Math. Log. Quart. 48 (2002) 2, 189 – 193

Mathematical LogicQuarterly

c© WILEY-VCH Verlag Berlin GmbH 2002

Adjoining Almost Disjoint Permutations

Yi Zhang1)

Department of Mathematics, University of Helsinki,00014 University of Helsinki, Finland2)

Dedicate to Natalia Lessiv

Abstract. We show that it is consistent with ZFC + ¬CH that there is a maximal almostdisjoint permutation family A ⊆ Sym(N) such that A is a proper subset of an eventually

different family E ⊆ NN and |A| < |E|. We also ask several questions in this area.

Mathematics Subject Classification: 03E35, 20A15, 20B07, 20B35.

Keywords: almost disjoint families; eventually different reals; forcing.

1 Introduction

We consider two kinds of mathematical structures in this paper: almost disjointpermutation families in Sym(N) and eventually different families of functions in N

N.We say that two permutations f, g ∈ Sym(N) are almost disjoint (a. d.) with each

other if and only if |f ∩ g | < ω, that is, if {n ∈ N : f(n) = g(n)} is finite. An almostdisjoint permutation family A ⊆ Sym(N) is a subset of Sym(N) such that f, g area. d. with each other for any f, g ∈ A.

FollowingE. van Douwen and A. Miller (see, e. g., [3], [6] and [7]), we say thattwo functions f, g ∈ N

N are eventually different (e. d.) with each other if and only if|f ∩ g | < ω. An eventually different family E ⊆ N

N is a subset of NN such that f, g

are e. d. with each other for any f, g ∈ E.There are several common results about this two families. For example:

(1) Any maximal almost disjoint (m. a. d.) permutation family in Sym(N) has sizeat least the cardinality of the smallest non-meager set of reals, and the same does anymaximal eventually different (m. e. d.) families in N

N (see [1] for details).(2) If we adjoin κ Cohen reals to a ground model of ZFC+GCH, then(2a) there exists a m. a. d. permutation family A ⊆ Sym(N) such that |A| = ℵ1 (see,

e. g., [10]);(2b) there exists a m. e. d. family in E ⊆ N

N such that |E| = ℵ1 (see, e. g., [9, 10]).

. . . etc.1)The research is partially supported by a visiting grant from the Institute Mittag-Leffler, Royal

Academy of Science, Sweden.2)e-mail: zhang@mat-92.ps.helsinki.fi

c© WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2002 0942-5616/02/0205-0189 $17.50+.50/0

190 Yi Zhang

N o t e . For more aspects about these two families, the reader can consult [1, 8, 9].This paper itself is a continuation of them.

In this paper, we are more interested in finding the difference between thesetwo structures. Of course, it is easily seen that any m. a. d. permutation familyA ⊆ Sym(N) is a proper subset of a m. e. d. family in N

N. Here we are interestedto know the answer of the following question:

Q u e s t i o n 1.1. Is it consistent that there exists a m. a. d. permutation familyA ⊆ Sym(N) such that A is a proper subset of a m. e. d. family E ⊆ N

N and |A| < |E| ?We shall give an “yes” answer to this question in Section 2. In Section 3, we shall

ask several open questions in this area.The set-theoretical notation that we use in this paper will follow [4] or [5]. Thus

if P is a notion of forcing and p, q ∈ P, then q ≤ p means that q is a strengtheningof p. M always denotes a countable transitive model of ZFC.

2 Adjoining almost disjoint permutations

In this section, we construct a forcing model which can positively answer Question 1.1.The forcing p. o. set which we will use to construct our model is given in Definition 2.5.We first prove several technical results (Lemma 2.2, and Corollary 2.4) which will beused later on.

D e f i n i t i o n 2.1. For any f ∈ NN and n ∈ N let Uf

n = {k ∈ Ev : f(k) = n},where Ev denotes the set of all even natural numbers.

L emma 2.2 (MA(κ)). Let E ⊆ NN with |E| ≤ κ < 2ℵ0 such that, for any

f ∈ E, either f ∈ Sym(N), or Ufn is infinite for any fixed n ∈ N. Then there exists a

g ∈ NN \E such that

(1) g ∩ f is finite for any f ∈ E,(2) Ug

n is infinite for any n ∈ N.P r o o f . For any E ⊆ N

N, let EE be the partial order which consists of allconditions of the form 〈s, F 〉 such that (a) s is a finite partial function from N into N

and (b) F is a finite subset of E, and where〈s2, F2〉 ≤ 〈s1 , F1〉 iff s2 ⊆ s1, F2 ⊆ F1, and f ∩ s1 ⊆ s2 for all f ∈ F2.

The p. o. set EE was first introduced by A. Miller in [6], and in [8] and [9] severalaspects of this p. o. set had been studied. For example, it is easily seen that EE fulfilsthe c. c. c. Moreover, it can be easily proved that the following sets are dense in EE :

An = {〈s, F 〉 ∈ EE : n ∈ dom(s) and n ∈ ran(s)}, where n ∈ N,

Bf = {〈s, F 〉 ∈ EE : f ∈ F }, where f ∈ E.

Thus EE adjoins a function g =⋃{s : 〈s, F 〉 ∈ G}, where G is a filter in EE , such

that g is e. d. from any f ∈ E.Now, for n, m ∈ N let

Cnm = {〈s, F 〉 ∈ EE : there exists k ≥ m such that k ∈ Ev and s(k) = n}.

By the assumption about E we obtain that Cnm is dense in EE for any m, n ∈ N. So

we know that the EE-generic function g satisfies condition (2), i. e., Ugn is infinite for

any n ∈ N.

Adjoining Almost Disjoint Permutations 191

Thus let D = {An : n ∈ N} ∪ {Bf : f ∈ E} ∪ {Cnm : m, n ∈ N}. Since |D| ≤ κ,

by MA(κ) there is a generic filter G∗ in EE such that G∗ ∩ d �= ∅ for any d ∈ D.Therefore we get that g∗ =

⋃{s : 〈s, F 〉 ∈ G∗} satisfies (1) and (2). ✷

The following result was proved by P. Neumann (see e. g. [2, Proposition 10.4]for details).

T h e o r em 2.3. There exists an a. d. permutation group G ⊆ Sym(N) such that|G| = 2ℵ0 .

Let ℵ1 ≤ κ < 2ℵ0. By Theorem 2.3 we know that there exists an almost disjointgroup G ⊆ Sym(N) with the cardinality κ. It is easily seen that G is also an a. d. familyin Sym(N). Thus by repeatedly applying of MA, we can prove the following statement:

C o r o l l a r y 2.4 (MA + ¬CH). Assume that ℵ1 ≤ κ < 2ℵ0 . Then there exist ana. d. family A ⊆ Sym(N) and an e. d. family E ⊆ N

N such that(1) A ⊆ E with κ = |A| < |E| ≤ 2ℵ0 ,(2) Uf

n is infinite for any f ∈ E \ A and any n ∈ N.De f i n i t i o n 2.5. Let A ⊆ Sym(N) be an a. d. family and let E ⊆ N

N be ane. d. family such that A ⊆ E and A �= E. Then the p. o. set PA,E consists of allconditions of the form 〈s, FA, FE〉 such that (i) s is a one-to-one partial function fromN into N, (ii) FA is a finite subset of A, and (iii) FE is a finite subset of E \ A,and where the relation 〈s2, FA,2, FE,2〉 ≤ 〈s1, FA,1, FE,1〉 is defined by (a) s1 ⊆ s2,FA,1 ⊆ FA,2, and FE,1 ⊆ FE,2, (b) s2 ∩ f ⊆ s1 for any f ∈ FA,1, and (c) s2 ∩ f ⊆ s1

for any f ∈ FE,1.Notice that since 〈s1 , FA,1, FE,1〉 and 〈s2 , FA,2, FE,2〉 are compatible whenever

s1 = s2, PA,E fulfils the c. c. c.L emma 2.6. Let G be PA,E-generic over M . Then M [G] contains a permutation

g ∈ Sym(N) such that(1) A ∪ {g} is an a. d. family in Sym(N);(2) E ∪ {g} is an e. d. family in N

N.P r o o f . It is easily seen that the following subsets of PA,E are dense:

Bn = {〈s, FA, FE〉 ∈ PA,E : n ∈ dom(s)}, where n ∈ N,Cn = {〈s, FA, FE〉 ∈ PA,E : n ∈ ran(s)}, where n ∈ N,Dg = {〈s, FA, FE〉 ∈ PA,E : g ∈ FA}, where g ∈ A,Ef = {〈s, FA, FE〉 ∈ PA,E : f ∈ FE}, where f ∈ E.

If G is PA,E -generic over M , then let g =⋃{s : 〈s, FA, FE〉 ∈ G}. Then g satisfies

(1) and (2) by construction. ✷

L emma 2.7. Let A be an a. d. family in Sym(N), let E ⊆ NN be an e. d. family

such that A ⊆ E and Ufn is infinite for any f ∈ E \ A and for any n ∈ N. If

f ∈ Sym(N)\ (A ∪ E), A ∪ {f} is an a. d. family in Sym(N), and g is a PA,E-genericpermutation, then |f ∩ g | = ω.

P r o o f . Consider the following subset of PA,E :D′

n = {〈s, FA, FE〉 ∈ PA,E : there exists m ≥ n such that f(m) = s(m)}for any n ∈ N. We can prove that D′

n is a dense subset of PA,E . Thus, if g is aPA,E-generic permutation, then |f ∩ g | = ω. ✷

192 Yi Zhang

Th e o r em 2.8. Let M � (ZFC + MA + ¬CH). Let κ, λ ∈ M be cardinals suchthat ℵ1 ≤ κ < 2ℵ0 = λ. Then there exists a c. c. c. notion of forcing P such that thefollowing statements hold in MP:

(1) 2ℵ0 = λ.

(2) There exists a m. a. d. family A ⊆ Sym(N) of cardinality κ.

(3) There exists an e. d. family E ⊆ NN such that A ⊆ E and |A| < |E|.

P r o o f . Since M � (ZFC +MA+ ¬CH), there exist an a. d. family A ⊆ Sym(N)and an e. d. family E ⊆ N

N such that (a) A ⊆ E with κ = |A| < |E| ≤ 2ℵ0 and (b)Uf

n is infinite for any f ∈ E \ A and for any n ∈ N. Now we define a finite supportiteration forcing P of length ω1 as follows: At the 0th step, we take G0 = G andE0 = E. At step α, we assume that we have constructed an a. d. permutation familyAα ⊆ Sym(N) and an e. d. familyEα = Aα ∪E. At this step, we use the forcing notionPα = PAα ,Eα . By Lemma 2.6, we get a new gα ∈ Sym(N) such that (i) gα /∈ Eα,(ii) Aα+1 = Aα ∪ {gα} is an a. d. family in Sym(N), and (iii) Eα+1 = Aα+1 ∪ E ⊆ N

N

is an e. d. family. Since Pα is a c. c. c. forcing, our iterated forcing is c. c. c.

For each Aα there exists a bijection from Aα onto κ + α, and for each Eα \ Aα

there exists a bijection from Eα \ Aα to E∗ = {β ∈ On : κ + α ≤ β < λ}. We cantake Pα consisting of all triples 〈s, FAα , FEα〉, where s is a finite one-to-one partialfunction from N into N, FAα is a finite subset of κ+ α and let η ∈ FAα stand for thecorresponding permutation, and FEα is a finite subset of E∗ and let η ∈ FEα standfor the corresponding function in N

N. Thus each Pα consists of a set in M (while itspartial order is not necessarily in M), and the cardinality of Pα is λ. Hence |P| = λ.Since P is a c. c. c. forcing, P preserves cardinals. Thus 2ℵ0 is the same cardinal inM [H ] and M , where H is P-generic over M .

We claim that Aℵ1 is a m. a. d. permutation family in Sym(N).

Assume not. Let f ∈ Sym(N) \ Aℵ1 in M [H ] be such that Aℵ1 ∪ {f} is ana. d. permutation family. Let f be a nice name of f . For each 〈n, m〉 ∈ N × N thereexists a maximal antichain A〈n,m〉 of P which decides whether f(n) = m. Since P isc. c. c., A〈n,m〉 is countable. Let A =

⋃〈n,m〉∈f A〈n,m〉. Then A is countable. Since P

is an ω1-length forcing with finite support, it is easily seen that there exists an α < ω1

such that supp(p) ⊂ α for any p ∈ A. If Hα is the component of H in the iteratedforcing up to (but not including) α, then we have f ∈ M [Hα]. If Aα ∪ {f} is ana. d. permutation family in Sym(N), then Lemma 2.7 implies that |f ∩ gα| = ℵ0. Weget a contradiction. Thus Aℵ1 is a m. a. d. permutation family in M [H ]. ✷

We conclude that it is consistent with ZFC+¬CH that there exists a m. a. d. per-mutation family A ⊆ Sym(N) which is a proper subset of an a. d. family E ⊆ N

N suchthat |A| < |E| ≤ 2ℵ0 .

3 Some open problems

P r ob l em 3.1. Let ae be the least λ such that there exists a m. e. d. family E ⊆ NN

with |E| = λ. Let ap be the least λ such that there exists a m. a. d. permutation familyA ⊆ Sym(N) with |A| = λ. Can we prove the consistency of ae �= ap ?

Adjoining Almost Disjoint Permutations 193

P r o b l em 3.2. Let a be the least λ such that there exists a m. a. d. family F ⊆ ℘(N)with |F | = λ. Then we can prove the consistency of a < ae, ap (see e. g. [9] and [1]).Can we prove the consistency of ae, ap < a?

P r o b l em 3.3. Is there any cardinal invariants which is the upper bound of ae

and ap ?

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(Received: January 4, 2001; Revised: March 19, 2001)