Download 1. Almost Disjoint Families We Study

Bulletin of the Section of Logic
Volume 30/1 (2001), pp. 1–13
Yi Zhang
REMARKS ON A CLASS OF ALMOST DISJOINT
FAMILIES ∗
Abstract
We study the relationship between several closely related m.a.d. families in several forcing models. We also prove some ZFC result concerning the lower bound
of these m.a.d. families, which answers a question suggested by Stevo Todorcevic.
In the last section, we state several open problems in this area.
1. Almost Disjoint Families We Study
Studying the concept of almost disjoint (a.d.) families can be traced back
to Hausdorff in his [6], P. Erdös, A. Hajnal [5] and A. Tarski [13] also
investigated in this area. After P. Cohen invented the forcing method, almost disjoint family becomes a central concept in combinatorial set theory.
There are not only many interesting results but also many open problems
in this area (see, e.g. [14] or [12]).
In this paper, we shall mainly study the relationship between several
closely related maximal almost disjoint (m.a.d.) families. The set theoretical notations we use in this paper are standard, see, e.g. [8]. Thus if P is a
notion of forcing and p, q ∈ P, then q ≤ p means that q is a strengthening
of p. M always denotes a countable transitive model of ZF C.
Let me first introduce two well-known m.a.d. families in set theory.
∗ 1991
Mathematics Subject Classification 03E35, 20A15
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Yi Zhang
Definition 1.1. If x, y are two infinite sets, x and y are almost disjoint
iff |x ∩ y| is finite. An a.d. family is an A ⊂ ℘(N) such that for any
x ∈ A, |x| = ℵ0 and any two distinct elements of A are a.d.. Let a be the
least λ such that there exists an infinite m.a.d. family F ⊆ ℘(N) of size λ.
Definition 1.2. We say that F ⊆ N × N is a m.a.d. family of graphs iff
(1) for any f ∈ F , f is a graph of partial function from N to N; and
(2) for any f, g ∈ F , |f ∩ g| is finite; and
(3) for any g ∈ N × N \ F , which is a graph of a partial function, there
exists a f ∈ F such that |f ∩ g| is infinite.
Let as be the least λ such that there exists a m.a.d. family F of graphs of
partial functions from N to N with |F| = λ.
Note. Both as and a are well-known cardinal invariants. There are lots
of parallel results can be proved for them (e.g. see [4], and [14], etc.). Here
we are interested in the difference between the two families. It is easily
seen that a ≤ as . In [11], by a very complicated forcing argument, Shelah
proved that it is consistent with ZFC that
a = ℵ1 < as = ℵ2 = 2ℵ0 .
Here we prove a slightly stronger result by a much easier argument as
follows.
Theorem 1.3. It is consistent with ZFC that a = ℵ1 < as = κ = 2ℵ0
where κ is an uncountable cardinal such that cof(κ) > ℵ0
Proof. Let M |= (ZF C + GCH). Let E be the partially ordered set
(p.o.set) consists of all conditions hs, F i such that
• s is a finite partial function from N to N; and
• F is a finite subset of N N,
where hs1 , F1 i ≤ hs2 , F2 i iff
(s2 ⊆ s1 ) and (F2 ⊆ F1 ) and ∀s ∈ F2 (f ∩ s1 ⊆ s2 ).
It is easily seen that E has the c.c.c.
We proceed with a system of finite support iterated forcing of length
κ, where κ ≥ ℵ2 and cof (κ) > ℵ0 , as follows.
3
Remarks on a Class of almost Disjoint Families
Define Eα for α < κ in the following way:
(a). E0 = EM ;
˙ M Eα = E).
˙ MEα is an Eα –name such that k– ((E)
(b). Eα+1 = Eα ∗ (E)
Eα
Note. This forcing p.o.set had been firstly studied by A. Miller in [9].
The following lemma which was proved in [9] is important for us to achieve
our result.
Lemma 1.4. Suppose that k– τ ∈ ω. If F ∈ [ω]<ω and nα < ω and
sα ∈ ω <ω for every α ∈ F , then there is a H ∈ [ω]<ω such that
∀q ∈ Eκ (supp(q) = F ∧∀α ∈ F (nqα = nα ∧sqα = sα ) → ∃p ≤ q(p k– τ ∈ H)).
Claim. There is no m.a.d. family F in N × N in M [Gκ ] such that F has
cardinality less than κ.
Proof of the Claim. Assume that there exists a m.a.d. family of
graphs F ∗ which has cardinality λ < κ. Let g ∈ F ∗ . For each n ∈ dom(g),
there is a maximal antichain Bng which decides the value of ġ(n), where ġ
is a nice name of g. Since Eκ is c.c.c., then |Bng | ≤ ℵ0 . Let
B∗ =
[
(
[
Bng ).
g∈F n∈dom(g)
Obviously, |B ∗ | ≤ λ < κ. For each p ∈ B ∗ , supt(p) is a finite subset of κ,
hence there exists an α < κ such that for any p ∈ B ∗ , supp(p) ⊂ α < κ. If
Gα is the compenent of Gκ in the iterated forcing up to (not including α),
then we have F ∗ ∈ M[Gα ]. For each gβ ∈ F ∗ , we can extend gβ to a full
function ḡβ . Then we get
F̄ ∗ = {ḡβ | β < λ}.
For any γ such that γ < κ, the Eγ –generic function fγ is almost disjoint
with any function in F̄ ∗ . This contradicts the assumption that F ∗ is a
m.a.d. family in (N × N)M [Gκ ] .
2
Notice that E is a susline c.c.c. forcing, by absoluteness (see e.g. [15]),
we know that
[
M [Gκ ] ∩ ℘(ω) = {M [Hα ] ∩ ℘(ω) | α < ℵ1 , Hα ∈ M [Gκ ],
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Yi Zhang
and Hα is Eα –generic over M }.
Thus the following claim will complete the proof of this theorem.
Claim. Let M |= (ZF C + GCH). There is a m.a.d. family F in M such
that for any α < ω1 , if Gα is Eα –generic over M , F remains maximal in
M [Gα ].
Proof of the Claim. Within M , we shall define a m.a.d. family.
By CH, let
E = {hpξ , τξ , ξi | ω ≤ ξ < ω1 }
enumerate all triples hp, τ, αi such that α < ω1 , p ∈ Eα and τ is a Eα –nice
name for subset of ω. By recursion, pick infinite Dξ ⊂ ω as follows:
Let {Dn | n < ω} be any almost disjoint set in ℘(ω).
If ω ≤ ξ < ω1 , and we have Dη for η < ξ, choose Dξ so that
• ∀η < ξ(|Dη ∩ Dξ | < ω), and
• if pξ k–Eξ (|τξ | = ω), and ∀η < ξ(pξ k–Eξ |τξ ∩ Dη | < ω),
then for any n ∈ ω and for any q ≤ pξ with q ∈ Eξ there exists some r ≤ q
with r ∈ Eξ and there exists some m ≥ n such that
m ∈ Dξ and r k–Eξ ṁ ∈ τξ .
Now let F = {Dξ | ξ < ω1M }.
We claim that F is a m.a.d. family in M [Gα ] for any α < ω1 .
We assume that there exists some α < ω1 such that Gα is Eα –generic
over M , and F fails to be maximal in M [Gα ]. Then there would be a
hpξ , τξ , ξi with ξ = α such that
pξ k–Eα |τξ | = ω, and
∀η < ξ(pξ k–Eα |τξ ∩ Dη | < ω̇).
Thus the hypothesis of (2) holds at ξ, but also
pξ k–Eα |τξ ∩ Dξ | < ω.
This is a contradiction. Therefore F is a m.a.d. family in M [Gα ].
Remarks on a Class of almost Disjoint Families
5
Next we prove that Dξ may be so chosen. Assume that the hypothesis
of (2) holds, since if it fails, then only (1) need to be considered, and we
can simply apply the fact that there is no m.a.d. family of size ω.
Let {En | n < ω} re-enumerate {Dη | η < ξ}.
We shall build an Dξ such that ∀n ∈ ω(|Dξ ∩ En | < ω) and
pξ k–Eξ (|τξ ∩ Dξ | = ω).
We shall build Dξ in stage as follows:
Dξ =
[
Kn
n<ω
where Kn ∈ ℘(ω) and Kn is finite. When we choose each Ki , i ∈ ω,
we will make sure that, for any p ∈ Eξ with p ≤ pξ , there exists a q ∈
Eξ with q ≤ p, such that ∃m ∈ Dξ (m ≥ n and q k–Eξ m ∈ τξ ), i.e.,
pξ k–Eξ (|Dξ ∩ τξ | = ω).
Let supp(pξ ) = {αiξ ∈ ω1 | 1 ≤ i < mξ }.
ξ
Let s̄ = hsα1 , ..., sαk i and ¯l = hlα1 , ..., lαk i with {α1ξ , ..., αm
} ⊆
ξ
⊆ {α1 , ...., αk }, where all αi ≤ ξ.
We list the countably many tuples
hᾱn , s̄n , ¯ln , mn i ∈ (ξ + 1)k × (ω <ω )k × ω k × ω for some k < ω
with infinitely many repetitions, such that
ξ
{α1ξ , ..., αm
} ⊆ ᾱn , and
ξ
∀i ≤ k(αi ≤ ξ),
ξ
j
∀αiξ ∈ supp(pξ )(spξ (αξ ) ⊆ sα
for some 1 ≤ j ≤ k).
n , if αi = αj
i
S
S
Let τ n = min{τξ \ max{τξ ∩ i<n Ei , ṁn , i<n Ki }}. To define Kn ,
αk
αk α1
1
we consider hsα
n , ..., sn , ln , ..., ln , mn i. By Lemma 1.4, there exists a set
<ω
Hn ∈ [ω]
such that for all q ∈ Eω2 with
• supp(q) = {α1 , ..., αk }, and
α
ω ln i
i
• for each 1 ≤ i ≤ k, k– q(αi ) = hsα
,
n , Fi i for some Fi ∈ [ ω]
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Yi Zhang
∃p ≤ q(p k–Eξ τ n ∈ Hn ).
We define
[
Kn = Hn \
Ei .
i≤n
And we set
[
Dξ =
Kn .
n<ω
By construction Dξ ∩ En ⊆
S
i<n
Ki for any n < ω, i.e.,
|Dξ ∩ En | < ω
for any n < ω.
We show that for any p ≤ pξ , p k–Eξ Dξ ∩ τξ is infinite.
Suppose otherwise, there exist some p ≤ pξ and some m < ω such
that
p k–Eξ Dξ ∩ τξ ⊆ m + 1.
We may assume that
supp(p) = {αi | 1 ≤ i ≤ k}, and
αi
ω ln
i
), and
k– (p(αi ) = hsα
n , F i for some F ∈ [ ω]
m = mn .
Then there exists some r ≤ p such that there exists t > mn and t ∈ Kn
and r k–Eξ (t ∈ τξ ). We have reached a contradiction. Thus we proved the
lemma.
2
Therefore, we have proved that
M [Gκ ] |= (a = ℵ1 < as = κ = 2ℵ0 ),
where κ ≥ ℵ2 and cof (κ) > ℵ0 .
2
Remarks on a Class of almost Disjoint Families
7
The third almost disjoint family was suggested by Simon Thomas as
follows.
Definition 1.5. Two permutations f, g ∈ Sym(N) are a.d. iff |f ∩ g| is
finite, i.e.,
|{n ∈ N | f (n) = g(n)}| < ℵ0 .
Let ap be the least λ such that there exists a m.a.d. family F of permutations with |F| = λ.
In [15] and [16], certain properties about ap were studied. For example, in [16],we proved the following theorem.
Theorem 1.6. It is consistent with ZFC that
a = ℵ1 < ap = ℵ2 = 2ℵ0 .
Some Explaination about the Theorem 1.6. Using the same idea
as the proof of theorem 1.3. We can improve this result to any κ such that
κ ≥ ℵ2 and cof (κ) 6= ℵ0 . However, due to the fact that the space Sym(N)
equipped with its natual topology is not compact (see, e.g. [16] for details),
the proof is much harder and lengthier than the one of Theorem 1.3.
2. Cohen Forcing
In this section, we prove that in Cohen forcing, as and ap stay small, i.e.,
we force with a c.c.c. partially ordered set
F n(I, 2) = {p | p is a finite function, dom(p) ⊆ I, rang(p) ⊆ 2}
on the groud model M which satisfies ZF C + GCH. We shall prove that
the generic model M F n(I,2) satisfies a = as = ap = ℵ1 . Thus it is consistent
with ZF C that
a = as = ap = ℵ1 < κ = 2ℵ0 ,
where ℵ2 ≤ κ = κω .
We first state a well-known lemma about Cohen forcing.
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Yi Zhang
Lemma 2.1. Suppose I, S ∈ M . Let G be F n(I, 2)–generic over M , and
let X ⊆ S with X ∈ M [G]. Then X ∈ M [G ∩ F n(I0 , 2)] for some I0 ∈ M
and (|I0 | ≤ |S|)M .
Proof. See [8].
2
Now we prove that M F n(I,2) |= as = ℵ1 . By the above lemma, it is
sufficient to construct some m.a.d. family of graphs A ⊆ N×N in extensions
via I0 = ω which are countable in M .
Theorem 2.2. Let M |= (ZF C + GCH). There is a m.a.d. family of
graphs A of size ω1 in M such that for any Cohen generic G over M , A
remains to be a m.a.d. family of graphs in M [G].
Proof. Since F n(ω, 2) has c.c.c. and M |= GCH, there are at most
|NN | = ℵ1 different antichains, there are at most (2ℵ0 )ℵ0 = ℵ1 nice names
for reals. In M , construct a m.a.d. family of graphs A ⊆ N × N as follows.
Let hpα , τα i for ω ≤ α < ω1 enumberate all pairs hp, τ i such that
p ∈ F n(ω, 2) and τ is a nice name for a graph in N × N.
By recursion, pick graph fα ∈ N × N in the following way.
Let {fn | n < ω} be an a.d. graph set in N × N.
If ω ≤ α < ω1 , and we have graphs fβ for β < α, choose a graph fα
so that
• ∀β < α(| {n | fα (n) = fβ (n)} |< ω),
• if pα k– (τα is a graph in N × N), and ∀β < α(pα k– τα and fβ are
a.d.),
then pα k– (τα and fα are not a.d. ).
To see that fα may be so chosen, let
Aα = {fβ | β < α}.
Let EAα be the partially ordered set consists of all conditions hs, F i such
that
• s is a finite partial function from N to N, and
• F is a finite subset of Aα ,
where hs1 , F1 i ≤ hs2 , F2 i iff
(s2 ⊆ s1 ) and (F2 ⊆ F1 ) and ∀f ∈ F2 (f ∩ s1 ⊆ s2 ).
Remarks on a Class of almost Disjoint Families
9
We consider the following dense subsets of EAα :
Df = {hs, F i ∈ EAα | f ∈ F },
En = {hs, F i ∈ EAα | n ∈ dom(s)},
Cn,q = {hs, F i ∈ EAα | ∃x ∈ ω∃m ≥ n∃r ≤ q(s(x) = m and r k– τα (ẋ) = ṁ)},
where q ≤ pα .
It is easily seen that Df and En are dense in EAα .
We prove that Cn,q is dense in EAα .
Given hs, F i ∈ EAα for some n ∈ N and q ≤ pα . By assumption of
(2), since |F | < ω and q ≤ pα ,
q k– (∃t ∈ ω)(∀z ≥ t)(τα (ż) 6= fβ (ż), for all β ∈ F )).
Then there exist q0 ≤ q and t ∈ ω with n ≤ t ∈ ω such that
q0 k– (∀z ≥ t)(τα (ż) 6= fβ (ż), for all β ∈ F )).
Therefore, there exist r ≤ q0 and x ≥ t and x 6= dom(s) and m ≥ n such
that
r k– τα (ẋ = ṁ).
Also, this gives τα (x) 6= fβ (x) for all β ∈ F . Hence, if s0 = s ∪ {hx, mi},
then hs0 , F i ≤ hs, F i and hs0 , F i ∈ Cn,q . Thus Cn,q is dense in EA,α .
Let D = {Df | f ∈ Aα } ∪ {En | n ∈ ω} ∪ {Cn,q | n ∈ ω and q ≤ pα }.
Then |D| ≤ ω. By MA(ω), there is a filter Gα ⊆ EAα such that for any
d ∈ D, Gα ∩ d 6= ∅.
S
Let fα = {s | hs, F i ∈ Gα }. Then fα satisfies (1) and (2).
Now let A = {fα | α < ω1M }.
We claim that A is the desired m.a.d. family of graphs in N × N.
Let G be F n(I, 2) over M . Suppose towards contradiction that A is
not maximal in M [G]. Then there is a hpα , τα i such that pα ∈ G,
pα k– (τα is a graph in N × N), and
pα k– ∀f ∈ A(τα ∩ f˙ is finite).
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Yi Zhang
Thus the condition of (2) holds at α and
pα k– (τα ∩ f˙α is finite).
This implies
pα k– (τα ∩ f˙α ⊆ n × n), for some n.
But this contradicts that pα k– (τα and fα are not a.d.).
2
Note. Similarly, we can prove that M F n(I,2) |= ap = ℵ1 by repeatedly
using the following c.c.c. p.o.set.
Assume that A ⊆ Sym(N). Let PA be the p.o.set consists of all
conditions hs, F i such that
• s is a finite 1–1 partial function from N to N, and
• F is a finite subset of A,
where hs1 , F1 i ≤ hs2 , F2 i iff
(s2 ⊆ s1 ) and (F2 ⊆ F1 ) and ∀f ∈ F2 (f ∩ s1 ⊆ s2 ).
Thus we conclude this section with the following
Theorem 2.3. It is consistent with ZF C that
a = as = ap = ℵ1 < κ = 2ℵ0
where ℵ2 ≤ κ = κω .
3. The lower bound of
as
and
ap
Knowing several results in [16] and [15], S. Todorcevic suggested to me the
following question ([Question 4.6, Z1]).
Question 3.1. It is easily seen that a ≤ as . Except the obvious lower
bound, ℵ1 for ap , can we find some other cardinal invariant, say m, such
that m ≤ as (or m ≤ ap )?
In [3], we answered this question by proving the following results.
Theorem 3.2. N on(M) ≤ as , ap , where N on(M) is the cardinality of
the smallest non-meager set of reals.
Remarks on a Class of almost Disjoint Families
11
Note. Our result are based on the results in [2], [7] and [10], the proofs
of these results are very complicated. Martin Goldstein once showed me
a weaker result but the proof of his result is quite elegant. We state his
result with the proof as follows.
Theorem 3.3.
b ≤ ap , where b is the size of the least cardinal of unbounded family in hN N, ≤∗ i.
Note. It is well-known that b ≤ a ≤ as , for proof, see e.g. [4].
Proof of Theorem 3.3. Let A be an almost disjoint family of permutations of ω of size < b. We will show that A is not maximal by giving a
permutation that is almost disjoint to every member of A.
Let N be a sufficiently closed set (elementary submodel of some H(χ))
containing A. Let f be a strictly monotone function dominating all functions in N . To simplify the argument below we will assume that all values
of f are odd numbers.
Let N 0 be a sufficiently closed set containing N and f , and let g
be a strictly monotone function dominating all functions in N 0 . Let a =
{g(0), g(1), . . .}.
Define a functions h0 and h1 as follows: dom(h0 ) = ω\a, h0 (n) = f (n)
for all n ∈
/ a; dom(h1 ) = a, h1 (n) = min(ω \ (rang(h0 ) ∩ {h1 (i) : i < n})
for n ∈ a.
Let h = h0 ∪ h1 .
Claim. h is a permutation almost disjoint from every element in A.
Note. The idea is that h0 grows too fast to have an infinite intersection
with funtions in A, and h1 grows too slow to have such an intersection.
Proof of the Claim. h0 dominates all functions from N , so h0 is almost
disjoint from all elements of A.
Note that h1 is strictly increasing. Moreover, the range of h1 includes
all even numbers, hence h1 (g(n)) ≤ 2n for all n.
Every number which is not in rang(h0 ) is in rang(h1 ), so h is onto.
It is also easy to see that h is 1-1. So h is a permutation.
Finally we have to check that h1 is disjoint from all elements of A.
For any p ∈ A, let p∗ (k) = max{i : p(i) ≤ k}.
If p(g(n)) = h1 (g(n)), then p(g(n)) ≤ 2n, so p∗ (2n) ≥ g(n). This can
hold for only finitely many n, since g dominates A.
2
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Yi Zhang
4. Problems
Problem 4.1. Can we prove a ≤ ap ? Or, can we prove the consistency
of ap < a?
Problem 4.2. Can we separate the two closely related cardinals ap and
as in any forcing model? And is there any ZFC inequality between ap and
as ?
Problem 4.3. (B. Velickovic) Except the obvious upper bound 2ℵ0 , can
we find any cardial invariant which is the upper bound of ap or as ?
Note. Theorem 3.2 makes this problem very hard to answer.
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Remarks on a Class of almost Disjoint Families
13
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Department of Mathematics and Computer Science
Istanbul Bilgi University
Kustepe, Sisli, 80310, Istanbul, Turkey
e-mail: cyzhangbilgi.edu.tr