Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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 2 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 ], 4 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] 6 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. 8 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). 10 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 12 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. References [1] A. Blass, Combinatorial cardinal characteristics of the continuum (to appear), [in:] Handbook of Set Theory (ed. A. Kananmori), Kluwer. [2] T. Bartoszynsky and H. Judah, Set Theory, On the structure of the real line, A.K.Peters, Wellesley, Massachusetts, 1995. [3] J. Brendle, O. Spinas and Y. Zhang, Uniformity of the meager ideal and maximal cofinitary groups, Journal of Algebra (to appear). [4] E. van Douwen, The integers and topology, [in:] Handbook of Set Theoretic Topology (ed. K.Kunen and J. Vaughan), 1984, NorthHolland, Amsterdam, pp. 111–167. [5] P. Erdös and A. Hajnal, On a property of families of sets, Acta. Math. Acad. Sci. Hungar, Vol. 12 (1961), pp. 87–123. [6] F. Hausdorff, Grundzüge der Mengenlehre, Leipzig, 1914. [7] M. Kada, The Baire category theorem and the evasion number, Proceedings of the American Mathematical Society 126 (1998), pp. 3381–3383. [8] K. Kunen, Set theory. An introduction to independence proofs, 1980, North Holland, Amsterdam. [9] A. Miller, Some properties of measure and category, Transactions of the American Mathematical Society 266, Number, 1 (July 1981), pp. 93–114. [10] S. Scheepers, Meager sets and infinite games, Contemporary Mathematics 192 (1996), pp. 77–89. Remarks on a Class of almost Disjoint Families 13 [11] S. Shelah, On cardinal invariants of the continuum, [in:] Axiomatic Set Theory (ed. J. E. Baumgartner, D. A. Martin, and S. Shelah), Contemporary Mathematics 31 (1984), pp. 183–207, American Mathematical Society, Providence. [12] S. Shelah, On what I do not know, preprint. [13] A. Tarski, Sur la décomposition des ensembles end sous ensembles preseque disjoint, Fund. Math. Vol. 14 (1929), pp. 205–215. [14] J. E. Vaughan, Small uncountable cardinals and topology, Open Problems in Topology (ed. J. van Mill and G. M. Reed) 1990, pp. 197–218, North-Holland, Amsterdam. [15] Y. Zhang, Permutation Groups and Coverring Properties, Journal of London Marthematical Society (2000), to appear. [16] Y. Zhang, A Class of MAD families, Journal of Symbolic Logic 64, Number 2 (1999), pp. 737–746. Department of Mathematics and Computer Science Istanbul Bilgi University Kustepe, Sisli, 80310, Istanbul, Turkey e-mail: cyzhangbilgi.edu.tr