On Properties of Families of Sets Lecture 1

On properties of families of sets Lecture 1

Lajos Soukup

Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences

http://www.renyi.hu/∼soukup

7th Young Set Theory Workshop “On properties of families of sets” “On properties of families of sets”

• Miller, Edwin W. On a property of families of sets. C. R. Soc. Sci. Varsovie 30, 31-38 (1937). “On properties of families of sets”

• Miller, Edwin W. On a property of families of sets. C. R. Soc. Sci. Varsovie 30, 31-38 (1937). • Erdos,˝ Paul; Hajnal, András On a property of families of sets. Acta Math. Acad. Sci. Hung. 12, 87-123 (1961). Outline

• Property B and its Friends-and-Relations: Outline

• Property B and its Friends-and-Relations: • chromatic number, transversal property, property B(κ), essential disjoint property, conﬂict free chromatic number, weak Freese nation property, Outline

• Property B and its Friends-and-Relations: • chromatic number, transversal property, property B(κ), essential disjoint property, conﬂict free chromatic number, weak Freese nation property,

• Elementary submodels Outline

• Property B and its Friends-and-Relations: • chromatic number, transversal property, property B(κ), essential disjoint property, conﬂict free chromatic number, weak Freese nation property,

• Elementary submodels • Shelah’s Singular Cardinal Compactness Theorem Outline

• Property B and its Friends-and-Relations: • chromatic number, transversal property, property B(κ), essential disjoint property, conﬂict free chromatic number, weak Freese nation property,

• Elementary submodels • Shelah’s Singular Cardinal Compactness Theorem • Forcing construction Outline

• Property B and its Friends-and-Relations: • chromatic number, transversal property, property B(κ), essential disjoint property, conﬂict free chromatic number, weak Freese nation property,

• Elementary submodels • Shelah’s Singular Cardinal Compactness Theorem • Forcing construction • Combinatorial principles Hilbert’s ﬁrst problem Hilbert’s ﬁrst problem

Hilbert, 1900: The investigations of Cantor ... suggest a very plausible theorem: Every system of inﬁnitely many real numbers . . . is either equivalent to the assemblage of natural integers, . . . or to the assemblage of all real numbers and therefore to the continuum . . . Cantor, 1885: Every perfect subset of the reals has size continuum. Hilbert’s ﬁrst problem

Theorem (Bernstein, 1908) There is X ⊂ R such that neither X nor R \ X contain a perfect subset. Hilbert’s ﬁrst problem

Theorem (Bernstein, 1908) There is X ⊂ R such that neither X nor R \ X contain a perfect subset. The family of perfect subsets of the reals has property B: Hilbert’s ﬁrst problem

Theorem (Bernstein, 1908) There is X ⊂ R such that neither X nor R \ X contain a perfect subset. The family of perfect subsets of the reals has property B: ∃X ⊂ R ∀ perfect P P ∩ X 6= empt and P \ X 6= ∅ Property B

Bernstein: The family of perfect subsets of the reals has property B. Property B

Bernstein: The family of perfect subsets of the reals has property B.

Deﬁnition: Let A⊂P(X) be a family of sets. Property B

Bernstein: The family of perfect subsets of the reals has property B.

Deﬁnition: Let A⊂P(X) be a family of sets. A has property B iff Property B

Bernstein: The family of perfect subsets of the reals has property B.

Deﬁnition: Let A⊂P(X) be a family of sets. A has property B iff • ∃Y ∀A ∈ A A ∩ Y 6= ∅ and A \ Y 6= ∅ Property B

Bernstein: The family of perfect subsets of the reals has property B.

Deﬁnition: Let A⊂P(X) be a family of sets. A has property B iff • ∃Y ∀A ∈ A A ∩ Y 6= ∅ and A \ Y 6= ∅ • χ(A)= 2 Property B

Bernstein: The family of perfect subsets of the reals has property B.

Deﬁnition: Let A⊂P(X) be a family of sets. A has property B iff • ∃Y ∀A ∈ A A ∩ Y 6= ∅ and A \ Y 6= ∅ • χ(A)= 2 where the chromatic number of A:

χ(A)=min{λ | there is a coloring of X with λ colors s. t. the elements of A are not monochromatic }, Property B

Bernstein: The family of perfect subsets of the reals has property B.

Deﬁnition: Let A⊂P(X) be a family of sets. A has property B iff • ∃Y ∀A ∈ A A ∩ Y 6= ∅ and A \ Y 6= ∅ • χ(A)= 2 where the chromatic number of A:

χ(A)=min{λ | there is a coloring of X with λ colors s. t. the elements of A are not monochromatic },

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

Bernstein: The family of perfect subsets of the reals has property B.

Deﬁnition: Let A⊂P(X) be a family of sets. A has property B iff • ∃Y ∀A ∈ A A ∩ Y 6= ∅ and A \ Y 6= ∅ • χ(A)= 2 where the chromatic number of A:

χ(A)=min{λ | there is a coloring of X with λ colors s. t. the elements of A are not monochromatic },

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

Question: Which families have property B? Almost disjoint families

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

A has property B iff ∃B ∀A ∈ A A ∩ B 6= ∅ 6= A \ B iff χ(A)= 2 Almost disjoint families

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

A has property B iff ∃B ∀A ∈ A A ∩ B 6= ∅ 6= A \ B iff χ(A)= 2

• A disjoint family of inﬁnite sets has property B. Almost disjoint families

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

A has property B iff ∃B ∀A ∈ A A ∩ B 6= ∅ 6= A \ B iff χ(A)= 2

• A disjoint family of infinite sets has property B. • Definition: A family A is almost disjoint iff A ∩ A′ is finite for A 6= A′ ∈ A. Almost disjoint families

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

A has property B iff ∃B ∀A ∈ A A ∩ B 6= ∅ 6= A \ B iff χ(A)= 2

• A disjoint family of infinite sets has property B. • Definition: A family A is almost disjoint iff A ∩ A′ is finite for A 6= A′ ∈ A. • Does every almost disjoint family of infinite sets has property B? Almost disjoint families

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

A has property B iff ∃B ∀A ∈ A A ∩ B 6= ∅ 6= A \ B iff χ(A)= 2

Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Proof (the hard way): Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Proof (the hard way): Theorem (Balcar, Vojtáš, 1980) Every non-principal ultraﬁlter U on ω has an almost disjoint reﬁnement, i.e. a family AU = {AU : U ∈U} such that ω • AU ∈ U for U ∈U

• AU ∩ AV is finite for U 6= V ∈U Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Proof (the hard way): Theorem (Balcar, Vojtáš, 1980) Every non-principal ultrafilter U on ω has an almost disjoint refinement, i.e. a family AU = {AU : U ∈U} such that ω • AU ∈ U for U ∈U

• AU ∩ AV is ﬁnite for U 6= V ∈U

Clearly χ(AU )= ω: Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Proof (the hard way): Theorem (Balcar, Vojtáš, 1980) Every non-principal ultraﬁlter U on ω has an almost disjoint reﬁnement, i.e. a family AU = {AU : U ∈U} such that ω • AU ∈ U for U ∈U

• AU ∩ AV is ﬁnite for U 6= V ∈U

Clearly χ(AU )= ω: if c : ω → n, then there is U ∈U with c[U]= {i}, and so c[AU ]= {i}. Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Theorem (Balcar, Vojtáš, 1980) Every non-principal ultrafilter U on ω has an almost disjoint refinement. Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Theorem (Balcar, Vojtáš, 1980) Every non-principal ultrafilter U on ω has an almost disjoint refinement. Direct proof: Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Theorem (Balcar, Vojtáš, 1980) Every non-principal ultrafilter U on ω has an almost disjoint refinement. Direct proof: • Let Q = {P ⊂ Q : P is a dense subset of some interval (a, b)} Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Theorem (Balcar, Vojtáš, 1980) Every non-principal ultrafilter U on ω has an almost disjoint refinement. Direct proof: • Let Q = {P ⊂ Q : P is a dense subset of some interval (a, b)} ′ • For each P ∈Q pick xP ∈ P s.t. xP 6= xP′ for P 6= P . Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Theorem (Balcar, Vojtáš, 1980) Every non-principal ultrafilter U on ω has an almost disjoint refinement. Direct proof: • Let Q = {P ⊂ Q : P is a dense subset of some interval (a, b)} ′ • For each P ∈Q pick xP ∈ P s.t. xP 6= xP′ for P 6= P .

• For each P ∈Q pick AP ⊂ P \{xP} s.t. AP → xP . Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Theorem (Balcar, Vojtáš, 1980) Every non-principal ultraﬁlter U on ω has an almost disjoint reﬁnement. Direct proof: • Let Q = {P ⊂ Q : P is a dense subset of some interval (a, b)} ′ • For each P ∈Q pick xP ∈ P s.t. xP 6= xP′ for P 6= P .

• For each P ∈Q pick AP ⊂ P \{xP} s.t. AP → xP .

• Let A = {AP : P ∈ Q}. Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Theorem (Balcar, Vojtáš, 1980) Every non-principal ultraﬁlter U on ω has an almost disjoint reﬁnement. Direct proof: • Let Q = {P ⊂ Q : P is a dense subset of some interval (a, b)} ′ • For each P ∈Q pick xP ∈ P s.t. xP 6= xP′ for P 6= P .

• For each P ∈Q pick AP ⊂ P \{xP} s.t. AP → xP .

• Let A = {AP : P ∈ Q}. • A is an almost disjoint refinement of Q. Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Theorem (Balcar, Vojtáš, 1980) Every non-principal ultrafilter U on ω has an almost disjoint refinement. Direct proof: • Let Q = {P ⊂ Q : P is a dense subset of some interval (a, b)} ′ • For each P ∈Q pick xP ∈ P s.t. xP 6= xP′ for P 6= P .

• For each P ∈Q pick AP ⊂ P \{xP} s.t. AP → xP .

• Let A = {AP : P ∈ Q}. • A is an almost disjoint refinement of Q. • If c : Q → n, then there is P ∈Q s.t. c[P]= {i}. Almost disjoint families with large chromatic number Theorem (E. W. Miller, 1937) ω There is an almost disjoint A⊂ ω with χ(A)= ω. Theorem (Balcar, Vojtáš, 1980) Every non-principal ultrafilter U on ω has an almost disjoint refinement. Direct proof: • Let Q = {P ⊂ Q : P is a dense subset of some interval (a, b)} ′ • For each P ∈Q pick xP ∈ P s.t. xP 6= xP′ for P 6= P .

• For each P ∈Q pick AP ⊂ P \{xP} s.t. AP → xP .

• Let A = {AP : P ∈ Q}. • A is an almost disjoint reﬁnement of Q. • If c : Q → n, then there is P ∈Q s.t. c[P]= {i}.

• Then c[AP ]= {i}. Which families have almost disjoint refinements? Non-principal ultrafilters on ω have almost disjoint refinements. Which families have almost disjoint refinements? Non-principal ultrafilters on ω have almost disjoint refinements.

• If A is AD, and X ⊂ S B for some finite B ⊂ A, then you can not expect A ∈ A with A ⊂ X. Which families have almost disjoint refinements? Non-principal ultrafilters on ω have almost disjoint refinements.

• If A is AD, and X ⊂ S B for some finite B ⊂ A, then you can not expect A ∈ A with A ⊂ X. • Definition (Hechler, 1971): An infinite almost disjoint family ω A⊂ ω is completely separable (saturated) iff for every subset X ⊂ ω either there is A ∈ A with A ⊂ X or there is a finite subfamily B ⊂ A with X ⊂ S B . Erdos,˝ Shelah, 1972 : Is there a completely separable MAD in ZFC? Theorem (Shelah, 2009) ω If 2 < ℵω, then there is an completely separable MAD. Con (there is no completely separable MAD) implies Con(there is a measurable cardinal) Theorem (Brendle, Balcar-Pazak, 2008 ) If V ⊂ W are models of ZFC, W contains a new real r, then V ∩ [ω]ω has an almost disjoint refinement in W . Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) . Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well

Stage α: Let γα = min{γ ≤ α : B(xγ)xα contains a perfect subset} Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well

Stage α: Let γα = min{γ ≤ α : B(xγ)xα contains a perfect subset}

• Pick bα∈ (B(xγ )xα \ V ) \{bζ : ζ < α} and let aα= bα ∩ xα. Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well

Stage α: Let γα = min{γ ≤ α : B(xγ)xα contains a perfect subset}

• Pick bα∈ (B(xγ )xα \ V ) \{bζ : ζ < α} and let aα= bα ∩ xα.

• Since bα ∈ B(xγ)xα the set aα is inﬁnite. Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well

Stage α: Let γα = min{γ ≤ α : B(xγ)xα contains a perfect subset}

• Pick bα∈ (B(xγ )xα \ V ) \{bζ : ζ < α} and let aα= bα ∩ xα.

• Since bα ∈ B(xγ)xα the set aα is inﬁnite.

Claim If α 6= β then |aα ∩ aβ| <ω. Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well

Stage α: Let γα = min{γ ≤ α : B(xγ)xα contains a perfect subset}

• Pick bα∈ (B(xγ )xα \ V ) \{bζ : ζ < α} and let aα= bα ∩ xα.

• Since bα ∈ B(xγ)xα the set aα is inﬁnite.

Claim If α 6= β then |aα ∩ aβ| <ω.

• Case 1: γα = γβ. Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well

Stage α: Let γα = min{γ ≤ α : B(xγ)xα contains a perfect subset}

• Pick bα∈ (B(xγ )xα \ V ) \{bζ : ζ < α} and let aα= bα ∩ xα.

• Since bα ∈ B(xγ)xα the set aα is inﬁnite.

Claim If α 6= β then |aα ∩ aβ| <ω.

• Case 1: γα = γβ.

• Then bα 6= bβ ∈ [T (xγα )]. Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well

Stage α: Let γα = min{γ ≤ α : B(xγ)xα contains a perfect subset}

• Pick bα∈ (B(xγ )xα \ V ) \{bζ : ζ < α} and let aα= bα ∩ xα.

• Since bα ∈ B(xγ)xα the set aα is inﬁnite.

Claim If α 6= β then |aα ∩ aβ| <ω.

• Case 1: γα = γβ.

• Then bα 6= bβ ∈ [T (xγα )].

• So bα ∩ bβ is ﬁnite, and aα ∩ aβ ⊂ bα ∩ bβ. Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well

Stage α: Let γα = min{γ ≤ α : B(xγ)xα contains a perfect subset}

• Pick bα∈ (B(xγ )xα \ V ) \{bζ : ζ < α} and let aα= bα ∩ xα.

• Since bα ∈ B(xγ)xα the set aα is inﬁnite.

Claim If α 6= β then |aα ∩ aβ| <ω.

• Case 2: γα <γβ. Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well

Stage α: Let γα = min{γ ≤ α : B(xγ)xα contains a perfect subset}

• Pick bα∈ (B(xγ )xα \ V ) \{bζ : ζ < α} and let aα= bα ∩ xα.

• Since bα ∈ B(xγ)xα the set aα is inﬁnite.

Claim If α 6= β then |aα ∩ aβ| <ω.

• Case 2: γα <γβ.

• γβ = min{γ ≤ β : B(xγ)xβ contains a perfect subset} Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well

Stage α: Let γα = min{γ ≤ α : B(xγ)xα contains a perfect subset}

• Pick bα∈ (B(xγ )xα \ V ) \{bζ : ζ < α} and let aα= bα ∩ xα.

• Since bα ∈ B(xγ)xα the set aα is inﬁnite.

Claim If α 6= β then |aα ∩ aβ| <ω.

• Case 2: γα <γβ.

• γβ = min{γ ≤ β : B(xγ)xβ contains a perfect subset}

• γα <γβ so |B(xγα )xβ |≤ ω so B(xγα )xβ ⊂ V Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well

Stage α: Let γα = min{γ ≤ α : B(xγ)xα contains a perfect subset}

• Pick bα∈ (B(xγ )xα \ V ) \{bζ : ζ < α} and let aα= bα ∩ xα.

• Since bα ∈ B(xγ)xα the set aα is inﬁnite.

Claim If α 6= β then |aα ∩ aβ| <ω.

• Case 2: γα <γβ.

• γβ = min{γ ≤ β : B(xγ)xβ contains a perfect subset}

• γα <γβ so |B(xγα )xβ |≤ ω so B(xγα )xβ ⊂ V so bα ∈/ B(xγα )xβ Thm: If V ⊂ W are models of ZFC, W contains a new real r, and ω {xα : α<κ} =[ω] ∩ V , then there is an almost disjoint family ω {aα : α<κ}⊂ ω in W s.t. aα ⊂ xα. ω • For all x ∈ ω ∩ V let T (x) be a binary tree on x ω • If y ∈ ω ∩ V , then B(x)y = {b ∈ [T (x)] : |b ∩ y| = ω} is Gδ

• Hence |B(x)y |≤ ω OR B(x)y contains a perfect set. W ω V • If B(x)y contains a perfect set then |(B(x)y ) \ V |≥ κ = (2 ) .

• Contstruct {aα : α<κ} inductively. We will deﬁne γα, bα as well

Stage α: Let γα = min{γ ≤ α : B(xγ)xα contains a perfect subset}

• Pick bα∈ (B(xγ )xα \ V ) \{bζ : ζ < α} and let aα= bα ∩ xα.

• Since bα ∈ B(xγ)xα the set aα is inﬁnite.

Claim If α 6= β then |aα ∩ aβ| <ω.

• Case 2: γα <γβ.

• γβ = min{γ ≤ β : B(xγ)xβ contains a perfect subset}

• γα <γβ so |B(xγα )xβ |≤ ω so B(xγα )xβ ⊂ V so bα ∈/ B(xγα )xβ

• Thus bα ∩ xβ is ﬁnite. So aα ∩ aα ⊂ bα ∩ xβ is also ﬁnite. Almost disjoint families with prescribed chromatic number Almost disjoint families with prescribed chromatic number Theorem ( ) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1. Almost disjoint families with prescribed chromatic number Theorem (Erdos,˝ Shelah, 1972) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1. Almost disjoint families with prescribed chromatic number Theorem (Erdos,˝ Shelah, 1972) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1.

• Let (A1,..., An+1) be a partition of ω into inﬁnite sets Almost disjoint families with prescribed chromatic number Theorem (Erdos,˝ Shelah, 1972) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1.

• Let (A1,..., An+1) be a partition of ω into inﬁnite sets i ω ω • {Fα : α< 2 }⊂ Ai almost disjont family for 1 ≤ i ≤ n + 1 Almost disjoint families with prescribed chromatic number Theorem (Erdos,˝ Shelah, 1972) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1.

• Let (A1,..., An+1) be a partition of ω into inﬁnite sets i ω ω • {Fα : α< 2 }⊂ Ai almost disjont family for 1 ≤ i ≤ n + 1 1 n ω • {(Dα,..., Dα): α< 2 )} partitions of ω into n sets. Almost disjoint families with prescribed chromatic number Theorem (Erdos,˝ Shelah, 1972) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1.

• Let (A1,..., An+1) be a partition of ω into inﬁnite sets i ω ω • {Fα : α< 2 }⊂ Ai almost disjont family for 1 ≤ i ≤ n + 1 1 n ω • {(Dα,..., Dα): α< 2 )} partitions of ω into n sets. i j • for all 1 ≤ i ≤ n + 1 there is 1 ≤ j ≤ n s.t. Fα ∩ Dα is inﬁnite Almost disjoint families with prescribed chromatic number Theorem (Erdos,˝ Shelah, 1972) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1.

• there is 1 ≤ j ≤ n and there are 1 ≤ i1 < i2 ≤ n such that i1 j i2 j both Fα ∩ Dα and Fα ∩ Dα are inﬁnite. Almost disjoint families with prescribed chromatic number Theorem (Erdos,˝ Shelah, 1972) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1.

• there is 1 ≤ j ≤ n and there are 1 ≤ i1 < i2 ≤ n such that i1 j i2 j both Fα ∩ Dα and Fα ∩ Dα are inﬁnite.

j i1 i2 • Let Gα= Dα ∩ (Fα ∪ Fα ). Almost disjoint families with prescribed chromatic number Theorem (Erdos,˝ Shelah, 1972) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1.

• there is 1 ≤ j ≤ n and there are 1 ≤ i1 < i2 ≤ n such that i1 j i2 j both Fα ∩ Dα and Fα ∩ Dα are inﬁnite.

j i1 i2 • Let Gα= Dα ∩ (Fα ∪ Fα ). ω Claim: G= {Gα : α< 2 } is almost disjoint and χ(G)= n + 1. Almost disjoint families with prescribed chromatic number Theorem (Erdos,˝ Shelah, 1972) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1.

• there is 1 ≤ j ≤ n and there are 1 ≤ i1 < i2 ≤ n such that i1 j i2 j both Fα ∩ Dα and Fα ∩ Dα are inﬁnite.

j i1 i2 • Let Gα= Dα ∩ (Fα ∪ Fα ). ω Claim: G= {Gα : α< 2 } is almost disjoint and χ(G)= n + 1. i i i i • Gα ∩ Gβ ⊂ S1≤i≤n+1(Fα ∩ Fβ), and Fα ∩ Fβ is ﬁnite. Almost disjoint families with prescribed chromatic number Theorem (Erdos,˝ Shelah, 1972) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1.

• there is 1 ≤ j ≤ n and there are 1 ≤ i1 < i2 ≤ n such that i1 j i2 j both Fα ∩ Dα and Fα ∩ Dα are inﬁnite.

j i1 i2 • Let Gα= Dα ∩ (Fα ∪ Fα ). ω Claim: G= {Gα : α< 2 } is almost disjoint and χ(G)= n + 1. i i i i • Gα ∩ Gβ ⊂ S1≤i≤n+1(Fα ∩ Fβ), and Fα ∩ Fβ is ﬁnite. j 1 n • Gα ⊂ Dα, so (Dα,..., Dα) can not prove χ(G) ≤ n. Almost disjoint families with prescribed chromatic number Theorem (Erdos,˝ Shelah, 1972) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1.

• there is 1 ≤ j ≤ n and there are 1 ≤ i1 < i2 ≤ n such that i1 j i2 j both Fα ∩ Dα and Fα ∩ Dα are inﬁnite.

j i1 j i1 • Gα ∩ Ai1 ⊃ Dα ∩ Fα 6= ∅ and Gα ∩ Ai2 ⊃ Dα ∩ Fα 6= ∅ Almost disjoint families with prescribed chromatic number Theorem (Erdos,˝ Shelah, 1972) ω ∀n ≥ 2 there is an almost disjoint family G ⊂ ω with χ(G)= n + 1.

• there is 1 ≤ j ≤ n and there are 1 ≤ i1 < i2 ≤ n such that i1 j i2 j both Fα ∩ Dα and Fα ∩ Dα are inﬁnite.

j i1 j i1 • Gα ∩ Ai1 ⊃ Dα ∩ Fα 6= ∅ and Gα ∩ Ai2 ⊃ Dα ∩ Fα 6= ∅ • So (A1,..., An+1) ⊢ χ(G) ≤ n + 1 Disjoint sets in almost disjoint families Disjoint sets in almost disjoint families

Which almost disjoint families have property B? Disjoint sets in almost disjoint families

Which almost disjoint families have property B? Theorem (Erdos,˝ Shelah, 1972) ≥ω If A⊂ X is almost disjoint, and A ∩ A′ 6= ∅ for all A, A′ ∈ A, then χ(A)= 2. Disjoint sets in almost disjoint families

Which almost disjoint families have property B? Theorem (Erdos,˝ Shelah, 1972) ≥ω If A⊂ X is almost disjoint, and A ∩ A′ 6= ∅ for all A, A′ ∈ A, then χ(A)= 2. Proof: Disjoint sets in almost disjoint families

Which almost disjoint families have property B? Theorem (Erdos,˝ Shelah, 1972) ≥ω If A⊂ X is almost disjoint, and A ∩ A′ 6= ∅ for all A, A′ ∈ A, then χ(A)= 2. Proof: ≥ω • Indirect: Assume that A⊂ X is almost disjoint, A ∩ A′ 6= ∅ for all A, A′ ∈ A, and χ(A) > 2. Disjoint sets in almost disjoint families

• Fix A ∈ A. ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B A

• Fix A ∈ A. Color X = B ∪ R, ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B2 B4 B A

R R1 R3 R5

• Fix A ∈ A. Color X = B ∪ R, choose B0, R1, B2, R2,... from A ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B2 B4 B A

R R1 R3 R5

• Fix A ∈ A. Color X = B ∪ R, choose B0, R1, B2, R2,... from A pick r0 from B0, b1 from R1, . . . s.t. ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B2 B4 B A

R R1 R3 R5

• Fix A ∈ A. Color X = B ∪ R, choose B0, R1, B2, R2,... from A pick r0 from B0, b1 from R1, . . . s.t.

• B2m ∩ R = B2m ∩ A ∩ R = {r2m}, R2m+1 ∩ B = R2m+1 ∩ A ∩ B = {b2m}. ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B2 B4 Y B A

R R1 R3 R5

• Fix A ∈ A. Color X = B ∪ R, choose B0, R1, B2, R2,... from A pick r0 from B0, b1 from R1, . . . s.t.

• B2m ∩ R = B2m ∩ A ∩ R = {r2m}, R2m+1 ∩ B = R2m+1 ∩ A ∩ B = {b2m}. • B ∪ R is not a good coloring of A: ∃Y ∈ A Y ⊂ B. ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B2 B4 Y B A

R R1 R3 R5

• Fix A ∈ A. Color X = B ∪ R, choose B0, R1, B2, R2,... from A pick r0 from B0, b1 from R1, . . . s.t.

• B2m ∩ R = B2m ∩ A ∩ R = {r2m}, R2m+1 ∩ B = R2m+1 ∩ A ∩ B = {b2m}. • B ∪ R is not a good coloring of A: ∃Y ∈ A Y ⊂ B.

• Y ∩ R2m+1 6= ∅, so b2m+1 ∈ Y ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B2 B4 Y B A

R R1 R3 R5

• Fix A ∈ A. Color X = B ∪ R, choose B0, R1, B2, R2,... from A pick r0 from B0, b1 from R1, . . . s.t.

• B2m ∩ R = B2m ∩ A ∩ R = {r2m}, R2m+1 ∩ B = R2m+1 ∩ A ∩ B = {b2m}. • B ∪ R is not a good coloring of A: ∃Y ∈ A Y ⊂ B.

• Y ∩ R2m+1 6= ∅, so b2m+1 ∈ Y

• {b2m+1 : m ∈ ω}⊂ Y , so Y ∩ A is inﬁnite. Contradiction. ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B2 B4 B A

R R1 R3 R5

• Fix A ∈ A. Color X = B ∪ R, choose B0, R1, B2, R2,... from A pick r0 from B0, b1 from R1, . . . s.t.

• B2m ∩ R = B2m ∩ A ∩ R = {r2m}, R2m+1 ∩ B = R2m+1 ∩ A ∩ B = {b2m}. ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B A

R R1

• B0 and R1 are constructed, r0∈ B0 ∩ A, b1∈ R1 ∩ A ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B A

R R1

• B0 and R1 are constructed, r0∈ B0 ∩ A, b1∈ R1 ∩ A

• R = (R1 \{b1}) ∪{r0} and B = (B0 \{r0}) ∪{b1} ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B A

R R1

• B0 and R1 are constructed, r0∈ B0 ∩ A, b1∈ R1 ∩ A

• R = (R1 \{b1}) ∪{r0} and B = (B0 \{r0}) ∪{b1}

• Need B2∈ A and r2∈ B2 ∩ A. ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B A

D R R1

• B0 and R1 are constructed, r0∈ B0 ∩ A, b1∈ R1 ∩ A

• R = (R1 \{b1}) ∪{r0} and B = (B0 \{r0}) ∪{b1}

• Need B2∈ A and r2∈ B2 ∩ A.

• c = A ∩ (B0 ∪ R1) is ﬁnite, so ∃D ∈ A s.t. D ∩ c = ∅. ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B A

r0 r2

D R R1

• B0 and R1 are constructed, r0∈ B0 ∩ A, b1∈ R1 ∩ A

• R = (R1 \{b1}) ∪{r0} and B = (B0 \{r0}) ∪{b1}

• Need B2∈ A and r2∈ B2 ∩ A.

• c = A ∩ (B0 ∪ R1) is ﬁnite, so ∃D ∈ A s.t. D ∩ c = ∅.

• Pick r2 ∈ D ∩ A. ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B b0 A

r0 r2

D R R1

• B0 and R1 are constructed, r0∈ B0 ∩ A, b1∈ R1 ∩ A

• R = (R1 \{b1}) ∪{r0} and B = (B0 \{r0}) ∪{b1}

• Need B2∈ A and r2∈ B2 ∩ A.

• c = A ∩ (B0 ∪ R1) is ﬁnite, so ∃D ∈ A s.t. D ∩ c = ∅.

• Pick r2 ∈ D ∩ A.

• Pick b0 ∈ B0 \ (A ∪ D). ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B b0 A

r0 r2

D R R1

• r0∈ B0 ∩ A, b1∈ R1 ∩ A, b0∈ B0 \ (A ∪ D), r2 ∈ A ∩ D \ (B0 ∪ R1) ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B b0 A

r0 r2

D R R1

• r0∈ B0 ∩ A, b1∈ R1 ∩ A, b0∈ B0 \ (A ∪ D), r2 ∈ A ∩ D \ (B0 ∪ R1)

• e = (D ∪ A ∪ B0 ∪ R1) \{b0, b1, r2}. ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B2 B b0 A

r0 r2

D R R1

• r0∈ B0 ∩ A, b1∈ R1 ∩ A, b0∈ B0 \ (A ∪ D), r2 ∈ A ∩ D \ (B0 ∪ R1)

• e = (D ∪ A ∪ B0 ∪ R1) \{b0, b1, r2}.

• ∃B2 ∈ A s.t. B2 ∩ e = ∅. ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B2 B b0 A

r0 r2

D R R1

• r0∈ B0 ∩ A, b1∈ R1 ∩ A, b0∈ B0 \ (A ∪ D), r2 ∈ A ∩ D \ (B0 ∪ R1)

• e = (D ∪ A ∪ B0 ∪ R1) \{b0, b1, r2}.

• ∃B2 ∈ A s.t. B2 ∩ e = ∅.

• Then D ∩ B2 = {r2}. R1 ∩ B2 = {b1} ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B2 B b0 A

r0 r2

D R R1

• r0∈ B0 ∩ A, b1∈ R1 ∩ A, b0∈ B0 \ (A ∪ D), r2 ∈ A ∩ D \ (B0 ∪ R1)

• e = (D ∪ A ∪ B0 ∪ R1) \{b0, b1, r2}.

• ∃B2 ∈ A s.t. B2 ∩ e = ∅.

• Then D ∩ B2 = {r2}. R1 ∩ B2 = {b1}

• So A ∩ B2 = {b1, r2} ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B2 B b0 A

r0 r2

D R R1

• r0∈ B0 ∩ A, b1∈ R1 ∩ A, b0∈ B0 \ (A ∪ D), r2 ∈ A ∩ D \ (B0 ∪ R1)

• e = (D ∪ A ∪ B0 ∪ R1) \{b0, b1, r2}.

• ∃B2 ∈ A s.t. B2 ∩ e = ∅.

• Then D ∩ B2 = {r2}. R1 ∩ B2 = {b1}

• So A ∩ B2 = {b1, r2}

• Let R = (R1 \{b1}) ∪{r0, r2} and B = (B0 \{r0}) ∪{b0} ∪ (B2 \{r2}) ≥ω A⊂ X is a.d., A ∩ A′ 6= ∅ for A 6= A′ ∈ A, but χ(A) > 2.

B0 B2 B4 B A

R R1 R3 R5

• Fix A ∈ A. Color X = B ∪ R, choose B0, R1, B2, R2,... from A pick r0 and b0 from B0, r1 and b1 from R1, . . . s.t.

• B2m ∩ R = B2m ∩ A ∩ R = {r2m}, R2m+1 ∩ B = R2m+1 ∩ A ∩ B = {b2m}. Almost disjoint families with large chromatic number Almost disjoint families with large chromatic number

Erdos,˝ Hajnal, 1961: Do there exist almost disjoint systems of countable sets with arbitrarily large chromatic number? Almost disjoint families with large chromatic number

Erdos,˝ Hajnal, 1961: Do there exist almost disjoint systems of countable sets with arbitrarily large chromatic number? Gy. Elekes, Gy Hoffman, 1973: YES. Almost disjoint families with large chromatic number

Erdos,˝ Hajnal, 1961: Do there exist almost disjoint systems of countable sets with arbitrarily large chromatic number? Gy. Elekes, Gy Hoffman, 1973: YES. Theorem (Komjáth) ω ω There is an almost disjoint family A⊂ R which reﬁnes R 1 (i.e. ω for all X ∈ R 1 there is A ∈ A with A ⊂ X), and so χ(A)= 2ω. Almost disjoint families with large chromatic number

ω • Let Q = {P ∈ R : P is dense-in-itself.} Almost disjoint families with large chromatic number

ω • Let Q = {P ∈ R : P is dense-in-itself.} ′ • For each P ∈Q pick xP ∈ P s.t. xP 6= xP′ for P 6= P . Almost disjoint families with large chromatic number

ω • Let Q = {P ∈ R : P is dense-in-itself.} ′ • For each P ∈Q pick xP ∈ P s.t. xP 6= xP′ for P 6= P .

• For each P ∈Q pick AP ⊂ P \{xP} s.t. AP → xP . Almost disjoint families with large chromatic number

ω • Let Q = {P ∈ R : P is dense-in-itself.} ′ • For each P ∈Q pick xP ∈ P s.t. xP 6= xP′ for P 6= P .

• For each P ∈Q pick AP ⊂ P \{xP} s.t. AP → xP .

• Let A = {AP : P ∈ Q}. Almost disjoint families with large chromatic number

ω • Let Q = {P ∈ R : P is dense-in-itself.} ′ • For each P ∈Q pick xP ∈ P s.t. xP 6= xP′ for P 6= P .

• For each P ∈Q pick AP ⊂ P \{xP} s.t. AP → xP .

• Let A = {AP : P ∈ Q}. ω ω • If X ∈ R 1 , then there is a dense-in-itself P ∈ X . Almost disjoint families with large chromatic number

ω • Let Q = {P ∈ R : P is dense-in-itself.} ′ • For each P ∈Q pick xP ∈ P s.t. xP 6= xP′ for P 6= P .

• For each P ∈Q pick AP ⊂ P \{xP} s.t. AP → xP .

• Let A = {AP : P ∈ Q}. ω ω • If X ∈ R 1 , then there is a dense-in-itself P ∈ X .

• So AP ⊂ X. E. W. Miller’s result

ω Question: Which uncountable families A⊂ X have property B?

It is not enough to assume that A is almost disjoint. E. W. Miller’s result

ω Question: Which uncountable families A⊂ X have property B?

It is not enough to assume that A is almost disjoint.

Deﬁnition: A is n-almost-disjoint iff |A ∩ A′| < n for all A 6= A′ ∈ A E. W. Miller’s result

ω Question: Which uncountable families A⊂ X have property B?

It is not enough to assume that A is almost disjoint.

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. E. W. Miller’s result

ω Question: Which uncountable families A⊂ X have property B?

It is not enough to assume that A is almost disjoint.

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.

ω n • The case A⊂ ω is trivial: |A| ≤ |ω | = ω. E. W. Miller’s result

ω Question: Which uncountable families A⊂ X have property B?

It is not enough to assume that A is almost disjoint.

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.

ω n • The case A⊂ ω is trivial: |A| ≤ |ω | = ω. ω • Consider the next special case: A⊂ ω1 (and so |A| = ω1), E. W. Miller’s result

ω Question: Which uncountable families A⊂ X have property B?

It is not enough to assume that A is almost disjoint.

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.

ω n • The case A⊂ ω is trivial: |A| ≤ |ω | = ω. ω • Consider the next special case: A⊂ ω1 (and so |A| = ω1), • The proof is the prototype of the proofs of many positive results. E. W. Miller’s result

ω Question: Which uncountable families A⊂ X have property B?

It is not enough to assume that A is almost disjoint.

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.

ω n • The case A⊂ ω is trivial: |A| ≤ |ω | = ω. ω • Consider the next special case: A⊂ ω1 (and so |A| = ω1), • The proof is the prototype of the proofs of many positive results. • Convention: we say that N is a elementary submodel iff N is an elementary submodel of H(ϑ) for some large enough regular cardinal ϑ ω An n-almost disjoint family A ⊂ ω1 has property B. ω An n-almost disjoint family A ⊂ ω1 has property B.

• A ∈ M1 ≺ M2 ···≺ Mα ≺ Mα+1 ≺···≺hH(κ), ∈,⊳i (α<ω1) ω An n-almost disjoint family A ⊂ ω1 has property B.

• A ∈ M1 ≺ M2 ···≺ Mα ≺ Mα+1 ≺···≺hH(κ), ∈,⊳i (α<ω1) • continuous chain of countable elementary submodels ω An n-almost disjoint family A ⊂ ω1 has property B.

• A ∈ M1 ≺ M2 ···≺ Mα ≺ Mα+1 ≺···≺hH(κ), ∈,⊳i (α<ω1)

• continuous chain of countable elementary submodels (M0 = ∅) ω An n-almost disjoint family A ⊂ ω1 has property B.

• A ∈ M1 ≺ M2 ···≺ Mα ≺ Mα+1 ≺···≺hH(κ), ∈,⊳i (α<ω1)

• continuous chain of countable elementary submodels (M0 = ∅) • Partition A into countable pieces: ∗ A = A , where A = A ∩ (M + \ M ). Sα<ω1 α α α 1 α ω An n-almost disjoint family A ⊂ ω1 has property B.

• A ∈ M1 ≺ M2 ···≺ Mα ≺ Mα+1 ≺···≺hH(κ), ∈,⊳i (α<ω1)

• continuous chain of countable elementary submodels (M0 = ∅) • Partition A into countable pieces: ∗ A = A , where A = A ∩ (M + \ M ). Sα<ω1 α α α 1 α • Key observation: If A ∈ Mα+1 \ Mα then A ⊂ Mα+1 and |A ∩ Mα| < n ω An n-almost disjoint family A ⊂ ω1 has property B.

• A ∈ M1 ≺ M2 ···≺ Mα ≺ Mα+1 ≺···≺hH(κ), ∈,⊳i (α<ω1)

Mα Mα+1 ω An n-almost disjoint family A ⊂ ω1 has property B.

• A ∈ M1 ≺ M2 ···≺ Mα ≺ Mα+1 ≺···≺hH(κ), ∈,⊳i (α<ω1)

X A

Mα Mα+1 ω An n-almost disjoint family A ⊂ ω1 has property B.

• A ∈ M1 ≺ M2 ···≺ Mα ≺ Mα+1 ≺···≺hH(κ), ∈,⊳i (α<ω1)

X A

Mα Mα+1 ω An n-almost disjoint family A ⊂ ω1 has property B.

• A ∈ M1 ≺ M2 ···≺ Mα ≺ Mα+1 ≺···≺hH(κ), ∈,⊳i (α<ω1)

• continuous chain of countable elementary submodels (M0 = ∅) α− • Partition A into countable pieces: ∗ A = A , where A = A ∩ (M + \ M ). Sα<ω1 α α α 1 α • Key observation: If A ∈ Mα+1 \ Mα then A ⊂ Mα+1 and |A ∩ Mα| < n

Aα Aα+1 Mα Mα+1 ω An n-almost disjoint family A ⊂ ω1 has property B.

• A ∈ M1 ≺ M2 ···≺ Mα ≺ Mα+1 ≺···≺hH(κ), ∈,⊳i (α<ω1)

B X

Aα Aα+1 Mα Mα+1 ω An n-almost disjoint family A ⊂ ω1 has property B.

• A ∈ M1 ≺ M2 ···≺ Mα ≺ Mα+1 ≺···≺hH(κ), ∈,⊳i (α<ω1)

X \ Mα+1 B \ Mα

C \ Mα

Aα Aα+1 Mα Mα+1 ω An n-almost disjoint family A ⊂ ω1 has property B.

• A ∈ M1 ≺ M2 ···≺ Mα ≺ Mα+1 ≺···≺hH(κ), ∈,⊳i (α<ω1)

X \ Mα+1 B \ Mα

C \ Mα

Aα Aα+1 Mα Mα+1 cα cα+1 2 2 To be continued . . .