.

• There is function fn s.t. Fn ⊂ R(αn)(fn):

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

e(P, x0) = e(P)

Q

e(Q, x0) = e(Q)

Rα0(f0)



• Let Fn={P : e(P) = e(P, xn)}.


• Let fn= {R(−αn)(P) : P ∈ Fn}

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

e(P, x0) = e(P)

Q

e(Q, x0) = e(Q)

Rα0(f0)



• Let Fn={P : e(P) = e(P, xn)}.



• So R2 =⋃

n∈ω R(αn)(fn).

x0 = R(α0)(x)

x1 = R(α1)(x)

P

e(P, x0)

e(P, x1)

e(P, x0) = e(P)

Q

e(Q, x0) = e(Q)

Rα0(f0)



• Let Fn={P : e(P) = e(P, xn)}.



• So R2 =⋃

n∈ω R(αn)(fn).

• E is 2-almost disjoint.

Beyond Property B.

Beyond Property B.

DefinitionA family B has transversal property iff there is an injective choicefunction on B.

Beyond Property B.


Enough: every 2-almost disjoint family of infinite sets has thetransversal property.

Beyond Property B.



• We have seen: An n-almost disjoint A ⊂[

X]ω

has a property B.

Beyond Property B.



• We have seen: An n-almost disjoint A ⊂[

X]ω

has a property B.

• Same proof: An n-almost disjoint A ⊂[

X]ω

has a transversal.

Beyond Property B and transversals

Beyond Property B and transversalsDefinition: A family A is essentially disjoint (ED) iff for each A ∈ A

there is F (A) ∈[

A]<ω

such that {A \ F (A) : A ∈ A} is disjoint.

A

C

B


there is F (A) ∈[

A]<ω


A \ F (A)

C \ F (C)

B \ F (B)


there is F (A) ∈[

A]<ω


• An essentially disjoint A has a transversal. If t(A) ∈ A \ F (A),then t is an injective choice function.

A \ F (A)

C \ F (C)

B \ F (B)


there is F (A) ∈[

A]<ω



A \ F (A)

C \ F (C)

B \ F (B)

t(A)

t(C)

t(B)


there is F (A) ∈[

A]<ω



A \ F (A)

C \ F (C)

B \ F (B)

t(A)

t(C)

t(B)

• There is a coloring c such that c[X ] = ω for all X ∈ A.Enough: c[X \ F (X)] = ω.

Property ED

Definition: A family A is essentially disjoint (ED) iff for each A ∈ A

there is F (A) ∈[

A]<ω


Property ED


there is F (A) ∈[

A]<ω


Theorem (Komjáth, 1984)Every n-almost disjoint family A ⊂

[

X]ω

is essentially disjoint.

Property ED


there is F (A) ∈[

A]<ω



[

X]ω


• The case A = {An : n < ω} is trivial: let F (An) = An ∩ (⋃

i<n Ai).

Property ED


there is F (A) ∈[

A]<ω



[

X]ω



i<n Ai).

• Consider the case A = {Aζ : ζ < ω1} ⊂[

ω1]ω

Property ED


there is F (A) ∈[

A]<ω



[

X]ω



i<n Ai).


ω1]ω

• A ∈ M1 ≺ M2 · · · ≺ Mα ≺ Mα+1 ≺ · · · ≺ 〈H(κ),∈, ⊳〉 (α < ω1)

Property ED


there is F (A) ∈[

A]<ω



[

X]ω



i<n Ai).


ω1]ω

• A ∈ M1 ≺ M2 · · · ≺ Mα ≺ Mα+1 ≺ · · · ≺ 〈H(κ),∈, ⊳〉 (α < ω1)

• continuous chain of countable elementary submodels

Property ED


there is F (A) ∈[

A]<ω



[

X]ω



i<n Ai).


ω1]ω

• A ∈ M1 ≺ M2 · · · ≺ Mα ≺ Mα+1 ≺ · · · ≺ 〈H(κ),∈, ⊳〉 (α < ω1)

• continuous chain of countable elementary submodels (M0 = ∅)

Property ED


there is F (A) ∈[

A]<ω



[

X]ω



i<n Ai).


ω1]ω

• A ∈ M1 ≺ M2 · · · ≺ Mα ≺ Mα+1 ≺ · · · ≺ 〈H(κ),∈, ⊳〉 (α < ω1)


• Partition A into countable pieces:A =

⋃∗

α<ω1Aα, where Aα = A∩ (Mα+1 \ Mα).

Property ED


there is F (A) ∈[

A]<ω



[

X]ω



i<n Ai).


ω1]ω

• A ∈ M1 ≺ M2 · · · ≺ Mα ≺ Mα+1 ≺ · · · ≺ 〈H(κ),∈, ⊳〉 (α < ω1)


• Partition A into countable pieces:A =

⋃∗


• Key observation:If A ∈ Mα+1 \ Mα then A ⊂ Mα+1 and |A ∩ Mα| < n

How to guarantee ED?

• A ⊂[

ω1]ω

is n-AD,A ∈ M1 ≺ M2 · · · ≺ Mα ≺ Mα+1 ≺ · · · ≺ 〈H(κ),∈, ⊳〉

• A =⋃∗


• Key observation: If A ∈ Mα+1 \ Mα then A ⊂ Mα+1 and|A ∩ Mα| < n


• A ⊂[

ω1]ω

is n-AD,A ∈ M1 ≺ M2 · · · ≺ Mα ≺ Mα+1 ≺ · · · ≺ 〈H(κ),∈, ⊳〉

• A =⋃∗



Mβ

AαAβ

Mα Mα+1

A

B

C

X


• A ⊂[

ω1]ω

is n-AD,A ∈ M1 ≺ M2 · · · ≺ Mα ≺ Mα+1 ≺ · · · ≺ 〈H(κ),∈, ⊳〉

• A =⋃∗



Mβ

AαAβ

Mα Mα+1

A \ Mα

B \ Mα

C \ Mα

X \ Mβ


• A ⊂[

ω1]ω

is n-AD,A ∈ M1 ≺ M2 · · · ≺ Mα ≺ Mα+1 ≺ · · · ≺ 〈H(κ),∈, ⊳〉

• A =⋃∗



Mβ

AαAβ

Mα Mα+1

A \ Mα

B \ Mα \ A

C \ Mα \ (A ∪ B)

X \ Mβ

How to guarantee Property B?

A is essentially disjoint iff for each A ∈ A there is F (A) ∈[

A]<ω

suchthat {A \ F (A) : A ∈ A} is disjoint.

If A ⊂[

X]ω

is ED then A has property B .

Miller: Every n-almost disjoint family A ⊂[

X]≥ω

has property B

Komjáth: Every n-almost disjoint family A ⊂[

X]ω

is essentially disjoint



A]<ω


If A ⊂[

X]ω



X]≥ω

has property B


X]ω


• Komjáth’s result implies Miller’s Theorem:

• if a family A refines a family B, then χ(B) ≤ χ(A).

• Def: A refines B iff ∀B ∈ B ∃A ∈ A A ⊂ B.



A]<ω


If A ⊂[

X]ω



X]≥ω

has property B


X]ω





Theorem (Gy. Hoffmann, P. Komjáth, 1976:)If a family of infinite sets has a transversal, then it has Property B.



A]<ω


If A ⊂[

X]ω



X]≥ω

has property B


X]ω





Theorem (Gy. Hoffmann, P. Komjáth, 1976:)If a family of infinite sets has a transversal, then it has Property B.

essentially disjoint // transversal // property B

If a family A of infinite sets has a transversal, then it has Property B.

• We can assume that A ⊂[

X]ω

. Let t : A → X be a transversal(injective choice function)



X]ω


• Let P = {c ∈ Fn(X , 2) : t(A) ∈ dom(c) implies c[A] = 2}.



X]ω



• By the Zorn lemma, 〈P,⊂〉 has a maximal element c.



X]ω




• We claim that t(A) ∈ dom(c) for all A ∈ A.



X]ω





• Case 1: there is A ∈ A s.t t(A) /∈ dom(c) and A ∩ dom(c) 6= ∅.



X]ω






• Pick x ∈ A ∩ dom(c). Then c′ = c ∪ {〈t(A), 1 − c(x)〉} ∈ P.Contradiction.



X]ω







• Case 2: t(A) /∈ dom(c) implies A ∩ dom(c) = ∅.



X]ω








• Y0 = {t(A)} s.t. t(A) /∈ dom c



X]ω








• Y0 = {t(A)} s.t. t(A) /∈ dom c

• Yn+1 = Yn∪⋃

{A ∈ A : t(A) ∈ Yn}.



X]ω








• Y0 = {t(A)} s.t. t(A) /∈ dom c

• Yn+1 = Yn∪⋃

{A ∈ A : t(A) ∈ Yn}.

• Y =⋃

n∈ω Yn ∈[

X \ dom c]ω

such that t(A) ∈ Y implies A ⊂ Y .



X]ω








• Y0 = {t(A)} s.t. t(A) /∈ dom c

• Yn+1 = Yn∪⋃

{A ∈ A : t(A) ∈ Yn}.

• Y =⋃

n∈ω Yn ∈[

X \ dom c]ω


• Choose d : Y → 2 s. t. d [A] = 2 whenever t(A) ∈ Y .



X]ω








• Y0 = {t(A)} s.t. t(A) /∈ dom c

• Yn+1 = Yn∪⋃

{A ∈ A : t(A) ∈ Yn}.

• Y =⋃

n∈ω Yn ∈[

X \ dom c]ω


• Choose d : Y → 2 s. t. d [A] = 2 whenever t(A) ∈ Y .

• Then c ( c ∪ d ∈ P. Contradiction.

Colorings

Colorings

• A ⊂ P(X) a family of sets,

Colorings

• A ⊂ P(X) a family of sets, f : X → ρ function

AB

CC

X

Colorings


• f is a proper coloring of A iff |f ′′A| ≥ 2 for each A ∈ A.

AB

CC

X

Colorings



• f is a conflict free coloring iff ∀A ∈ A ∃ξ ∈ ρ ∃!a ∈ A f (a) = ξ.

AB

C

X

Colorings



• f is a conflict free coloring iff ∀A ∈ A ∃ξ ∈ ρ ∃!a ∈ A f (a) = ξ.

• χCF(A)= min{ρ : ∃f : X → ρ conflict free coloring}

AB

C

X

Conflict free colorings

• c is a conflict free coloring iff ∀A ∈ A ∃ξA ∃!a ∈ A c(a) = ξA.

• χCF(A)= min{ρ : ∃c : X → ρ conflict free coloring}




• χ(A) ≤ χCF(A)





• If a family A ⊂[

λ]ω

is ED, then χCF(A) ≤ ω






λ]ω


A \ F (A)B \ F (B)

C \ F (C)






λ]ω


A \ F (A)B \ F (B)

C \ F (C)

• There is f : λ → ω such that f ↾ A \ F (A) is 1-to-1 for all A ∈ A.






λ]ω


A \ F (A)B \ F (B)

C \ F (C)

• There is f : λ → ω such that f ↾ A \ F (A) is 1-to-1 for all A ∈ A.• f ↾ A is almost 1-to-1 for all A ∈ A.

Conflict free colorings of n-ad families



• If A ⊂[

λ]ω

is ED, then χCF(A) ≤ ω.




• If A ⊂[

λ]ω


• If A ⊂[

λ]ω

is n-ad, then A is ED, so χCF(A) ≤ ω.




• If A ⊂[

λ]ω


• If A ⊂[

λ]ω


• (Hajnal, Juhász, S, Szentmiklóssy, 2009)

If B ⊂[

λ]κ

is n-ad, then χCF(B) ≤ ω.




• If A ⊂[

λ]ω


• If A ⊂[

λ]ω


• (Hajnal, Juhász, S, Szentmiklóssy, 2009)

If B ⊂[

λ]κ


• “A refines B” does not imply χCF(A) ≥ χCF(B)


Thm: If A ⊂[

λ]ω

is n-ad, χCF(A) ≤ ω.

TheoremIf B ⊂

[

λ]κ



Thm: If A ⊂[

λ]ω


TheoremIf B ⊂

[

λ]κ


• Definition. A family B has property B(ω) iff there is a set X suchthat 0 < |X ∩ B| < ω for each B ∈ B.


Thm: If A ⊂[

λ]ω


TheoremIf B ⊂

[

λ]κ



• Theorem (Erdos-Hajnal,1961) Any n-almost disjoint familyB ⊂

[

λ]κ

has property B(ω).


Thm: If A ⊂[

λ]ω


TheoremIf B ⊂

[

λ]κ




[

λ]κ

has property B(ω).

• Construct pairwise disjoint sets X0,X1, . . . ⊂ λ such that for allB ∈ B we have 0 < |(B \

⋃

i<n Xn)∩Xn| < ω.


Thm: If A ⊂[

λ]ω


TheoremIf B ⊂

[

λ]κ




[

λ]κ

has property B(ω).


⋃

i<n Xn)∩Xn| < ω.

• Let Y =⋃

n∈ω Xn. Then |Y ∩ B| = ω for all B ∈ B.


Thm: If A ⊂[

λ]ω


TheoremIf B ⊂

[

λ]κ




[

λ]κ

has property B(ω).


⋃

i<n Xn)∩Xn| < ω.

• Let Y =⋃


• Let f : Y → ω \ {0} be a conflict-free coloring for {B∩Y : B ∈ B}.


Thm: If A ⊂[

λ]ω


TheoremIf B ⊂

[

λ]κ




[

λ]κ

has property B(ω).


⋃

i<n Xn)∩Xn| < ω.

• Let Y =⋃


• Let f : Y → ω \ {0} be a conflict-free coloring for {B∩Y : B ∈ B}.

• Define g : λ → ω s.t. f ⊂ g and g(x) = 0 for all x ∈ λ \ Y .

Conflict free colorings of inhomogeneous n-ad familiesIf B ⊂

[

λ]κ


Any n-almost disjoint family B ⊂[

λ]κ

has property B(ω).

B has property B(ω) iff ∃X such that 0 < |X ∩ B| < ω for each B ∈ B.


[

λ]κ



λ]κ

has property B(ω).


• there is 2-ad A ⊂ [ω1]≥ω s.t A does not have property B(ω).


[

λ]κ



λ]κ

has property B(ω).



ω

ω1


[

λ]κ



λ]κ

has property B(ω).



ω

ω1

An

An = ω1 × {n}


[

λ]κ



λ]κ

has property B(ω).



ω

ω1

An

An = ω1 × {n}

Bζ

Bζ = {ζ} × ω


[

λ]κ



λ]κ

has property B(ω).



ω

ω1

An

An = ω1 × {n}

Bζ

Bζ = {ζ} × ω

A = {An,Bζ : n < ω, ζ < ω1}


[

λ]κ



λ]κ

has property B(ω).



ω

ω1

An

An = ω1 × {n}

Bζ

Bζ = {ζ} × ω

A = {An,Bζ : n < ω, ζ < ω1}

Theorem (Komjáth, 2012)χCF(A) ≤ ω for each n-almost disjoint A ⊂

[

κ]≥ω

.

Stepping up

Stepping up

• Every n-almost disjoint family A ⊂[

X]ω

is ED.

Stepping up


X]ω

is ED.

• χCF(B) ≤ ω for every n-almost disjoint family of infinite sets.

Stepping up


X]ω

is ED.


• Stepping up: What about almost disjoint families of uncountablesets? In particular, what about A ⊂

[

X]ω1?

Stepping up


X]ω

is ED.



[

X]ω1?

• Assume that A ⊂[

X]ω1 is almost disjoint.

Stepping up


X]ω

is ED.



[

X]ω1?



• χCF(A) ≤ ω1?

Stepping up


X]ω

is ED.



[

X]ω1?



• χCF(A) ≤ ω1?• Does A have the transversal property?

Stepping up


X]ω

is ED.



[

X]ω1?



• χCF(A) ≤ ω1?• Does A have the transversal property?• Is A ω1-essentially disjoint?

Stepping up


X]ω

is ED.



[

X]ω1?



• χCF(A) ≤ ω1?• Does A have the transversal property?• Is A ω1-essentially disjoint?

• A is ω1-essentially disjoint iff for each A ∈ A there isF (A) ∈

[

A]<ω1 such that {A \ F (A) : A ∈ A} is disjoint.

Stepping up: easy implications

A is ω1-essentially disjoint iff for each A ∈ A there is F (A) ∈[

A]≤ω




A]≤ω


DefinitionA has property B(ω1) iff ∃X such that 0 < |X ∩ A| < ω1 for all A ∈ A.



A]≤ω



Consider an arbitrary family B of uncountable sets.



A]≤ω



Consider an arbitrary family B of uncountable sets.

ω1-ED

TP B(ω1) χCF ≤ ω

B

χ ≤ ω

How to prove positive results?


• Every n-ad A ⊂[

ω1]ω

is ED.



ω1]ω

is ED.

• Consider a continuous chain 〈Mα : α < ω1〉 of elementarysubmodels . . .



ω1]ω

is ED.


• Key observation:If A ∈ M ≺ H(θ), and A ∈ A \ M, then |A ∩ M| < n (|A ∩ M| < ω).



ω1]ω

is ED.



• If M ≺ H(θ), X is a set, and X ∩ M is infinite, then for all n ∈ ω

we have M ∩[

X]n

6= ∅.



ω1]ω

is ED.




we have M ∩[

X]n

6= ∅.

• Want to prove: Every ω-ad A ⊂[

κ]ω1 is ω1-ED.



ω1]ω

is ED.




we have M ∩[

X]n

6= ∅.


κ]ω1 is ω1-ED.

• Need: there is a continuous chain 〈Mα : α < κ〉 of elementarysubmodels s.t. if X is a set, and X ∩ Mα is uncountable, thenMα ∩

[

X]ω

6= ∅.



ω1]ω

is ED.




we have M ∩[

X]n

6= ∅.


κ]ω1 is ω1-ED.


[

X]ω

6= ∅.

• “GCH + �∗∗λ for all λ > cf (λ) = ω” ⊢ ∃ suitable chains.



ω1]ω

is ED.




we have M ∩[

X]n

6= ∅.


κ]ω1 is ω1-ED.


[

X]ω

6= ∅.

• “GCH + �∗∗λ for all λ > cf (λ) = ω” ⊢ ∃ suitable chains.

• GCH is not enough. If GCH holds, then for all κ there is acontinuous chain 〈Mα : α < κ〉 of elementary submodels s.t. if Xis a set, and |X ∩ Mα| ≥ ω2, then Mα ∩

[

X]ω

6= ∅.

Positive results concerning almost disjoint families

V=L: ∃ 〈Mα : α < κ〉 s.t. ( if |X ∩ Mα| ≥ ω1 then Mα ∩[

X]ω

6= ∅)

GCH: ∃ 〈Mα : α < κ〉 s.t. ( if |X ∩ Mα| ≥ ω2 then Mα ∩[

X]ω

6= ∅)



X]ω

6= ∅)


X]ω

6= ∅)

1. (Komjáth, 1984)If V=L, then every AD subfamily of

[

λ]ω1 is ω2-ED

If GCH holds, then every AD subfamily of[

λ]ω2 is ω1-ED



X]ω

6= ∅)


X]ω

6= ∅)


[

λ]ω1 is ω2-ED


λ]ω2 is ω1-ED

2. Corollary:If V=L, then χCF(A) ≤ ω1 for all AD A ⊂

[

λ]κ

for all ω1 ≤ κ ≤ λ

If GCH, then χCF(A) ≤ ω2 for all AD A ⊂[

λ]κ




X]ω

6= ∅)


X]ω

6= ∅)


[

λ]ω1 is ω2-ED


λ]ω2 is ω1-ED


[

λ]κ



λ]κ


(Erdos, Hajnal, 1961) If GCH, every AD subfamily of[

λ]κ

hasproperty B(ω2) for all ω2 ≤ κ ≤ λ.



X]ω

6= ∅)


X]ω

6= ∅)


[

λ]ω1 is ω2-ED


λ]ω2 is ω1-ED


[

λ]κ



λ]κ



λ]κ


If κ ≥ ω1, A ⊂[

λ]κ

is AD, then ∃X s.t {A ∩ X : A ∈ A} ⊂[

X]ω1 .



X]ω

6= ∅)


X]ω

6= ∅)


[

λ]ω1 is ω2-ED


λ]ω2 is ω1-ED


[

λ]κ



λ]κ



λ]κ


If κ ≥ ω1, A ⊂[

λ]κ


X]ω1 .

If κ ≥ ω2, A ⊂[

λ]κ


X]ω2 .

Negative results

If V=L, then χCF(A) ≤ ω1 for all AD A ⊂[

λ]κ



λ]κ


Negative results


λ]κ



λ]κ


1. (Hajnal, Juhász, Shelah (2000) ) It is consistent (modulo asupercompact cardinal), that GCH holds and

Negative results


λ]κ



λ]κ



• there is an almost disjoint family A ⊂[

ℵω+1]ω1 with

χ(A) = (χCF(A)) = ℵω+1

Negative results


λ]κ



λ]κ




ℵω+1]ω1 with

χ(A) = (χCF(A)) = ℵω+1

2. (Hajnal, Juhász, S, Szentmiklóssy, (2010) It is consistent(modulo a supercompact cardinal), that GCH holds and

Negative results


λ]κ



λ]κ




ℵω+1]ω1 with

χ(A) = (χCF(A)) = ℵω+1

2. (Hajnal, Juhász, S, Szentmiklóssy, (2010) It is consistent(modulo a supercompact cardinal), that GCH holds and

• there is an almost disjoint family B ⊂[

ℵω+1]ω2 with

χCF(B) = ω2.

Inhomogeneous systems

• every n-ad subfamily of[

λ]ω

is ED

• If V=L, then every almost disjoint subfamily of[

λ]ω1 is ω1-ED.

• If GCH holds, then every almost disjoint subfamily of[

λ]ω2 is

ω2-ED.

• χCF(A) ≤ ω for all n-almost disjoint family A ⊂[

λ]≥ω

.



λ]ω

is ED


λ]ω1 is ω1-ED.


λ]ω2 is

ω2-ED.


λ]≥ω

.

Expected results: If V=L, then χCF(A) ≤ ω1 for all ω-almost disjointfamily A ⊂

[

λ]≥ω1 . If GCH, then χCF(A) ≤ ω2 for all ω-almost disjoint

family A ⊂[

λ]≥ω2 .



λ]ω

is ED


λ]ω1 is ω1-ED.


λ]ω2 is

ω2-ED.


λ]≥ω

.

Expected results: If V=L, then χCF(A) ≤ ω1 for all ω-almost disjointfamily A ⊂

[

λ]≥ω1 . If GCH, then χCF(A) ≤ ω2 for all ω-almost disjoint

family A ⊂[

λ]≥ω2 .

Theorem (S, 2012)If every µ-almost disjoint subfamily of

[

λ]κ

is κ-ED, then χCF(A) ≤ κ

for all µ-almost disjoint A ⊂[

λ]≥κ

.

So the expected results hold.

An unexpected result


ω1-ED


B

χ ≤ ω


ω1-ED


B

χ ≤ ω

Theorem (Komjáth, 1984)If every ω-almost disjoint subfamily of

[

λ]ω1 has the transversal

propertythen every ω-almost disjoint subfamily of

[

λ]ω1 is ω-essentially

disjoint.

Properties of µ-almost disjoint families for µ > ω


Theorem (S, 2010)Let µ < iω ≤ λ.



• If A ⊂[

λ]iω is a µ-almost disjoint family, then A is

iω-essentially disjoint.



• If A ⊂[



• If A ⊂[

λ]≥iω is µ-almost disjoint, then χCF(A) ≤ iω.



• If A ⊂[



• If A ⊂[


Definitionρ[ν] = ρ iff there is a family B ⊂

[

ρ]≤ν

of size ρ such that for all

u ∈[

ρ]ν

there is P ∈[

B]<ν

such that u = ∪P .



• If A ⊂[



• If A ⊂[


Definitionρ[ν] = ρ iff there is a family B ⊂

[

ρ]≤ν

of size ρ such that for all

u ∈[

ρ]ν

there is P ∈[

B]<ν

such that u = ∪P .

Theorem (Shelah, Revised GCH)If ρ ≥ iω, then ρ[ν] = ρ for each large enough regular ν < iω.

The questions

The questions

If every ω-almost disjoint subfamily of[

λ]ω1 has the transversal prop-

ertythen every ω-almost disjoint subfamily of

[

λ]ω1 is ω-essentially disjoint.

If every µ-almost disjoint subfamily of[

λ]κ



λ]≥κ

.

The questions

If every ω-almost disjoint subfamily of[

λ]ω1 has the transversal prop-

ertythen every ω-almost disjoint subfamily of

[

λ]ω1 is ω-essentially disjoint.

If every µ-almost disjoint subfamily of[

λ]κ



λ]≥κ

.

What are the right questions?


• Let weak and strong be properties of families of sets.

∀A(

strong(A) =⇒ weak(A))

.



∀A(


.

• You can not expect: ∀A(

weak(A) =⇒ strong(A))

.



∀A(


.



.

• The right question/a better question:Let A be a reasonable collection of families of sets.Prove/disprove

∀A ∈ A weak(A) =⇒ ∀A ∈ A strong(A)



∀A(


.



.



• Cheap solution: ZFC ⊢ ∀A ∈ A strong(A), or



∀A(


.



.




ZFC ⊢ ∃A ∈ A ¬weak(A)



∀A(


.



.





• Need methods to investigate: M?

|= ∀A ∈ A weak(A)



∀A(


.



.





• Need methods to investigate: M?


M?

|= ∀A ∈ A strong(A)

What does “reasonable” mean?

• Let weak and strong be properties of families of sets,strong(A) =⇒ weak(A).

• Let A be a reasonable collection of families of sets.Prove/disprove






• A1 = { µ-almost disjoint subfamilies of[

λ]κ

}.






λ]κ

}.

• A2 = { ladder systems }.






λ]κ

}.


Definition. Let S ⊂ κ.A family A = {Aα : α ∈ S} is a ladder system on S iff

Aα is a cofinal subset of α (with order type cf (α)).






λ]κ

}.


Definition. Let S ⊂ κ.A family A = {Aα : α ∈ S} is a ladder system on S iff

Aα is a cofinal subset of α (with order type cf (α)).

A ladder system is a ladder system on some S.

Degustation

DegustationProve/disprove: ∀A ∈ A weak(A) =⇒ ∀A ∈ A strong(A)


• Prove implication theorems. Are there any?



• Prove separation theorems.




• Forcing: not hopeless up to continuum. But . . .




• Forcing: not hopeless up to continuum. But . . .• Forcing: more compliceted above c. No nice iterations.




• Forcing: not hopeless up to continuum. But . . .• Forcing: more compliceted above c. No nice iterations.• Preservation theorems.





Given a model M, decide M?








Compactness argument:








• At singulars: Shelah’s Singular Compactness Theorem.









• At regulars:









• At regulars:

• Independence result: large cardinals vs boxes









• At regulars:

• Independence result: large cardinals vs boxes• Shelah’s Revised GCH









• At regulars:

• Independence result: large cardinals vs boxes• Shelah’s Revised GCH• Combinatorial principles

Problems around ω-almost disjoint families A ⊂[

λ]ω1


λ]ω1

ω-ED

TP B(≤ ω) χCF ≤ ω1

λ 6→ [A]1ω1

B

χ ≤ ω1


λ]ω1

ω-ED


λ 6→ [A]1ω1

B

χ ≤ ω1

λ 6→ [A]1ω1


λ]ω1

ω-ED


λ 6→ [A]1ω1

B

χ ≤ ω1

λ 6→ [A]1ω1

λ 6→ [A]1ω1iff there is f : λ → ω1 s.t. f [A] = ω1 for all A ∈ A.

To be continued . . .

On properties of families of sets - Uniwersytet Wrocławski

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Transcript of On properties of families of sets - Uniwersytet Wrocławski