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Math. Log. Quart. 45 (1999) 2, 183 - 188
Mathematical Logic Quarterly
@ WILEY-VCH Verlag Berlin GmbH 1999
Towards a Problem of E. van Douwen and A. Miller
Yi Zhang')
Institute of Mathematics, Academica Sinica, Beijing, 100080, P. R. China')
Abstract. forcing models.
Mathematics Subject Classification: 03335, 03E40, 03E50.
Keywords: Eventually different functions, Almost disjoint sets, Maximal almost disjoint family, Martin's axiom, Continuum hypothesis.
We discuss a problem asked by E. van Douwen and A. Miller [5] in various
1 Introduction
We say that two functions f , g E " w are eventually different (e. d.) iff I f n g1 < w . In this paper, we shall mainly discuss a problem asked by E. VAN DOUWEN and A. MILLER (see [5]) as follows:
The coniinuum hypothesis CH implies that there exists a set A " w which is maximal with respect to eventually different functions and which is also maximal with respect to infinite partial functions. Is there always such an A ? What is a,, the cardinality of the smallest such A ?
We shall also discuss the relationship between a, and a, which is defined as follows. D e f i n i t i o n 1.2. Sets z,y c w are almost disjoint (a.d.) iff Iz f l y( < w . An
almost disjoint family is a family A c p(w) such that for any z E A , 121 = w and any two distinct elements of A are a .d . Let a be the least infinite cardinal A such that there exists a maximal almost disjoint family (m. a. d. f .) F c p(w) of size A .
Q u e s t i o n 1.1.
There are several well-known results about a (see [l]). (1) Martin's Axiom MA implies that a = 2". (2) Assume w1 5 K 5 2". It is consistent with ZFC that there exists a m. a. d. f.
p(w) such that Jdl = K .
(3) Let M k ZFC+CH. Then a will stay small in M [ H ] which is obtained by adding A
n Cohen reals.
*)The research is partially supported by a grant from "Chinese Climbing Project Foundation for Natural Science" and CNSF-69573037. I would like to give my thanks to Professor SIMON THOMAS for his consistent and patient help, to Professors MARTIN GOLDSTEIN, RENLING JIN, and BOBAN VELICKOVI~ for the discussions which I had with them about this research project. Thanks are also given to Professor QIEYUAN HUANG for her efforts to find grant support for part of this research project.
')Current address: Mathematics Department, Rutgers University, New Brunswick, N. J. 08903, U.S.A. e-mail: cyzhangOmath.rutgers.edu
184 Yi Zhang
In Sections 2 - 4 we shall prove the corresponding results of ( l ) , (2) and (3) for a,. In Section 5 we shall prove that it is consistent with ZFC that a = w1 and a, = w2 = 2w.
We use P to denote the set of infinite partial functions from w to w . By A4 we always denote a countable transitive model of ZFC. Our forcing notations are standard, which can be found in either [2] or [3].
2 Martin's Axiom
The following c. c. c. partially ordered set for adding eventually different reals was introduced by A. MILLER in [4], which we will use frequently in this paper.
D e f i n i t i o n 2.1. For any A C " 'w, define a partial order EA which consists of all conditions of the form (s, F ) , such that s is a finite partial function from w to w , F is a finite subset of A, and
( ~ 1 , Fi) I (SZ, Fz) iff sz C si A Fz C FI A (Vf E Fz)(f n si C ~2). L e m m a 2.2 (MA(IE)). Let A 2 '"w, where IAI 5 K and w 5 IE < 2 w . Assume
h @ A is an infinite partial function of w , such that Ih n f I < w for any f E A. Then there exists g E '"w such that (1) )g rl f l < w for any f E A and (2) )g n hl = w .
P r o o f . Consider the partial ordering EA . We first define several dense sets in IEA . For each f E A we define DAJ = {(s, F ) E EA : f E F } , and for any n E w let
EA,n = {(s, F ) E EA : n E dom(s)}, CA,n = {(S, F ) E EA (3m > n)((m, h(m)) E S)}.
Let 27 = { D A J : f E A) U {EA,, : n E W } U ( C A , ~ : n E w } . It is easy to see that ID1 5 IE. By MA(n), there is a filter G* in EA such that G* n d # 0 for any d E V. Then by the density of E A , n we know that g* = u { s : (3F c A)( ( s , F ) E G*)} is a function in '"w. By the density of DA,f we know that )g* r l f l < w for any f E A. By
0
T h e o r e m 2.3 (MA). There exists a f a m i l y d '"w which is maximal with respect t o e. d . functions and which is also maximal with respect to infinite partial functions.
P r o o f . Let (fa : w 5 Q < 2'") be an enumeration of all functions in P. We prove the theorem by induction on Q.
Let An C '"w, n < w , be any set of e. d. functions such that lAnl = w , and let An, = An, for any n l , 122 < w . Assume Ap has been constructed, where p < Q < 2w, and IAp( < 2w. Let Ab, = Up<, Ap. Here [A;)= IE < 2"'. Consider TEAL with f,. We shall construct g, which satisfies the following conditions:
(1) go E ' "w; By Lemma 2.2, MA(IE) implies that we can construct such a 9,. Let A, = Ab, u {g,} and A = Ua<2Y A,. We claim that A is maximal with respect to eventually different functions and that A is also maximal with respect to infinite partial functions.
Assume it is not the case. Then there exists f E P such that )g n f ) < w for any g E A. We can assume that f = f a for some Q < 2'" in our enumerating list. Then If, n g) < w for any g E A,. But by our construction of A, we know that g, E A,
0
the density of CA,n we know that Ig* n h( = w .
(2) if )g n f,) < w for any g E Ab,, then 19, n f,l = w .
and If, n g, 1 = w . We get a contradiction. C o r o l l a r y 2.4 (MA). am = 2'".
Towards a Problem of E. van Douwen and A. Miller 185
3 It may be that a, is less than continuum
First we note L e m m a 3.1. Let M k ZFC. Let A
L e m m a 3.2. Let A
" w with A E M and G EA-generic over M . 0
" w be a set of pairwise e. d . functions and let f be an infinite partial function from w to w such that f 4 A and f n g is finite for any g E A. If g* is an EA-generic function, then 1g* n f l = w .
Then M[G] contains a function g E " w such that Ig n f l < w for any f E A.
P r o o f . We shall force with EA . For each n E w we define D A , n = ( ( 8 , F ) E E A (3Z E ~ ) ( 3 r n 2 n) ( f ( x ) = S ( X ) = m)}.
Fix ( s , F ) E EA and n E w . Since f n g is finite for any g E A and F C A , then for any g E F , f \ g is a cofinite subset of f . Also since IF1 < w , I f \ U g E F g1 = w . Consider all ( x , rn) E f \ U S E F g, where rn 2 n. Since Is1 < w , we can find an integer z dom(s) such that (z,rn) E f . Let s' = { ( x , r n ) } U s. Then ( s ' , F ) 5 ( s , F ) , and (s', F ) E D A , ~ . Therefore D A , ~ is dense for every n E w . Hence, if G is EA-generic
0
T h e o r e m 3.3. Let .M i= (ZFC + TCH). Let K , A E M be cardinals such that w1 5 K < 2" = A. Then there exists a c. c. c. notion of forcing P such that the following statements hold in M': (1) 2" = A. ( 2 ) There exists a set A* C " w of e. d. functions of cardinality K which is maximal with respect to e. d. functions and to infinite partial functions.
P r o o f . Without loss of generality, let M i= (MA + K < 2"). By Corollary 2.4 there exists a sequence (g,, : q < K ) E M of functions from w to w such that
Now we define a finite support iterated forcing of length w1 as follows. At step a , we assume that we have constructed a sequence (go : 7 < K + a) of pairwise e. d. functions in "w. We let A, = {g,, : q < K + (Y} and we use the forcing notion EA,. By Lemma 3.1, we introduce a new function gn+, E " w which is e. d. t o each member of A,. Since EA, is a c. c. c. forcing, our iterated forcing is c. c. c.
We can take EA, to consist of all pairs (s, F ) , where s is a finite partial function from w to w and F is a finite subset of K + a, and let each q E F stand for the corresponding function gs. Thus each EA, will consist of a set in M (while its partial order is not necessarily in M ) , and the cardinality of E A ~ is therefore max(a, K ) = K. For the iterated forcing we can use only standard names in the set E of conditions. Hence IIE I = K . Since IE is a c. c. c. forcing, IE preserves cardinals. Thus 2" is the same cardinal in M[G] and M .
We claim that {g,, : 71 < K + w1} is a set of e. d. functions which is maximal with respect to e. d. functions and to infinite partial functions. Let g E P in M[G] be such that g # g,, for any q < x: + w1. Let B be a nice name of g. For each (A, 7iz) E w x w , there exists a maximal antichain A(n,m) of IE which decides whether j ( n ) = rn. Since E is c. c. c., A(n,m) is countable. Let A = U(,,, Eg A(n,m). Then IAl 5 w . Since IE
support(p) c (Y for any p E A. If G, is the component of G in the iterated forcing up to (but not including) a, then we have that g E M[G,]. If 1g n g,,l < w for every q < K + (Y, then Lemma 2.2 implies that 1g n gal = w . Thus {gs : 17 < K: + w 1 ) is
over M , then DA,,, n G # 0 for any n E w .
(Vv, c < 4 (v # c + Isrl n SCI < w>.
is a wl-length forcing with finite support, it is c 1 ear that there is a < w1 such that
186 Yi Zhang
set of e. d. functions which is maximal with respect to e. d. functions and to infinite partial functions. 0
C o r o l l a r y 3.4. If Con(ZFC), then Con(ZFC+a, < 2w).
4 Cohen forcing
In this section we will prove that a, stays small in Cohen forcing by Theorem 4.2. We first state a Lemma about Cohen forcing.
L e m m a 4.1 (KUNEN [3, p. 2561). Suppose I , S E M . Let G be Fn(I12)-generic over M I and let X c S with X E M[G]. Then X E M[G n Fn(lo,2)] for some l o c I
0
The idea of the proof is as follows. Lemma 4.1 implies that it is sufficient to construct in M a maximal e .d . family A s ww such that A remains a maximal e. d. family in any forcing extension MFn(’o~2), where A is maximal with respect to e: d. functions and A is also maximal with respect to infinite partial functions, and where I0 E M is countable. For any f E ww in MFn(’i2), f is an element of MFn(’oi2)
for some countable 10 C I . When l o is finite, then MFn(Ion2) = M ; and when 1101 = w , then Fn(10,2) is isomorphic to Fn(w, 2) in M . Since isomorphic partially ordered sets yield the same generic extensions (see e.g. [3, p. 220]), it is sufficient to construct A so that whenever G is Fn(w, 2)-generic over MI then there does not exist a function g E P n M[G] such tha.t g is e. d. from f for all f E A.
T h e o r e m 4.2. Let M I= (ZFC + CH). There is a maximal e. d. family A E M of size w1 in M such that for any Cohen generic G over M I A remains a maximal e. d. family in M[GIl where A is maximal with respect t o e. d. functions and A is also maximal with respect to infinite partial functions.
P r o o f . Since Fn(w, 2) has c. c. c. and M I= CH, there are at most ww = 2w = w1 different antichains. Hence there are at most (2w)w = w1 nice names for reals. In M we define a maximal set4 of e. d. functions A of size w1 as follows:
Let ( ( p , , ~ , ) : w 5 a < w1) be an enumeration of all pairs ( p , ~ ) such that p E Fn(w, 2) and T is a nice name for an infinite partial function in P. By recursion, pick functions fa E ww as follows: Let {fn : n < w} be any e.d. family in ww. If w 5 a < w1 and we have functions f p for ,L? < a, choose a function fa so that
such that l o E M and (1101 5
(1) I{n : f,(n) = fp(n)}l < w for all P < a, (2) if p , It- “ T~ is an infinite partial function of w ” and p a IF “T, and f p are e. d.”
for all p < a , then p , It- “T, and f, are not e.d.”. To see that f, may be so chosen, let A , = { f p : P < a}. We consider the c. c. c. par- tially ordered set EA, € MI defined in Definition 2.1 and the following dense sets:
Dj = { ( s , F ) E EA, : f E F } , En = { ( s , F ) E EA, : n E dom(s)},
Cn,q = {(S, F ) E EA, : (32 E w ) ( h 2 n)(3r 5 q ) ( s ( Z ) = vz A r IF ~ ~ ( d ) = h)},
where q 5 p , . We know that Dj and En are dense in EA,. We prove that Cn,q is dense in EA,. Let ( s , F ) E EA,, n E w , and q _ < p a . By the assumption of (2), since IF1 < w and q I pal q It- (3t E w)(Vz 2 t ) ( z E dom(.r,) + (VP E F ) ~,(i) # f p ( i ) ) . Hence there exist qo 5 q and n 5 t E w such that
qo It- (Vz L t ) ( z E dom(.r,) -+ (VP E F ) ~,( i) # f p ( i ) ) .
Towards a Problem of E. van Douwen and A. Miller 187
There exist r 5 qo , z 2 t , z 4 dom(s), and rn 2 n such that r It- r,(?) = riZ. We also have that T I l - (VP E F)r,(z) # fp(z). Hence, if s' = s U {(z ,rn)} , then (s', F ) 5 (s, F ) and (s', F ) E C,,,,,. Thus C,,,,, is dense in IEA,.
Let D = {Dj : f E A,} U {En : E w} U {Cn,,, : n E w A q 5 p a } . Then ID1 5 w. By MA(w), there is a filter G, C IEA, such that G, n D # 0 for any D E D. Let fa = u { s : ( s , F ) E G,}. Then fa satisfies (1) and (2).
Now let A = {f, : cr < w y } . Let G be Fn(w,2)-generic over M . Suppose that d is not maximal in M[G]. Then there exists ( p , , ~ , ) such that p , E G, p , It- "r , is an infinite partial function of w " , and p , , I t - (Vf E d) 17, n f ] < w. Thus the condition of (2) holds at a, and p , Il- 17, II f,l < w. But this contradicts
0 p , I t - "r, and fa are not e. d.".
5 It is consistent that a < am
In this section we shall force with the C . C . C . partially ordered set IE = IEA which we defined in Definition 1.1, where A = ,w. Our ground model M satisfies ZFC + GCH.
By a usual density argument, we can prove L e m m a 5.1. Let G be IE-generic over M . Then in M [ q , there is g E "w such
0
We proceed with a system of iterated forcing of length w2 with finite support. where is an
It is easily seen that M[G,,] i= "any maximal e. d. family in ww has size w2 = 2"".
The following Lemmas 5.2 and 5.3 were proved by A. MILLER in [4]. L e m m a 5.2. For any p E I&,,, there is q 5 p such that for all (Y E support(q) there
are a finite s, E w<" and a natural n, such that It-lc, (3G E [Ww]nu) q(a) = (s, , G ) . We shall say that a condition q E I&,,, is canonical if it satisfies the conclusion of
Lemma 5.2. From now on we shall assume that all conditions in I&,,, are canonical. (This is harmless since the canonical conditions are dense in &, .)
L e m m a 5.3. Suppose Il- r E M , and given F E [u2]<", n, < w, and s, E w<" f o r every a E F . Then there is H E [MI<'" such that for all q E I&,,,, i f support(q) = F
0
r E P \ " w . Then for any A E ,w n M , i f p It,, (Vf E A ) I f n 7) < w and g, i s an Ea generic function, where 0 5 (Y < w2,
then f o r any q 5 p in IE there exists some r 5 q such that T ll-u, 1r n gal = w. P r o o f . Let C,,,,, = { ( s , F ) E IE, :ME- t= (3r 5 q)(3rn > n)(rIt- r(m) = ~ ( r n ) ) } ,
where q 5 p in I&,,,. We claim that C,,,q is dense in E,. Fix ( s , F ) E IE,. Since F is finite and q 5 p in llL, , q I l - (3 E w)(Vz 2 t ) ( z E dom(r) 4 (Vf E F ) r (z ) # f(z)). Hence there exist qo 5 q and n 5 t E w such that
Let ( 2 , r ( i ) ) E T such that i. = min{z E w \ t : r(i) # f(i) for all f E F } . By Lem- ma 5.3, there exist a finite subset K of w x w and TO 5 qo such that T O I t - ( 2 , ~ ( 2 ) ) E A'. Wecanseparate Kintofinitelymanyfinitefunctionstl,tz,. . . , t ko fw. Lets: = s U t j ,
that I f n g1 < w for all f E '"w n M .
Define IE, for a < w2 as follows: IE, name such that l t - ~ ~
Next we will prove that a, is wz in M[G,,].
= E M ; IE,+l = IE, * = IE).
and n$ = n, and s$ = s, for all (Y E F , then p I t - r E H for some p 5 q. Using Lemma 5.3 we can prove L e m m a 5.4. Suppose p E I&,,2 and p
QO I t - (Vz L t) ( z E dom(r) + (Vf E F ) r ( z ) # f(z)) .
188 Yi Zhang
1 5 j 5 k. Then PO It- ( 3 j E {1,2,. . . , k}) ( s i ( i ) = r ( i ) A (Vf E F ) s i ( i ) # f(2)). Hence P It- ( s i ( i ) = r ( i ) A (Vf E F ) $(i) # f(i)) for some P 5 PO and some j in { 1,2,. . . , k}. Therefore, (si, F ) 5 (s, F ) and (si , F ) E Cn,q, i. e., Cn,p is dense in L2.
Thus, if ga is an &generic function, then for any n E w there exists z > n such 0
L e m m a 5.5. There exists an e. d. set A '"w in M [ q which is maximal with respect to both e. d . functions and infinite partial functions.
P r o o f . Let A0 C_ '"w n M be a maximal e. d. family in M with respect to both e. d. fuctions and infinite partial functions. Hence for any infinite partial function g in M , there exists some f E A0 such that I f n g1 = w . Let f, be an &-generic function. Then we know that ( I f , n f l < w ) for all f E Ao. Let
By the Axiom of Choice, A1 can be extended to a maximal family A with respect to e. d. functions. Assume f* is an infinite partial function which is e. d. from any function f E A. Since A0 E A we know that f' 6 '"w n M . Let r be a name of f * . Then there is p E lL2 such that (1) p I t - (Vf E A ) I f n T I < w , (2) p It- I f o n T I < w . But by (1) and Lemma 5.4 we know that for all q 5 p there is P 5 q such that
0
that for all q 5 p there is r 5 q with P It- r ( i ) = g a ( i ) .
A1 = A0 U { fn : a < w2 and fa is an &-generic function}.
r It- 17 n fol = w , i.e., p It- If0 n T I = w , contradicting (2). C o r o l l a r y 5.6. M[G,,] t= a, = w2 = 2'". L e m m a 5.7. M[G,,] t= a = w1.
P r o o f (sketch). By Lemma 5.7 we can prove that there is a m. a. d. f. F p ( w ) in M such that for any a < w l , if G, is &generic over M, then 3 remains maximal in M[G,]. In [4], MILLER proved the following statement for L2:
M[GJ n p(w) = U{M[H,] n ~ ( w ) : < ~ i , H , E M[G,,], and H , is &-generic over M } .
(For details, see [6, pp. 51 - 561 or [7].) 0
We conclude T h e o r e m 5.8. M[G,,] t= a, = w2 and M[G,,] t= a = w1. 0
References
[l] VAN DOUWEN, E., The integers and topology. In: Handbook of Set Theoretic Topology (K. KUNEN and J. VAUGHAN, eds.), North-Holland Publ. Comp., Amsterdam 1984, pp. 111 - 167.
[2] JECH, T., Set Theory. Academic Press, New York 1978. [3] KUNEN, K., Set Theory: An Introduction to Independence Proofs. North-Holland
[4] MILLER, A,, Some properties of measure and category. Trans. Amer. Math. SOC. 266
[5] MILLER, A., Some interesting problems (updated at 15th Oct, 1996). In: Set Theory of the Reah (HAM JUDAH, ed.), Israel Mathematical Conference Proceedings, vol. 6, American Math. Society, Providence, RI, 1993, pp. 645 - 654.
[6] ZHANG, Y., Cofinitary Groups and Almost Disjoint Families. Phd. thesis, Rutgers Uni- versity, New Brunswick 1997.
[7] ZHANG, Y., On a class of MAD families. J . Symbolic Logic (to appear).
Publ. Comp., Amsterdam 1980.
(1981), 93 - 114.
(Received: November 14, 1997; Revised: February 25, 1998)
