Arch. Math. Logic 44, 499–512 (2005) Digital Object Identifier (DOI): 10.1007/s00153-004-0263-x

Mathematical Logic

Miloˇs S. Kurili´c

Mad families, forcing and the Suslin Hypothesis Received: 26 May 2003 / Revised version: 20 June 2004 Published online: 8 October 2004 – © Springer-Verlag 2004 Abstract. Let κ be a regular cardinal and P a partial ordering preserving the regularity of κ. If P is (κ-Baire and) of density κ, then there is a mad family on κ killed in all generic extensions (if and) only if below each p ∈ P there exists a κ-sized antichain. In this case a mad family A on κ is killed (if and) only if there exists an injection from κ onto a dense subset of Ult(P) mapping the elements of A onto nowhere dense sets. If 2<κ = κ, then in each generic extension of V , in which κ is the minimal cardinal obtaining new subsets, some mad family on κ is killed or an independent subset of κ appears. Also, the κ-Suslin Hypothesis holds iff there exists a mad family A on κ which is killed in each generic extension containing new subsets of κ and preserving P (λ) for λ < κ.

1. Introduction If κ is an infinite cardinal, a family A ⊂ [κ]κ is said to be an almost disjoint family (a.d.f.) on κ if and only if |A ∩ B| < κ, for each different A, B ∈ A. An a.d.f. A is called a maximal almost disjoint family (or a mad family, or a m.a.d.f.) if and only if it is not properly contained in any a.d.f., that is, if for each X ∈ [κ]κ there exists an A ∈ A satisfying |A ∩ X| = κ. The aim of the paper is to investigate stability (and instability) of mad families with respect to forcing. At the beginning we mention some evident facts. Firstly, we will consider mad families on regular cardinals κ which are of size > κ, since κ-sized mad families on κ of size κ do not exist and < κ-sized are not interesting in this context. Secondly, if, for example, A is a mad family on ω1 and if in some generic extension V [G] the cardinal ℵ1 is collapsed to ℵ0 , then in V [G] A is a family of countable subsets of a countable ordinal and it is not an a.d.f. on ω1 any more. Finally, if in some extension, V [G], the cardinal 2κ is collapsed to κ, then each mad family on κ becomes a family of size κ and, hence, loses its maximality. To avoid problems with singular cardinals, we restrict our attention to regular values of κ and to extensions in which the regularity of κ is preserved. In the second section stable mad families are characterized inside the ground model in a combinatorial way. This result is used in the third section where killing of mad families is connected with cardinal properties of the considered forcing notion Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovi´ca 4, 21000 Novi Sad, Serbia and Montenegro e-mail: [email protected] Mathematics Subject Classification (2000): 03E35, 03E40, 03E65 Key words or phrases: Mad families – Forcing – The Suslin Hypothesis

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and where unstable mad families are described topologically. In Section 4, considering κ-Baire orderings of density κ, the relation “p forces A is not a m.a.d.f.” is characterized topologically. For such orderings a necessary and sufficient condition for the existence of unstable mad families is given in Section 5, where a forcing equivalent of the Suslin Hypothesis is obtained as well. In Section 6 killing of mad families is related to the appearance of independent sets. For example, if 2<κ = κ, then in each generic extension in which κ is the minimal cardinal obtaining new subsets, some mad family on κ is killed or an independent subset of κ appears. It is known that there is a mad family on ω which is killed in each generic extension containing new reals. The analogue of this statement for a regular cardinal κ is shown to be equivalent to the κ-Suslin Hypothesis, if 2<κ = κ. The notation used in the paper is mainly standard. So, if P, ≤ is a partial ordering, then the elements p, q ∈ P are called compatible if there exists r ∈ P such that r ≤ p, q. Otherwise, p and q are called incompatible, in notation p ⊥ q. Also, for p ∈ P we use the following notation: p ↓= {q ∈ P : q ≤ p} and p ↑= {q ∈ P : q ≥ p}. A subset A ⊂ P is an antichain if p ⊥ q for each different p, q ∈ A. The minimal cardinal κ such that P does not contain antichains of cardinality κ is denoted by cc(P). The set D ⊂ P is said to be dense if for each p ∈ P there exists q ∈ D such that q ≤ p. The density of P is the cardinal π(P) = min{|D| : D is dense in P}. A set D ⊂ P is said to be open if for each p ∈ D and q ≤ p there holds q ∈ D. Throughout the paper we assume that each considered partial ordering P has a largest element, denoted by 1P . 2. A combinatorial characterization of stability We remind the reader that, if P is a partial order, then a P-name of the form
ˇ × Aα , where Aα , α ∈ κ, are antichains in P, is called a nice name for τ = α∈κ {α} a subset of κ. ˇ By definability of forcing, the sets τp = {α ∈ κ : ∃q ≤ p q
αˇ ∈ τ }, p ∈ P, belong to V . Using elementary
 properties of forcing it is easy to prove that q ≤ p ⇒ τq ⊂ τp and τp = q≤p r≤q τr and that there holds ˇ p ∈ P and A ∈ V , Lemma 1. If τ is a nice name for a subset of κ, ˇ 1P
reg(κ), then (i) p
|Aˇ ∩ τ | = κˇ if and only if |A ∩ τq | = κ for each q ≤ p. (ii) p
|τ | = κˇ if and only if |τq | = κ for each q ≤ p. Theorem 1. Let A be a mad family on a regular cardinal κ and P a partial ordering preserving the regularity of κ. Then ˇ τ , such (a) 1P
“Aˇ is a m.a.d.f.” if and only if for each nice name for a subset of κ, that |τp | = κ, p ∈ P, the set Dτ = {q ∈ P : ∃A ∈ A ∀r ≤ q |A ∩ τr | = κ} is dense in P. (b) If p∗ ∈ P, then p ∗
“Aˇ is a m.a.d.f.” if and only if for each nice name for a subset of κ, ˇ τ , such that |τp | = κ, p ∈ P, the set Dτ is dense below p ∗ . Proof. We prove (a) and the proof of (b) is similar. Let 1P
“Aˇ is a m.a.d.f.”. ˇ κˇ ∃A ∈ Aˇ |A ∩ X| = κ, ˇ so for each P-name σ there Then 1P
∀X ∈ [κ]

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holds 1P
σ ⊂ κˇ ∧ |σ | = κˇ ⇒ ∃A ∈ Aˇ |A ∩ σ | = κ. ˇ If τ is a nice name for a subset of κˇ such that |τp | = κ, p ∈ P, then 1P
τ ⊂ κˇ and, by Lemma ˇ so 1P
∃A ∈ Aˇ |A ∩ τ | = κ. ˇ By Lemma 1 this implies 1, 1P
|τ | = κ, ∀p ∈ P ∃q ≤ p ∃A ∈ A ∀r ≤ q |A ∩ τr | = κ, so the set Dτ is dense in P. Conversely, suppose A is not a m.a.d.f. in some generic extension V [G]. Then, since in V [G] κ remains a cardinal, A is an a.d.f. but there exists X ⊂ κ such that |X|V [G] = κ and |A ∩ X|V [G] < κ, for all A ∈ A. Consequently, if σ is a P-name satisfying X = σG , then there is p∗ ∈ G such that p ∗
σ ⊂ κˇ ∧ |σ | = κˇ ∧ ∀A ∈ Aˇ |A ∩ σ | < κ. ˇ

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Clearly, 1P
∃x(x ⊂ κˇ ∧ |x| = κˇ ∧ (σ ⊂ κˇ ∧ |σ | = κˇ ⇒ x = σ )) (In V [G], if σG ∈ [κ]κ , we put x = σG ; otherwise, x = κ). By the Maximum Principle (see [3], ˇ (ii) 1P
|ρ| = κˇ and (iii) p. 226) there is a name ρ ∈ V P satisfying (i) 1P
ρ ⊂ κ; 1P
σ ⊂ κˇ ∧|σ | = κˇ ⇒ ρ = σ . According to Lemma VII.5.12 of [3], there exists a nice name for a subset of κ, ˇ denoted by τ , such that 1P
ρ ⊂ κˇ ⇒ ρ = τ thus, by (i), we have 1P
ρ = τ and from (ii) it follows 1P
|τ | = κˇ so by Lemma 1 we have |τq | = κ, for all q ∈ P. From (1) and (iii) we obtain p ∗
ρ = σ , ˇ that so p∗
σ = τ . Using (1) again we have p ∗
∀A ∈ Aˇ |A ∩ τ | < κ, is ∀A ∈ A ¬∃q ≤ p ∗ q
|Aˇ ∩ τ | = κˇ which is, by Lemma 1, equivalent to ¬∃q ≤ p ∗ ∃A ∈ A ∀r ≤ q |A ∩ τr | = κ. So, the set Dτ is not dense in P.   3. Necessary conditions Theorem 2. Let A be a mad family on a regular cardinal κ and P a partial ordering preserving the regularity of κ. If 1P
“Aˇ is not a m.a.d.f.”, then below each p ∈ P there exists a κ-sized antichain. Generally, if p∗ ∈ P and p ∗
“Aˇ is not a m.a.d.f.”, then below each p ≤ p ∗ there exists a κ-sized antichain. Proof. Under the assumptions 1P
∃X ∈ [κ] ˇ κˇ ∀A ∈ Aˇ |A ∩ X| < κ, ˇ so, by the Maximum Principle there is a name τ such that 1P
τ ⊂ κˇ ∧ |τ | = κˇ and 1P
∀A ∈ Aˇ ∃α < κˇ sup(A ∩ τ ) = α. Then the sets DA = {q ∈ P : ∃α < κ q
sup(Aˇ ∩ τ ) = α}, ˇ A ∈ A, are dense in P. Let p ∈ P. Suppose that for each A ∈ A the set SA = {α < κ : ∃q ≤ p q
ˇ ) = α} sup(A∩τ ˇ is of size < κ. Then for A ∈ A there is αA < κ such that SA ⊂ αA . Since the set DA is dense below p, the set {q ≤ p : q
Aˇ ∩ τ ⊂ αˇA } is too, thus p
Aˇ ∩ τ ⊂ αˇA . Clearly, τp = {α < κ : ∃q ≤ p q
αˇ ∈ τ } ∈ V and, since 1P
|τ | = κ, ˇ we have |τp | = κ. For α ∈ A ∩ τp there is q ≤ p such that ˇ q
αˇ ∈ A ∩ τ ⊂ αˇA , so α < αA . Thus |A ∩ τp | < κ for all A ∈ A, which is not true, since A is a mad family. So, there is A ∈ A such that |SA | = κ. For each α ∈ SA we pick qα ≤ p satisfying qα
sup(Aˇ ∩ τ ) = α. ˇ Clearly, {qα : α ∈ SA } is a κ-sized antichain below p.   If B is a complete Boolean algebra, by h2 (B) we denote the minimal cardinal λ such that B is not (λ, 2)-distributive.

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Theorem 3. A partial ordering P can kill a mad family on a regular cardinal κ only if h2 (r.o.(P)) ≤ κ < cc(P). Proof. Let a m.a.d.f. A be killed in some generic extension V [G]. Firstly suppose κ < h2 (r.o.(P)). Then in V [G] κ remains a cardinal and hence A is an a.d.f. so, some new subset of κ appears destroying the maximality of A. But this is impossible since P V [G] (κ) = P V (κ), by the (κ, 2)-distributivity of r.o.(P). Suppose κ ≥ cc(P). Then 1P
reg(κ) ˇ and, by Theorem 2, P must contain an antichain of cardinality κ. A contradiction.   A non-empty subset of a partial ordering P is called a filter on P iff (F1) p, q ∈ ⇒ ∃r ∈ r ≤ p, q and (F2) p ≥ q ∈ ⇒ p ∈ . If in addition is not properly contained in any filter on P, then is said to be an ultrafilter on P. The set of all ultrafilters on P will be denoted by Ult(P). A non-empty set C ⊂ P is said to be centered iff each finite subset of C has a lower bound. A centered set C ⊂ P is said to be maximally centered (see [11]) if it is not a proper subset of any centered subset of P. By C(P) we will denote the set of all maximal centered subsets of P. Fact 1. (a) Each filter is a centered family. (b) Each filter is contained in some ultrafilter. (c) Each centered family is contained in some maximally centered. (d) Ult(P) The sets Bp = { ∈ Ult(P) : p ∈ }, p ∈ P, form a base for a T1 -topology on C (P) the set Ult(P). (e) The sets Bp = {C ∈ C(P) : p ∈ C}, p ∈ P, form a subbase for a zero-dimensional topology on the set C(P). (f) The equality Ult(P) = C(P) holds if and only if each centered subset of P is contained in some filter. (In considerations of forcing this equality can be assumed without loss of generality, but it is not
used (and assumed) in the paper). (g) d(Ult(P)) = min{|F| : F ⊂ Ult(P) ∧ P = F}, where d denotes topological density. (h) d(Ult(P)) ≤ π(P). (i) If B is a Boolean algebra, then Ult(B \ {0}) = St(B) (the Stone space of B, which is a compact zero-dimensional space). (j) If P is a reversed tree, then Ult(P) is the set of its branches (maximal chains). Ult(P)

In this paper we will consider the spaces Ult(P) only, so instead of Bp write Bp , whenever confusion is not possible.

we will

Theorem 4. Let A be a mad family on a regular cardinal κ, P a partial order preserving the regularity of κ and π(P) = κ. Then 1P
“Aˇ is not a m.a.d.f.” implies there is a bijection g from κ onto a dense subset of Ult(P) such that the sets g[A], A ∈ A, are nowhere dense. Generally, if p∗ ∈ P, where π(p ∗↓) = κ and p∗
“Aˇ is not a m.a.d.f.”, then there exists a bijection g from κ onto a dense subset of Bp∗ such that the sets g[A], A ∈ A, are nowhere dense. ˇ = Proof. Since 1P
reg κˇ we have 1P
“Aˇ is an a.d.f.”. So, 1P
∃X(X ⊂ κ∧|X| κˇ ∧ ∀A ∈ Aˇ |A ∩ X| < κ) ˇ and by the Maximum Principle (see [3], p. 226) there exists a P-name σ such that 1P
σ ⊂ κˇ ∧ |σ | = κˇ ∧ ∀A ∈ Aˇ |A ∩ σ | < κ. ˇ Like in the proof of Theorem 1 we find a nice name for a subset of κ, ˇ τ , such that the sets τp = {α ∈ κ : ∃q ≤ p q
αˇ ∈ τ }, p ∈ P are of size κ and ∀A ∈ A ∀q ∈ P ∃r ≤ q |A ∩ τr | < κ.

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Claim 1. For each p ∈ P there holds π(p↓) = d(Bp ) = κ. Proof of Claim 1. Let p ∈ P. By Theorem 2, below p there is an antichain of size κ, so κ ≤ π(p ↓) ≤ π(P) and consequently π(p ↓) = κ. Since antichains in p↓ determine cellular families in Bp , we have d(Bp ) ≥ κ. Finally, By Fact 1(h), d(Bp ) ≤ π(p↓) thus d(Bp ) = κ. Claim 1 is proved. Let D ⊂ P be a κ-sized dense set and let D be a κ-sized dense subset of Ult(P). Let  and ≺ be some well-orderings of D and D respectively such that typeD,  = typeD, ≺ = κ. Using recursion on the well-ordered set D,  we define the function f : D → κ × D, where f (p) = ξp , p , as follows. Let p ∈ D and let ξp and p be defined for all p   p, then ξp = min∈ (τp \ {ξp : p  p}), qp = min {q ∈ D : q ≤ p ∧ q
ξˇp ∈ τ }, p = min≺ { ∈ D \ { p : p  p} : qp ∈ }. The definition of qp is correct. Namely ξp ∈ τp , so there is s ≤ p such that s
ξˇp ∈ τ . Since the set D is dense, there exists q ∈ D such that q ≤ s so q
ξˇp ∈ τ and q ≤ p. Also p is well-defined because by Claim 1 we have d(Bqp ) = κ so, since the set D ∩ Bqp is dense in Bqp , there holds |D ∩ Bqp | = κ. Consequently, D ∩ Bqp \ { p : p  p} = ∅. Let F = {ξp , p  : p ∈ D}. Then F ⊂ κ × D and for different p, q ∈ D we have ξp = ξq and p = q . Hence the set S = {ξp : p ∈ D} ⊂ κ is of cardinality κ and F : S → D is an injection. Claim 2. (a) The set F [S] is dense in Ult(P). (b) F [S] ∩ Bp ⊂ F [S ∩ τp ], for all p ∈ P. (c) ∀q ∈ P ∃r ≤ q |F [A ∩ S] ∩ Br | < κ, for each A ∈ A. (d) The sets F [A ∩ S], A ∈ A, are nowhere dense. Proof of Claim 2. (a) We prove that F [S] ∩ Bp = ∅, for each p ∈ P. Since the set D is dense, we can pick p1 ∈ D such that p1 ≤ p. Now qp1 ≤ p1 ≤ p and qp1 ∈ p1 ∈ F [S] and, consequently, p ∈ p1 ∈ F [S] ∩ Bp . (b) Let p ∈ P and ∈ F [S] ∩ Bp . Then, by the definition of F , = p for some p ∈ D and qp ∈ p so there exists r  ∈ p satisfying r  ≤ p, qp . Since qp
ξˇp ∈ τ and r  ≤ qp , we have ξp ∈ τr  ⊂ τp . Thus ξp ∈ S ∩ τp and = p = F (ξp ) ∈ F [S ∩ τp ]. (c) Let A ∈ A and q ∈ P. By (2) there exists r ≤ q such that |A ∩ τr | < κ. Using (b) and the fact that F is an injection we have F [A ∩ S] ∩ Br = F [A ∩ S] ∩ F [S] ∩ Br ⊂ F [A ∩ S] ∩ F [S ∩ τr ] = F [A ∩ S ∩ τr ]. Consequently |F [A ∩ S] ∩ Br | ≤ |A ∩ S ∩ τr | ≤ |A ∩ τr | < κ. (d) Let A ∈ A and X = F [A ∩ S]. Suppose Int X = ∅. Then there exists a basic open set Bs ⊂ Int X. According to (c) there is r ≤ s such that |X ∩ Br | < κ. Now Br ⊂ X. By Theorem 2, below r there is an antichain Z of size κ. Since X ∩ Bz = ∅, for all z ∈ Z, we obtain |X ∩ Br | ≥ κ. A contradiction. So, F [A ∩ S] is a nowhere dense set and Claim 2 is proved.

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So, in the previous construction we obtained the set S ∈ [κ]κ and the injection F : S → D, where D was an arbitrary dense subset of Ult(P). Now we construct the desired injection g : κ → Ult(P). Since A is a m.a.d.f. there exists a set A ∈ A such that |A ∩ S| = κ. By Claim 2 the set N = F [A ∩ S] is a nowhere dense subset of D of cardinality κ. Since removing a nowhere dense set from a dense set leaves a dense set, D = D \ N is a dense subset of Ult(P). Putting D instead of D in the construction given above we obtain a set S  ∈ [κ]κ and an injection F  : S  → D such that F  [S  ] is a dense subset of Ult(P) and the sets F  [A ∩ S  ], A ∈ A, are nowhere dense. Since |N | = κ we can pick an injection ϕ : κ \ S  → N . Finally, let g : κ → D, where g = ϕ ∪ F  . Then F  [S  ] ⊂ g[κ], so g[κ] is a dense subset of Ult(P). Also, for A ∈ A, the set g[A] = ϕ[A \ S  ] ∪ F  [A ∩ S  ] is nowhere dense (because it is the union of two nowhere dense sets).   4. A characterization in κ-Baire orderings of density κ The aim of this section is to prove the following statement. Theorem 5. Let A be a mad family on a regular cardinal κ and P a κ-Baire partial ordering of density κ. Then 1P
“Aˇ is not a m.a.d.f.” if and only if (i) Below each p ∈ P there is a κ-sized antichain; (ii) There exists a bijection g from κ onto a dense subset of Ult(P) such that the sets g[A], A ∈ A, are nowhere dense. Generally, let P be a partial ordering and p ∗ ∈ P, where p ∗↓ is a κ-Baire ordering of density κ. Then p ∗
“Aˇ is not a m.a.d.f.” if and only if (i) Below each p ≤ p∗ there is a κ-sized antichain; (ii) There exists a bijection g from κ onto a dense subset of Bp∗ such that the sets g[A], A ∈ A, are nowhere dense. In proof of Theorem 5 (given in the end of the section) we will use the following notions and assertions. Let κ be a regular cardinal. We will say that a family P ⊂ [κ]κ has the strong κ-intersection property (shortly sκip) if each subset of P of size < κ has intersection of size κ. A set X ∈ [κ]κ is a pseudointersection of the family P if X ⊂∗ P (that is |X \ P | < κ) for every P ∈ P. For example, if A is a mad family on κ of size > κ, then the family P = {κ \ A : A ∈ A} has the sκip and does not have a pseudointersection. So we can define pκ = min{|P| : P ⊂ [κ]κ has the sκip and does not have a pseudointersection}. Then we have Fact 2. If κ is a regular cardinal, then κ < pκ ≤ 2κ . κ Proof.  If P = {Pα : α < κ} ⊂ [κ] is a family having the sκip and if κξα = min( β≤α Pβ \ {ξβ : β < α}), α < κ, then P = {ξα : α < κ} ∈ [κ] is a pseudointersection of the family P. Thus κ < pκ and, obviously, pκ ≤ 2κ .  

Lemma 2. Let κ be a regular cardinal. A partial ordering P is κ-Baire (i.e. the intersection of < κ many open dense subsets of P is dense in P) iff 1P
κˇ ∈ Reg ∧ ∀Y ∈ [ ]<κˇ ∃p ∈ P p ≤ Y.

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(Here = {p, ˇ p : p ∈ P} is the canonical name for a P-generic filter over V and p ≤ Y means: p ≤ q, for every q ∈ Y .) Proof. (⇒) Let the ordering P be κ-Baire and let G be a P-generic filter over V . Firstly, by [3], p. 240, we have κ ∈ RegV [G] . Secondly, suppose λ < κ and Y = {qα : α < λ} ⊂ G. It is easy to show that the sets Dα = {p ∈ P : p ≤ qα ∨ p ⊥ qα }, α < λ, are open and dense in P. Also Dα ∈ V , α < λ, and moreover {Dα : α < λ} ∈ V , since each function from λ to V belongs to V (see [3], p. 240).  By the assumption, the set D = α<λ Dα is dense in P and, clearly, belongs to V . So there exists p ∈ G ∩ D. Now for each α < λ we have p ≤ qα (since p, qα ∈ G) which means that p ≤ Y . (⇐) Let condition (3) hold,λ < κ and let the sets Dα ⊂ P, α < λ, be open and dense. In order to prove D = α<λ Dα is a dense subset of P we pick p ∈ P. Let G be a P-generic filter over V and p ∈ G. If we choose qα ∈ G ∩ Dα , α < λ, then Y = {p} ∪ {qα : α < λ} ∈ [G]<κ so, by the assumption, there is r ∈ P satisfying r ≤ Y . Since the sets Dα are open, r ≤ qα implies r ∈ Dα , hence r ∈ D and r ≤ p. The set D is dense.   If (X, O) is a topological space, by Nd(X) we denote the collection of all nowhere dense subsets of X. Lemma 3. Let P be a partial ordering. Then the set N ⊂ Ult(P) is nowhere dense iff the set N = {q ∈ P : Bq ∩ N = ∅} is dense in P. Proof. (⇒) Let N ⊂ Ult(P) be a nowhere dense set. Suppose there is p ∈ P such that Bq ∩ N = ∅, for each q ≤ p. Then it is easy to show that Bp ⊂ N , so Int N = ∅, which is not true. (⇐) Let N be a dense subset of P and suppose Int N = ∅. Then there exists p ∈ P such that Bp ⊂ N . If q ≤ p and ∈ Bq , then ∈ N so Bq ∩ N = ∅ (since Bq is a neighborhood of ). So q ∈ N , for each q ≤ p. A contradiction.   Proposition 1. Let κ be a regular cardinal and P a κ-Baire partial ordering of density κ such that cc(p↓) > κ, for each p ∈ P. Then (a) d(Ult(P)) = κ; (b) For each κ-sized dense subset D of Ult(P) there holds ˇ κˇ ∀N ∈ (Nd(Ult(P))V )ˇ |X ∩ N | < κ. 1P
∃X ∈ [D] ˇ Proof. (a) Since in P there is an antichain of cardinality κ, the set P can not be covered by < κ ultrafilters, so, by Fact 1(g), d(Ult(P)) ≥ κ. On the other hand, d(Ult(P)) ≤ π(P) = κ, and the equality is proved. (b) In V , let D ⊂ Ult(P) be dense and |D| = κ and let D ⊂ P be dense and |D| = κ. Let G be a P-generic filter over V . Then by Lemma 2 we have κ ∈ RegV [G] . Clearly, the family P = {Bp ∩ D : p ∈ G ∩ D} belongs to V [G] and it is non-empty, since G ∩ D = ∅. By the assumption, below each p ∈ P there is a κ-sized antichain which determines a κ-sized cellular family in Bp , so, by density of D, we have |Bp ∩ D| = κ. Thus P ⊂ [D]κ and we prove that, in V [G], P has the sκip. Let Y ∈ V [G], Y ⊂ P, and |Y| < κ. Then there exists Y ∈ V [G] such that Y ⊂ G ∩ D, |Y | < κ and Y = {Bp ∩ D : p ∈ Y }. Since

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Y ∈ [G]<κ , according to Lemma 2 there exists p ∗ ∈  P satisfying p ∗  ≤ Y. Hence for every p ∈ Y we have Bp∗ ⊂ Bp so Bp∗ ∩D ⊂ p∈Y Bp ∩D = Y. Now |Bp∗ ∩ D| = κ implies | Y| = κ and thus the family P has the sκip. Since |P|V [G] ≤ |D|V [G] = κ < pVκ [G] in V [G] the family P has a pseudointersection, namely there is X ∈ V [G] such that X ⊂ D, |X |V [G] = κ and ∀p ∈ G ∩ D |X \ Bp |V [G] < κ.

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Let N ∈ Nd(Ult(P))V . By Lemma 3 the set N = {q ∈ P : Bq ∩ N = ∅} is dense in P. Clearly N ∈ V so there exists q ∈ G ∩ N . Since the set D is dense below q ∈ G, there is r ∈ G ∩ D, r ≤ q. By (4) we have |X \ Br |V [G] < κ and Br ⊂ Bq implies |X \ Bq |V [G] < κ. Since q ∈ N we have Bq ∩ N = ∅. So,   X ∩ N ⊂ X \ Bq and consequently |X ∩ N |V [G] < κ. Proof of Theorem 5. (⇒) Theorems 2 and 4 give (i) and (ii), even when the assumpˇ tion P is κ-Baire is replaced by the weaker assumption: 1P
reg(κ). (⇐) Let the assumptions hold and suppose (i) and (ii). By Proposition 1(a) we have d(Ult(P)) = κ. Since g[κ] is a dense subset of Ult(P) of size κ, according to Proposition 1(b) there holds ˇ κˇ ∀N ∈ (Nd(Ult(P))V )ˇ |X ∩ N | < κ. 1P
∃X ∈ [g[κ]] ˇ

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Let G be a P-generic filter over V . By (5) there exists X ∈ V [G] such that X ⊂ g[κ], |X |V [G] = κ and ∀N ∈ Nd(Ult(P))V |X ∩ N |V [G] < κ.

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Let A ∈ A. By (ii) we have g[A] ∈ Nd(Ult(P))V and by (6) there holds |X ∩ g[A]| < κ. The set X = g −1 [X ] belonging to V [G] is of size κ (in V [G]) and, since g is an injection, the set g −1 [X ∩ g[A]] = X ∩ A is of size < κ. So X ∈ ([κ]κ )V [G] and |X ∩ A| < κ, for each A ∈ A, thus the a.d.f. A is not maximal in V [G]. The proof of the general part easily follows from the next well known facts. (1) Ult(P) If p∗ ∈ P, then the mapping ϕ : Bp∗ → Ult(p ∗↓) defined by ϕ( ) = ∩ p∗↓, is a homeomorphism. (2) If G ⊂ P is a P-generic filter over V and p∗ ∈ G, then G ∩ p∗ ↓ is a p ∗ ↓-generic filter over V and Vp∗↓ [G ∩ p∗ ↓] = VP [G]. Then we generalize Proposition 1 and continue as above.   5. The existence of unstable mad families for κ-Baire orderings of density κ Theorem 6. Let κ be a regular cardinal and P a κ-Baire partial ordering of density κ. Then (a) There exists a m.a.d.f. A on κ which is killed in each generic extension if and only if below each p ∈ P there is a κ-sized antichain. (b) There exists a m.a.d.f. A on κ which is killed in some generic extension if and only if there exists p ∗ ∈ P such that below each p ≤ p∗ there is a κ-sized antichain.

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Proof. We prove (a) and the proof of (b) is similar. The implication (⇒) is Theorem 2. For the proof of (⇐) suppose cc(p↓) > κ, for all p ∈ P. Claim 1. Each κ-sized subset of Ult(P) has a nowhere dense κ-sized subset. Proof of Claim 1. Let X ∈ [Ult(P)]κ . If X is a nowhere dense set, the proof is over. Otherwise, by Lemma 3, there is p ∈ P such that Bq ∩ X = ∅, for all q ≤ p. Let A be a κ-sized antichain below p and let us pick a ∈ Ba ∩ X , for every a ∈ A. Then, by Lemma 3 and since cc(q ↓) > κ, for all q ≤ p, the set N = { a : a ∈ A} ∈ [X ]κ is nowhere dense. Claim 1 is proved. By Proposition 1, d(Ult(P)) = κ. Let D ∈ [Ult(P)]κ be dense. According to Claim 1 the set  = Nd(Ult(P)) ∩ [D]κ is an open and dense set in the partial ordering [D]κ , ⊂ so it contains a maximal antichain A∗ of [D]κ , ⊂, which is an adf on D. By Claim 1, A∗ is a mad family on D. Let g : κ → D be a bijection and A = {g −1 [A] : A ∈ A∗ }. Then conditions (i) and (ii) of Theorem 5 are satisfied so 1P
“Aˇ is not a m.a.d.f.”.   We will say that a partial ordering P is ramified iff below each p ∈ P there are two incompatible elements of P. This condition is natural in considerations of forcing since it ensures non-triviality of generic extensions (that is V [G] = V ). Now we have Corollary 1. Let P be a ramified partial ordering of density ω. Then (a) There exists a mad family on ω which is killed in each generic extension. (b) For each mad family A on ω there holds: 1P
“Aˇ is not a m.a.d.f.” if and only if there exists a bijection g from ω onto a dense subset of Ult(P) such that the sets g[A], A ∈ A, are nowhere dense. Proof. Clearly, each partial order is ω-Baire. Since P is ramified, below every p ∈ P we easily construct an infinite antichain. So, (a) follows from Theorem 6 and (b) from Theorem 5.   Remark. It is well known (see e.g. [9], p. 71) that each ramified partial ordering of density ℵ0 is forcing-equivalent to the partial ordering which adds a single Cohen real. So, Corollary 1 is in fact a statement about the Cohen forcing. In [5], Theorem 2, it is shown that a mad family A on ω is killed by the Cohen forcing if and only if there exists a bijection H : ω → Q such that the sets H [A], A ∈ A, are nowhere dense in R. By Corollary 1, in this characterization the real line can be replaced by the Cantor cube 2ω or by the Stone dual of the reduced Borel algebra. By (a) of Corollary 1 Cohen-unstable mad families on ω exist in ZFC, but it is an open question whether the existence of Cohen-stable mad families on ω is a theorem of ZFC. We remark that, by [5], if b = c or a < cov(M), where M is the meager ideal, then Cohen-stable mad families exist. Also, we note that the Cohen forcing Fn(κ, 2) kills mad families on ω only, since it is ccc and Theorem 3 holds. Concerning (a) of Corollary 1 we remark that the following stronger result is known (see [3], p. 289). This result is a consequence of Theorem 8 given in the next section.

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Fact 3. There exists a mad family on ω which is killed in each generic extension containing new subsets of ω. Must each κ-Baire partial ordering of density κ kill some m.a.d.f. (on κ)? Of course not, since forcing by the order P (κ) \ {∅}, ⊂ does not produce new sets at all. But the following modification of the question seems to be interesting: Does each ramified κ-Baire partial ordering of density κ kill some m.a.d.f.? By Corollary 1, for κ = ℵ0 the answer is “Yes”. For κ = ℵ1 the system ZFC does not provide the answer. Namely there holds Theorem 7. The following conditions are equivalent: (a) SH (i.e. there are no Suslin trees) (b) Each ramified ℵ1 -Baire partial ordering of density ℵ1 kills some mad family in some extension. (c) Each ramified partial ordering of density ℵ1 kills some mad family in some extension. Proof. (a ⇒ b) Suppose there exists a ramified, ℵ1 -Baire partial ordering P of density ℵ1 which does not kill mad families on ω1 . Then, by Theorem 6, there exists q ∈ P such that q ↓ is a ccc partial ordering. Clearly the ordering q ↓ is ℵ1 -Baire and, consequently, B = r.o.(q↓) is a complete, atomless (q↓ is ramified), ℵ0 -distributive ccc Boolean algebra, namely a Suslin algebra, which exists iff the Suslin Hypothesis fails (see [2], p. 220). (b ⇒ c) Let condition (b) hold and let P be a ramified partial ordering of density ℵ1 . Suppose P preserves all mad families. Then, by Fact 3, forcing by P does not produce new subsets of countable sets belonging to V . Consequently, P does not collapse ℵ1 (since otherwise a witness of collapse, a new subset of ω, would appear). By (b) the ordering P is not κ-Baire, so there exists a generic extension V [G] containing a new function f : ω → V . Then there exists a set B ∈ V such that f : ω → B (see [3], p. 240). Since P is ℵ2 -cc, by Lemma 6.8 of [3] in V there exists a function
F : ω → [B]≤ℵ1 such that f (n) ∈ F (n), for all n ∈ ω. Then ran(f ) ⊂ S = n∈ω F (n) ∈ V , that is f : ω → S and |S| ≤ ℵ1 . Since countable sets do not obtain new subsets, we have |S| = ℵ1 . Let ϕ : S → ω1 be a bijection belonging to V . The function h = ϕ ◦ f : ω → ω1 does not belong to V (since f = ϕ −1 ◦ h) so h[ω] is an unbounded subset of ω1 and, consequently, ℵ1 is collapsed in V [G]. A contradiction. (c ⇒ a) Suppose ¬ SH. Then there is a normal Suslin tree T and, if P denotes T with the reversed order, then P is a ramified partial ordering of density ℵ1 . Since h2 (r.o.(P)) = cc(P) = ℵ1 , according to Theorem 3 forcing by P does not kill any m.a.d.f.   The results of this section were obtained under the assumption π(P) = κ, which will be omitted in the sequel.

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6. Independent sets and killing mad families Let P be a partial ordering, κ a regular cardinal and V [G] a generic extension of V by P. A subset X of κ belonging to V [G] is called independent if for each A ∈ [κ]κ ∩ V there holds A ∩ X = ∅ and A \ X = ∅ (see [6]). We note that an independent subset of κ kills all bases of uniform non-principal ultrafilters on κ belonging to the ground model. In consideration of new sets of ordinals produced by forcing the following questions seem natural: 1) Must a cardinal κ obtain an independent subset, if some mad family on κ is killed in V [G]? 2) Must some mad family on κ be killed, if V [G] contains an independent subset of κ? 3) Must forcing kill a mad family on κ or produce an independent subset of κ, whenever κ is the minimal cardinal obtaining new subsets? The answer to the first two questions is ‘No’. Namely, if P is the Sacks forcing (see [8]) or Miller’s rational perfect set forcing, then, by Fact 3, some mad families are killed in each extension, but independent subsets of ω do not appear (see [1] and [7]). On the other hand, if P = Fn(ω1 , 2) (adding ℵ1 -many Cohen reals), then in each extension V [G] ω1 obtains independent sets (for example X = {α ∈ ω1 :
α, 1 ∈ G}), but P does not kill mad families on ω1 , since cc(P) = ℵ1 . A similar situation appears if Suslin trees exist (see Lemmas 4, 6 and Theorem 3). The answer to the third question is ‘Yes’ if, for example, the GCH holds. In order to show it, we firstly list some definitions. Let κ be a regular cardinal. A tree T is called a κ-Suslin tree if it is of size κ, but both chains and antichains of T are of size < κ. In the sequel we will work with reversed trees and ≤ will denote the reversed order. A reversed κ-Suslin tree T will be called normal (see [10], p. 243) if and only if (i) T has a single root (that is | Lev0 (T )| = 1); (ii) Nodes of limit height are singletons (here, like in [10], the node of the tree is the set of points having the same set of predecessors); (iii) If ht(t) < β < κ, then |t ↓ ∩ Levβ (T )| > 1; T will be called good if it satisfies (i) and (iii) and T will be called well-pruned (see [3], p. 71) if it satisfies (i) and (iv) If ht(t) < β < κ, then |t↓ ∩ Levβ (T )| > 0. Theorem 8. Let κ be a regular cardinal satisfying 2<κ = κ. Then there exists a mad family A on κ such that for each generic extension VP [G] in which κ is the minimal cardinal obtaining new subsets, there holds: A is killed in VP [G] or in V there exists a good κ-Suslin tree which in VP [G] obtains a cofinal branch. In the latter case some independent subset of κ appears. In the proof of the theorem we will use the following three lemmas. Lemma 4. Let κ be a regular cardinal. Then each reversed κ-Suslin tree has a subset which is a good reversed κ-Suslin tree. Proof. By Lemma II 5.11 of [3], without loss of generality we assume T is a well-pruned reversed κ-Suslin tree (i.e. T satisfies (i) and (iv)). Using a theorem of

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Kurepa from [4] (see [10], p. 242) we conclude that for each t ∈ T there exists an ordinal α(t) such that ht(t) < α(t) < κ and |t↓ ∩ Levα(t) (T )| ≥ ℵ0 . As in Proposition 2.8 of [10] we consider the set C = {δ ∈ Lim ∩κ : ∀t ∈ T
 δ α(t) < δ} ∪ {0} and easily prove C is a club subset of κ. Clearly T  C = δ∈C Levδ (T ) is a reversed κ-Suslin tree satisfying (i). For the proof of (iii) we suppose t ∈ T  C, htT (t) = δ and δ1 ∈ C, where δ < δ1 . Then t ∈ T  δ1 so α(t) < δ1 . Since |t↓ ∩ Levα(t) (T )| ≥ ℵ0 and since T satisfies (iv), we have |t ↓ ∩ Levδ1 (T )| ≥ ℵ0 , so (iii) holds.   Lemma 5. Let κ be a regular cardinal, T a reversed κ-Suslin tree and VP [G] a generic extension in which T obtains a cofinal branch H . Then H is a T -generic filter over V and VT [H ] ⊂ VP [G]. Proof. For genericity it is sufficient to prove that H intersects each maximal antichain A of T belonging to V . Since |A| < κ there exists α < κ such that A ⊂ T  α. Then t ∈ H ∩ Levα (T ) is compatible (that is comparable) with some a ∈ A, i.e. t ≤ a, and consequently a ∈ H ∩ A. Since H ∈ VP [G] we have VT [H ] ⊂ VP [G],   by the minimality of VT [H ]. Lemma 6. Let κ be a regular cardinal. Then forcing by every good reversed κ-Suslin tree T preserves the regularity of κ and produces independent subsets of κ in each generic extension. Proof. It is known (see [3], p. 249) that each well-pruned and, consequently, each good reversed κ-Suslin tree is κ-Baire (the intersection of < κ dense open subsets is dense) so, κ remains a regular cardinal in generic extensions by such trees. By [s] we denote the node
of the element s ∈ T . Let ϕ be a choice function for the family of nodes {[s] : s ∈ α<κ Levα+1 (T )} and let W = ran ϕ. By condition (iii) these nodes are of size > 1. Let G be a T -generic filter over V . We will prove that the set X = {α < κ : G ∩ Levα+1 (T ) ⊂ W } is an independent subset of κ. Let A ∈ ([κ]κ )V . Clearly, the set D = {t ∈ T : ∃α ∈ A t ∈ Levα+1 (T ) ∩ W } is dense in T . Now, for t ∈ G ∩ D there is α ∈ A such that t ∈ Levα+1 (T ) ∩ W , so G ∩ Levα+1 (T ) = {t} ⊂ W and hence α ∈ A ∩ X. Also, by condition (iii) the set  = {t ∈ T : ∃α ∈ A t ∈ Levα+1 (T ) \ W } is dense in T which implies A \ X = ∅.   Proof of Theorem 8. Instead of working on κ we define a mad family on the κ-sized set <κ 2 ordered reversely. Let  be the set of all almost disjoint families whose members are chains or antichains or good reversed κ-Suslin subtrees of the tree <κ 2. Then, by Zorn’s lemma, the poset , ⊂ has a maximal element, say A. If a set S ⊂ [<κ 2]κ contains a κ-sized chain or antichain, then |A ∩ S| = κ, for some A ∈ A. Otherwise |S ∩ α 2| < κ for all α < κ so, since κ is a regular cardinal, S is a subtree of the tree <κ 2 of height κ, moreover, a κ-Suslin tree. By Lemma 4 S contains a good κ-Suslin tree T , so |A∩S| = κ for some A ∈ A, by the maximality of A. Thus A is a mad family on <κ 2. Let κ be the minimal cardinal which in VP [G] obtains new subsets. Then κ is a cardinal in VP [G] (otherwise a witness of collapse would appear on some λ < κ).

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Let Y ∈ VP [G] be a new subset of κ. If f : κ → 2 is the characteristic function of Y , then the set Z = {f  α : α < κ} is a new branch of the tree (<κ 2)V [G] = (<κ 2)V . Suppose A is not killed by Z so, in particular, there exists a set A ∈ A such that |A ∩ Z|V [G] = κ. Clearly, A is not an antichain of
<κ 2. If A would be a chain, then we would have A ⊂ Z and consequently f = A ∈ V , which is not true. So A is a good reversed κ-Suslin tree (in V ). Since H = A ∩ Z is of size κ, it is a cofinal branch of A. By Lemma 5 H is an A-generic filter over V and VA [H ] ⊂ VP [G]. By Lemma 6 forcing by A produces independent subsets of κ, so they appear in VP [G].   Corollary 2. Let κ be a regular cardinal satisfying 2<κ = κ. Then in each generic extension, in which κ is the minimal cardinal obtaining new subsets, some mad family on κ is killed or an independent subset of κ appears. Corollary 3. Let κ be a regular cardinal satisfying 2<κ = κ. Then SHκ ⇔ There exists a mad family A on κ which is killed in each generic extension in which κ is the minimal cardinal obtaining new subsets. Proof. The direction ⇒ follows from the previous theorem. For ⇐ suppose ¬ SHκ . Then there exists a normal reversed κ-Suslin tree P. P is κ-Baire (see e.g. [3], p. 249) so forcing by P does not produce new subsets of λ < κ and clearly produces new subsets of κ. But P preserves mad families, since h2 (r.o.(P)) = cc(P) = κ and Theorem 3 holds.   Acknowledgements. The author would like to express his gratitude to the referee for constructive suggestions which provided more elegant proofs and improved the contents of the paper. Research supported by the MNTRS (Project 1768: Forcing, Model Theory and Set-theoretic Topology).

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[10] Todorˇcevi´c, S.: Trees and linearly ordered sets. In: K. Kunen, J.E. Vaughan (eds.), Handbook of Set-Theoretic Topology, Elsevier Science Publishers B.V., Amsterdam, 1984, pp. 235–293 [11] Weiss, W.: Versions of Martin’s Axiom. In: K. Kunen, J.E. Vaughan (eds.), Handbook of Set-Theoretic Topology, Elsevier Science Publishers B.V., Amsterdam, 1984, pp. 827–886