Andrzej Roslanowski and Saharon Shelah- Localizations of infinite subsets of omega

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    Localizations of infinite subsets of

    Andrzej Roslanowski

    Institute of Mathematics

    The Hebrew University of Jerusalem

    Jerusalem, Israel

    and

    Mathematical Institute of Wroclaw University

    50384 Wroclaw, Poland

    Saharon Shelah

    Institute of Mathematics

    The Hebrew University of Jerusalem

    Jerusalem, Israel

    and

    Department of Mathematics

    Rutgers University

    New Brunswick, NJ 08854, USA

    done: September 1992printed: October 6, 2003

    The research was partially supported by Polish Committee of Scientific Research, GrantKBN 654/2/91

    The second author would like to thank Basic Research Foundation of The Israel Academyof Sciences and Humanities for partial support. Publication number 501.

    0

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    0 Introduction0.1 Preliminaries

    When we say reals we mean one of the nicely defined Polish spaces or their(finite or countable) products like: the real line R, the Cantor space 2 , theBaire space or the space of infinite sets of integers [] .In the present paper we are interested in properties of forcing notions (or, gen-erally, extensions of models of ZFC) which measure in a sense the distancebetween the ground model reals and the reals in the extension. In particular welook at the ways the new reals can be localized (or: aproximated) by oldreals. There are two extreme cases here: there are no new reals and the old realsare countable. However, between these two extremes we have a wide spectrum

    of properties among which the localizations by slaloms seem to be the mostpopular. A systematic study of slaloms and related localization properties andcardinal invariants was presented in [Bar1].

    A slalom is a function S : []< such that (n)(|S(n)| = n + 1). Wesay that a slalom S localizes a function f whenever (n )(f(n) S(n)).In this situation we can think that the slalom S is an approximation of thefunction f. It does not determine the function but it provides some boundson possible values of f. Bartoszynski, Cichon, Kamburelis et al. studied thelocalization by slaloms and those investigations gave the following surprisingresult.

    Theorem 0.1 (Bartoszynski, [Bar2]) Suppose that V V are models ofZFC. Then the following conditions are equivalent:

    1. Any function from V can be localized by a slalom from V.

    2. Any Borel (Lebesgue) null set coded in V can be covered by a Borel nullset coded in V.

    On localizations by slaloms see Chapter VI of [Sh:b] too; other localizations ofslalom-like type appeared in [GoSh:448].A stronger localization property was considered in [NeRo]. Fix a natural numberk 2. By a k-tree on we mean a tree T

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    Definition 0.2 Assume that X, Y are Polish spaces and R X Y is a Borelrelation. Suppose that V V are models of ZFC and that all parameterswe need are in V. We say that the pair (V, V) has the property of the R-localization if

    (xXV)(y YV)((x, y) R)

    If x X V, y Y V and (x, y) R they we say that y R-localizes x.

    In the examples we gave earlier X was and Y was the space of slaloms orthe space of all k-trees, respectively. The respective relations should be obvious.Those localizations were to approximate functions in an extension by objectsfrom the ground model. They are not useful if we consider infinite subsets of .Though each member of [] can be identified with its increasing enumeration,the localization (either by slaloms or by k-trees) of the enumeration does not

    provide satisfactory information on successive points of the set. The localizationgives us candidates for the n-th point of the set but the same candidates canappear several times for distinct n. That led to a suggestion that we shouldconsider disjoint subsets of as sets of candidates for successive points of thelocalized set (the approach was suggested by B. Weglorz). Now we have twopossibilities. Either we can demand that each set from the localization containsa limited number of members of the localized set or we can postulate that eachintersection of that kind is large. Localizations of this kind are studied in section1. In the second section we investigate localizations of infinite subsets of bysets of integers from the ground model. These localizations might be thoughtas localizations by partitions of into successive intervals. A starting point forour considerations was the following observation.

    Proposition 0.3 Suppose that V V are (transitive) models of ZFC. Then:

    1. V is unbounded in V if and only if

    for every set X [] V there exists a set Y in[] V such thatinfinitely often between two successive points of Y there are at least 2points of X.

    2. V is dominating in V if and only if

    for every set X [] V there exists a set Y [] V such that forall but finitely many pairs of two successive points of Y there are atleast 2 points of X between them.

    Now we try to replace the quantifier for infinitely manyabove by stronger quan-tifiers (but still weaker than for all but finitely many), like for infinitely manyn, for both n and n + 1. Finally, in section 3 we formulate several corollaries tothe results of previous sections for cardinal invariants related to the notions westudy.

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    0.2 Notation

    Our notation is rather standard and essentially compatible with that of [Jec]and [Kun]. In forcing considerations, however, we will use the convention thata stronger condition is the greater one.

    Basic Definitions 0.4 1. A tree on is a setT < closed under initialsegments. For the tree T the body [T] of T is the set

    {x : (l )(xl T)}.

    If t T then succT(t) = {s T : t s & lh(t) + 1 = lh(s)}.

    2. By a model of ZFC we will mean a transitive model of (enough of) ZFC.

    Models of ZFC will be denoted by V, V

    etc.3. We will be interested in extensions of models, i.e. in pairs (V, V) of

    models such that V V. If a property of an extension is defined thenwe extend this definition to notions of forcing. We say that a notion offorcing P has the property whenever for any generic filter G P over Vthe extension V V[G] has the considered property.

    4. We will use the quantifiers (n) and (n) as abbreviations for

    (m )(n > m) and (m )(n > m),

    respectively.

    5. The Baire space of all functions from to is endowed with thepartial order :

    f g (n)(f(n) g(n)).

    A family F is unbounded in (, ) if

    (g )(f F)(f g)

    and it is dominating in (, ) if

    (g )(f F)(g f).

    6. The unbounded numberb is the minimal size of an unbounded family in(, ); the dominating number d is the minimal size of a dominatingfamily in that order.

    7. The size of the continuum is denoted by c, [] stands for the family ofinfinite co-infinite subsets of .

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    0.3 Acknowledgement

    We would like to thank Professor Uri Abraham for his helpful comments on thepaper.

    1 Rk, Rk - localizations

    In this section we show that a localization of infinite subsets of suggested byB. Weglorz implies that the considered extension adds no new real.

    Definition 1.1 1. A partition of into finite sets is a sequence Kn : n of disjoint finite sets such that

    n Kn = .

    2. Pk is the set of all partitions Kn : n of into finite sets such that(n )(|Kn| > k).

    [Note that Pk is a 02-subset of ([]< ) so it is a Polish space.]

    3. We define relations Rk [] Pk and Rk []

    Pk+1 by

    (X, Kn : n ) Rk (

    n)(|X Kn| k)

    (X, Kn : n ) Rk (

    n)(|X Kn| k).

    Their complements (in [] Pk, [] Pk+1) are denoted by cRk, cR

    k,

    respectively.

    If we want to approximate an infinite co-infinite subset of by an object in

    a given model we can look for a separation of distinct members of the set bya sequence of sets from the model. Thus we could ask if it is possible to finda partition of (in V) such that the localized set is a partial selector of thepartition. More generally we may ask for Rklocalization; recall definition 0.2.Thus the Rk-localization property means that for every infinite set of integers Xfrom the extension there exists a partition Kn : n Pk from the groundmodel such that for almost all n the intersection X Kn is of size at mostk. The following result shows that the Rk-localization fails if we add new reals.

    Theorem 1.2 Suppose that V V are models of ZFC such that V 2 =V 2. Then there is a set X [] V such that for no k there is apartition Kn : n V of such that

    (n )(|X Kn| k & |Kn| > k).

    Consequently the extension V V does not have the Rk-localization property(for any k).

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    Proof Let x 2 be a new real (i.e. x V \ V) and let

    X = < \ {xi : i }.

    As we can identify with < we may think that X []. Now the followingclaim works.

    Claim 1.2.1 Suppose that x 2, x / V. Let Kn : n V be asequence of disjoint subsets of < such that (n )(|Kn| > k), k . Thenfor some n the set Kn \ {xi : i } has at least k + 1 points.

    Proof of Claim: Suppose not. Thus for each n we have

    |Kn \ {xi : i }| k and thus Kn {xi : i } = .

    First note that each Kn is finite. If not then the tree

    {s 2< : (t Kn)(s t)}

    has exactly one infinite branch - the branch is x. As the tree is in V we wouldget x V.Let

    u(n) = max{lh(s) : s Kn} and d(n) = min{lh(s) : s Kn}

    (remember that each Kn is finite). Choose an increasing sequence n : 0 is given by the following ob-servation.

    Proposition 1.3 Let V V be an extension of models of ZFC. Then thefollowing conditions are equivalent:

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    1. For each k > 0 the pair (V, V) has the cRk

    -localization.

    2. For some k > 0 the pair (V, V) has the cRk-localization.

    3. V is unbounded in V .

    Proof 2 3 Let an increasing function f V be given. Take anincreasing function f V such that

    (n )(f(f(n)) + 1 < f(n + 1)).

    Consider the set rng(f) []. Let Kn : n V be the partitionof given for this set by the cRk-localization. Let kn = min Kn and putg(kn) = 1 + max Kn. Extend g to putting g(m) = 0 if m / {kn : n }.Clearly g V.Note that |Kn rng(f

    )| > k > 0 implies that f(kn) < g(kn). Consequently

    (m )(f(m) < g (m)).

    3 1 Given k > 0. Let X [] V. Define

    f(n) = min{m > n : |X [n, m)| > 2k} for n .

    Since V is unbounded in V we find an increasing function g V

    such that (n )(f(n) < g (n)). Let kn be defined by:

    k0 = 0, kn+1 = k + 1 + kn + g(kn).

    Put Kn = [kn, kn+1). Clearly Kn : n Pk V. Now suppose thatm Kn is such that f(m) < g(m). As g is increasing we have g(m) < g (kn+1) 2k and hence either |Kn X| > k or|Kn+1 X| > k . Hence (

    n )(|Kn X| > k).

    In a similar way one can prove the analogous result for the cRk-localization.

    Proposition 1.4 Let V V be models of ZFC. Then the following condi-tions are equivalent:

    1. For each k the pair (V, V) has the cRk-localization.

    2. For some k the pair (V, V) has the cRk-localization.

    3. V is a dominating family in V .

    For the Rk-localization we did not find a full description. First note that therequirement that members of the partition have to have at least k + 2 elements(i.e. Rk []

    Pk+1) is to avoid a trivial localization. If we divide intok + 1-element intervals then for each set X [] infinitely many intervalscontain at most k members of X.

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    Definition 1.5 Let V V be models of ZFC and let k < l < . A set X [] V is called (l, k)-large (over V) if for every sequence Kn : n Vof disjoint l-element subsets of we have: (n )(| Kn X | > k).[Note that we do not require that Kn : n is a partition of .]

    Theorem 1.6 All Rk-localizations (for k ) are equivalent, i.e. if an exten-sion V V has the Rk-localization property for some k then it has theRk-localization for each k .

    Proof Let V V be models of ZFC. Let k < l < .

    Claim 1.6.1 If X [] V is (l, k)-large, k + 1 < l < and m 1 then Xis (lm,lm (l k))-large.

    Proof of Claim: Let Kn : n V be a sequence of disjoint subsets of of the size lm. Let Kn = {an,i : i < lm} be the increasing enumeration

    and let KAn = {an,i : i A} for A lm. Fix a set A [lm]l and consider

    the sequence KAn : n []l. This sequence is in V and its members

    are disjoint. Thus we find N(A) such that (n N(A))(| KAn X | > k).

    Let N = max{N(A) : A [lm]l}. Then for each n N and A [lm]l wehave | KAn X | > k and hence | K

    An \ X | < l k. Hence we conclude

    that | Kn \ X | < l k for each n N (just take a suitable A). Thus| Kn X | > lm (l k) for each n N and the claim is proved.

    Claim 1.6.2 If X [] V is (l, k)-large, k + 1 < l < and m l k thenX is (m, m (l k))-large.

    Proof of Claim: Let Kn : n V be a sequence of disjoint m-elementsets. Put Kn = Kln Kln+1 . . . Kln+(l1). Clearly K

    n : n V,

    |Kn| = lm and the sets Kns are disjoint. It follows from 1.6.1 that the set X

    is (lm,lm (l k))-large and hence there is N such that

    (n N)(| Kn X | > lm (l k)).

    So | Kn \ X | < l k for n N and hence | Kn \ X | < l k for n lN. Thisimplies (n lN)(| Kn X | > m (l k)) and the claim is proved.

    Claim 1.6.3 Assume that the extension V V has the R0 -localization prop-erty. Then it has the Rk-localization for each k .

    Proof of Claim: Let k > 0 and assume that the Rk-localization fails. Then we

    have a set X [] witnessing it, i.e. such that

    (Kn : n Pk+1 V)(n )(| Kn X | > k).

    Then, in particular, the set X is (l, k)-large for each l > k + 1. By claim1.6.2 it is (l, 0)-large for each l 2. By the R0-localization we find a partition

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    Kn

    : n P1

    V such that (n )(Kn

    X = ). Each set Kn

    wepartition into 2- and 3-element sets:

    Kn =

    {K2n,i : i w2n}

    {K3n,i : i w

    3n}

    (everything should be done in V, of course). Next look at

    K2n,i : n , i w2n, K

    3n,i : n , i w

    3n V.

    These are sequences of disjoint 2- (or 3-, respectively) element subsets of . Atleast one of them is infinite; for simplicity we assume that both are infinite. AsX is both (2, 0)- and (3, 0)-large we find N such that for each n N, j {2, 3}and i wjn we have K

    jn,i X = . But this implies that for each n N the

    intersection Kn X is not empty - a contradiction to the choice of Kn : n .

    Claim 1.6.4 Assume that k , V V has the Rk-localization property.Then the extension V V has the R0-localization.

    Proof of Claim: Suppose that the R0 -localization fails and this is witnessed bya set X [] V. As earlier we conclude from this that the set X is (l, 0)-largeand hence, by 1.6.2, it is (l + k, k)-large for each l 2. By the Rk-localizationwe find Kn : n Pk+1 V such that (

    n )(| Kn X | k). Fori let Ai = {n : |Kn| = i} (some of these sets can be finite or evenempty). For n /

    i2k+2 Ai partition Kn into 2- and 3-element sets to have

    more than k pieces:

    Kn =

    {K2n,i : i w2n}

    {K3n,i : i w

    3n}, |w

    2n| + |w

    3n| > k

    (everything is done in V, of course). Consider the sequences

    K2n,i : i w2n, n /

    j2k+2 Aj,

    K3n,i : i w3n, n /

    j2k+2 Aj,

    Kn : n Ak+2, . . . ,Kn : n A2k+2

    (note that Ai = for i < k + 2). These are sequences of disjoint sets of thesizes 2, 3, k + 2, . . . , 2k + 2, respectively, and all sequences are in V. Since X is(l + k, k)-large for each l 2 and it is (2,0)- and (3,0)-large we find N such that

    (a) if n > N, n Ai, k + 2 i 2k + 2 then |Kn X| > k

    (b) if n > N, n /j2k+2 Aj , i w

    xn, x {2, 3} then K

    xn,i X = .

    The condition (b) implies that if n > N, n /j2k+2 Aj then | Kn X | > k(recall that we have more than k sets Kxn,i). Consequently | Kn X | > k for

    all n > N which contradicts the choice of Kn : n .

    The above proof suggests to consider (m, 0)-large sets (over V) and ask ifthe existence of such sets depends on m 2. The answer is given by the nextresult.

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    Proposition 1.7 Suppose V V are models of ZFC, m 2. Then thereexists an (m, 0)-large set over V if and only if there exists an (m + 1, 0)-largeset over V.

    Proof Clearly each (m, 0)-large set is (m + 1, 0)-large. So suppose now thatX [] V is (m + 1, 0)-large over V. If it is (m, 0)-large then we are done.So assume that X is not (m, 0)-large and this is witnessed by Kn : n V(so |Kn| = m, Kns are disjoint and (n )(Kn X = )). Let

    Y = {n : Kn X = }.

    Clearly Y is infinite co-infinite. We are going to show that Y is (2, 0)-large (andhence (m, 0)-large) over V.

    Suppose that Ln : n V is a sequence of disjoint 2-element sets.Let Kn =

    lLn

    Kl. Thus |Kn| = 2m and Kns are disjoint. Obviously the

    sequence Kn : n is in V. Since the set X is (m + 1, 0)-large (and hence,by 1.6.2, (2m, m 1)-large) we have

    (n )(| Kn X | > m 1).

    But Kn X = , Ln = {l0, l1} imply that either Kl0 X = or Kl1 X =

    and hence Ln X = . Consequently Y is (2,0)-large.

    Corollary 1.8 Let V V be models of ZFC, m 2, k . Then thefollowing conditions are equivalent:

    1. there is no (m, 0)-large set in []

    V

    over V2. there is no (2, 0)-large set in [] V over V

    3. V V has the R0 -localization property

    4. V V has the Rk-localization property.

    Remark: One can consider a modification of the notion of (m, 0)-largenessgiving (probably) more freedom. For an increasing function f V we saythat a set X [] is f-large over V if for every sequence Kn : n V ofdisjoint finite subsets of we have

    either (n )(|Kn| < f(n) + 2) or (n )(|Kn X| > f(n)).

    Proposition 1.9 LetV V be models of ZFC.a) If V 2 is not meager iv V then the pair (V, V) has the R0 -localizationproperty.b) If the pair(V, V) has the R0-localization property thenV

    is unboundedin V .

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    Proof a) If V 2 is not meager in V then

    () (fV)(Y V[])(g V)(n Y)(f(n) = g(n))

    and V is unbounded in V (see [Bar1]). By proposition 1.3, thepair (V, V) has the cR1 -localization property. Suppose that X []

    V.By the cR1 -localization we find a partition Kn : n V P1 such that(n )(| Kn \ X | 2). In V

    we define

    f(n) = Kn \ X []< (for n ),Y = {n : |f(n)| 2}.

    By () we find g V, g : []< such that g(n) [Kn] 2 and

    (n Y)(f(n) = g(n)).

    Since f(n) = g(n) implies g(n) = Kn \ X we get (n )(g(n) X = ).

    Hence we easily get that X can be R0 -localized by a partition from V.b) Since |K| > 1 & K ( \ X) = implies |K X| 2 we get that

    R0-localization implies the cR1 -localization. Now proposition 1.3 works.

    The next result gives some bounds on possible improvements of the previousone.

    Proposition 1.10 1. The Cohen forcing notion has the R0-localization prop-erty. Consequently, the R0 -localization does not imply that the old realsare a dominating family.

    2. The Random real forcing does not have the R0 -localization property. Con-sequently, the localization is not implied by the fact that there is no un-bounded real in the extension.

    Proof 1. As in the extensions via the Cohen forcing the ground modelreals are not meager, we may apply 1.9.

    2. The Random algebra B is the quotient algebra of Borel subsets of 2

    modulo the ideal of Lebesgue null sets. We define a B-name for an element of[] that cannot be localized:

    Let l0 = 0, lk+1 = lk + 2k2 (for k ).

    For each k fix disjoint Borel sets Am 2 for lk m < lk+1

    such that (Am) = 2k2

    , where is the Lebesgue measure on 2.X is a B-name for a subset of such that if m [lk, lk+1), k then [[m / X]]B = [Am].

    It should be clear that B (k )(| [lk, lk+1) \ X | = 1).

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    Suppose now that Kn

    : n P1

    V. Let k0n

    = min Kn

    and k1n

    = max Kn

    .Suppose that lm k0n < lm+1. Note that

    k1n < lm+1 [[k0n / X & k

    1n / X]]B = 0

    andk1n lm+1 ([[k

    1n / X]]B) 2

    (m+1)2.

    Hence

    ([[X Kn = ]]B) ([[k0n / X & k

    1n / X]]B) 2

    (m+1)2 .

    Consequently, for each m 0:

    ([[(n )(lm k0

    n & Kn

    X = )]]B) n

    ([[lm k0

    n & Kn

    X = ]]B) =

    =

    rm

    k0n[lr,lr+1)

    ([[X Kn = ]]B)

    rm

    2r2

    2(r+1)2

    =1

    3212m.

    Hence we can conclude that ([[(n )(Kn X = )]]B) = 0 which meansBKn : n does not R

    0 -localize X.

    Though the random real forcing is an example of a forcing notion adding a(2,0)-large set over V (without adding an unbounded real!) it does not seemto be the minimal one. A canonical example of a forcing notion without theR0-localization property is given below. (Recall that a forcing notion Q is -

    centered if it can be presented as a countable union of sets which all finitesubsets have upper bounds in Q.)

    Example 1.11 There is a-centered (Borel) forcing notionQ adding no dom-inating real and without the R0 -localization property.

    Proof The forcing notion Q consists of pairs (u, K) such that u []<

    and K is a finite set of families of disjoint 2-element subsets of (so F K

    F []2). The order ofQ is given by

    (u0, K0) (u1, K1) if and only ifu1 (1 + max u0) = u0, K0 K1 and if K F K0 and K u1then K u0.

    For u []< , m let

    Qu = {(u, K) : (u, K) Q} and Qmu = {(u, K) Qu : |K| = m}.

    Since each Qu is obviously centered we get that Q is -centered. Let w be aQ-name such that wG =

    {u : (K)((u, K) G)} for each generic filter G Q

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    over V. Note that Q w []. If K F K, (u, K) Q and max u < min Kthen (u, K) Q K\ w = . Hence we conclude that

    Q \ w is (2, 0)-large over V.

    So Q does not have the R0-localization property. Suppose now that is aQ-name for a member of .

    Claim 1.11.1 Letu []< , m, n . The there is f(u,m,n) < such that:

    for every p Qmu there is q = (uq, Kq) p such that q decides the

    value of (n) and max uq < f(u,m,n).

    Proof of Claim: The space X of all families F []2 of two-element disjoint

    subsets of can be equipped with a natural topology. For F X, N theN-th basic open neighbourhood of F is

    {F X : {K N : K F} = {K N : K F}}.

    This topology is compact. It introduces a (product) topology on Qmu such that ifq Qu, q p, p Qmu then for some open neighbourhood V of p (in Q

    mu ) each

    member of V has an extension in Qu . Applying this fact and the compactnessof Qmu we get the claim.

    Now we define a function g putting

    g(k) = 1 + max{l : (u k)(m, n k)(v)(u v f(u,m,n) a n d (q Qv)(q (n) = l))}.

    Given p Qmu , l . Take k > max{l,m, max u}. By the definition off(u,m,k) we find v such that u v f(u,m,k) and some q Qv, q pdecides the value of (k). By the definition of the function g, the condition qforces (k) < g(k).

    Remark: An example of a forcing notion P with the R0-localization propertyand such that

    P V 2 is meager

    is an application of a general framework of [RoSh:470] and will be presentedthere.

    2 Between dominating and unbounded reals

    In this section we are interested in some localizations which are between the cRk-localization and cRk-localization (so between not adding a dominating real andnot adding an unbounded real). The localizations are similar to that consideredin the previous section. The difference is that we will consider partitions of

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    into intervals and we will introduce quantifiers stronger than n but weakerthan n.

    For an infinite subset X of let X : X be the increasing enumerationof X. A set X [] can be identified with the partition

    [X(n), X(n + 1)) : n

    of \X(0), so essentially [] P0. Now, for k > 0 and an increasing function

    , we define relations Sk, S+, S+, S+ []

    []:

    (X, Y) Sk (n )(i < k)(| [Y(n + i), Y(n + i + 1)) X | 2)

    (X, Y) S+ (m )(n )(i < m)(| [Y(n + i), Y(n + i +1)) X | 2)

    (X, Y) S+ (

    n )(i < 2n

    )(| [Y(2n

    + i), Y(2n

    + i + 1)) X | 2)(X, Y) S+ (

    n )(i < (n))(| [Y(n + i), Y(n + i + 1)) X | 2).

    [The relation S+ appears here for historical reasons only: it determines a car-dinal invariant which was of serious use in [RoSh:475].]

    Note that S1 S2 S3 . . . S+ S+S+ (remember that is increasing).

    If the function is increasing fast enough (e.g. (n) > 22n) then S+ S+.

    It should be clear that if we consider S-localizations we could put any integergreater than 2 in place of 2 in the definitions above. However we do not knowif replacing 2 by 1 provides the same notions of localizations.

    Proposition 2.1 Let V V be models of ZFC.

    (a) The pair (V, V) has the S1-localization property if and only if V isunbounded in V .

    (b) If the pair (V, V) has the Sk+1-localization property then it has the Sk-localization. The S+-localization property implies the Sk-localization foreachk > 0 and is implied by both the S+ and the S+-localization properties( - an increasing function).

    (c) If V is dominating in V then the extension V V has the

    S+-localization property for every increasing function V .

    Proposition 2.2 If V V are models of ZFC such that V 2 is not meager

    in V

    , V

    is increasing then the pair (V, V

    ) has the S

    +-localizationproperty.

    Proof The proof is almost the same as that of 1.9(a). Suppose that X V []. Let X0 [X]

    be such that for each n

    | [X0(n), X0(n + 1)) X | > 3(X0 (n) + 4) + 6.

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    As V is unbounded in V we find a set Y0

    V such that theset

    X1 = {Y0(n) : n & 2 | [Y0(n), Y0(n + 1)) X0 |}

    is infinite. Let f : []< be such that for every Y0(n) X1 we have

    f(Y0(n)) = [Y0(n), Y0(n + 1)) X

    (so, for Y0(n) X1, | f(Y0(n)) | > 3(X0 (k) + 4) + 6 where k is the firstsuch that Y0(n) X0(k) < Y0(n + 1)). By () from the proof of 1.9 we finda function g V, g : []< such that

    (n )(Y0(n) X1 & g(Y0(n)) = f(Y0(n))).

    Next, using g and Y0 (both are in V) we define sets Y1, Y V []

    :

    Y1 =

    n

    g(Y0(n)) [Y0(n), Y0(n + 1))

    and Y(n) = Y1(3n) for each n. Note that ifn is such that Y0(n) X1 andg(Y0(n)) = f(Y0(n)) then

    Y1 [Y0(n), Y0(n + 1)) = X [Y0(n), Y0(n + 1))

    is of the size > 3(Y0(n) + 4) + 6 and consequently Y S+-localizes X.

    Proposition 2.3 1. The Cohen forcing notion has the S+-localization prop-

    erty for each increasing function .

    2. If V V V are models of ZFC, the pair (V, V) has the S+-localization property and V is dominating in V then theextension V V has the S+-localization property.

    3. The iteration of the Cohen forcing notion and the random real forcing hasthe S+-localization.

    Consequently, the S+-localization property implies neither that the ground modelreals are dominating in the extension nor that the old reals are not meager.

    Theorem 2.4 For each k > 0, the Sk-localization property does not imply theSk+1-localization property.

    Proof To prove the theorem we will define a forcing notion Qk which willpossess the Sk-localization property but not Sk+1. The forcing notion is similarto that used in [Sh:207], [BsSh:242] and is a special case of the forcing notionsof [RoSh:470].

    We start with a series of definitions.

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    1. A function n is a nice norm on A if:

    n : P(A) and it is monotonic

    (i.e. B C A n(B) n(C)),

    if B C A, n(C) > 0

    then either n(B) n(C) 1 or n(C\ B) n(C) 1,

    n(A) > 0, if a A then n({a}) 1.

    2. A creature is a tuple T = T, nor, L , R such that

    ) T < is a finite nonempty tree,

    ) for each t T, either succT(t) = or |succT(t)| = k or |succT(t)| > k

    [so we have three kinds of nodes in the tree T]) nor is a function with the domain

    dom(nor) = {t T : |succT(t)| > k}

    and such that for t dom(nor), nor(t) is a nice norm on succT(t)

    [nor stands for norm],

    ) if s succT(t) then either |succT(t)| = k or |succT(s)| = k

    [i.e. we do not have two successive k-ramifications in T],

    ) L, R : T are functions such that for each t T:

    L(t) R(t)

    if s succT(t) then [L(s), R(s)] [L(t), R(t)] if s1, s2 succT(t) are distinct

    then [L(s1), R(s1)] [L(s2), R(s2)] =

    if succT(t) = then L(t) = R(t)

    [L stands for left and R is for right].

    3. We will use the convention that if a, b are indexes and Tab is a creaturethen its components are denoted by Tab , nor

    ab , L

    ab and R

    ab , respectively.

    4. Let T = T, nor, L , R be a creature. We define its weight T and itscontribution cont(T):

    T = min{nor(t)(succT(t)) : t dom(nor) & |succT(t)| > k}

    [if dom(nor) = then we put T = 0]

    cont(T) = {L(t) : t T & succT(t) = }

    [recall that if t is a leaf in T then L(t) = R(t)].

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    5. Let Ti

    (i = 0, 1) be creatures. We say that the creature T1

    refines T0

    (wewrite: T0 T1) if:

    a) T1 T0, L1 = L0T1, R1 = R0T1,

    b) if t T1 then

    |succT1(t)| > k iff |succT0(t)| > k ,|succT1(t)| = k iff |succT0(t)| = k, and|succT1(t)| = iff |succT0(t)| =

    (in other words we keep the kind of nodes),

    c) nor1(t) = nor0(t)P(succT1(t)) for all t dom(nor1).

    6. Let T0,T1, . . . ,Tn be creatures. We say that the creature T is built of the

    creatures T0, . . . ,Tn (we will write it as T (T0, . . . ,Tn)) if there is amaximal antichain F of T such that for each t F, for some i n:

    {s T : t s} = {t r : r Ti},

    if s = t r T, r Ti then L(s) = Li(r), R(s) = Ri(r) and nor(s) =nori(r) (if defined).

    7. Ifn k, H : P(n + 1) is a nice norm on n + 1 (i.e. it satisfies theconditions listed in (1)) then SH(T0, . . . ,Tn) is the creature T such that

    T = {i t : i n, t Ti}, nor() = H, nor(i t) = nori(t)

    and similarly for L and R.

    Clearly SH(T0, . . . ,Tn) (T0, . . . ,Tn).8. We define gluing k creatures similarly to the operation SH above. Thus

    S(T0, . . . ,Tk1) = T is a creature such that

    T = {i t : i < k, t Ti}

    and nor, L, R are defined naturally. Once again, S(T0, . . . ,Tk1) (T0, . . . ,Tk1).

    9. For a creature T we define its upper halfTuh = Tuh, noruh, Luh, Ruh by:

    Tuh = T, Luh = L, Ruh = R but

    noruh(t)(A) = max{0, nor(t)(A) [ T2 ]}

    whenever t T, |succT(t)| > k and A succT(t). Above, [x] stands forthe integer part of x.

    [It is routine to check that Tuh is really a creature and that Tuh =T [ 1

    2T], cont(Tuh) = cont(T).]

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    10. For creaturesT0

    , . . . ,Tn

    , the closure of{T0

    , . . . ,Tn

    } under the operationsof shrinking (refining), taking the upper half and building creatures isdenoted by (T0, . . . ,Tn). Thus {T0, . . . ,Tn}

    (T0, . . . ,Tn), ifT (T0, . . . ,Tn) then

    Tuh (T0, . . . ,Tn), T T T (T0, . . . ,Tn),

    and ifT0, . . . ,Tm

    (T0, . . . ,Tn) then

    (T0, . . . ,Tm)

    (T0, . . . ,Tn).

    [Note that (T0, . . . ,Tn) is finite (up to isomorphism).]

    Now we may define our forcing notion Qk:

    Conditions are sequences w,T0,T1,T2, . . . such that w []< , Ti arecreatures, Ti and

    max(w) < L0() R0() < L1() R1() < . . .

    (recall that Ti = Ti, nori, Li, Ri).

    The order is given byw,T0,T1 . . . w

    ,T0,T1 . . . if and only if

    for some increasing sequence n0 < n1 < n2 < .. . <

    w w w

    i

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    Claim 2.4.2 Suppose that T = T, nor, L , R is a creature, T 15 andB []. Then there is a creature T T such that T T 14 and

    (n )(i k)(| cont(T) [B(n + i), B(n + i + 1)) | < 2).

    Proof of Claim: We prove this essentially by the induction on |T| (or the heightof T). We show how to eliminate and apply the inductive hypothesis to Tabove t (for t succT()).

    Case 1: k = |succT()|If, for each t succT(), succT(t) = then | cont(T) | = k and there is no

    problem. So there are t succT() such that |succT(t)| > k (here we use therequirement () of the definition of creatures); above each such t we can applythe inductive hypothesis and shrink suitably the tree T. However the problems

    coming from distinct t could accumulate. Therefore for each such t we firstchoose a set A = At succT(t) such that nor(t)(A) T 7 and one of thefollowing occurs:

    for some m, (s A)(B(m) L(s) R(s) B(m + 1))

    there are 1 m0 < m1 such that

    (s A)(B(m0) L(s) R(s) B(m1)),

    and if t0 succT(), t0 = t then

    either R(t0) < B(m0 1) or B(m1 + 1) < L(t0).

    To find such a set A we use the second property of nice norms. First we lookat the set

    A0 = {s succT(t) : (m )(B(m) L(s) R(s) B(m + 1))}.

    If nor(t)(A0) nor(t)(succT(t)) 1 T 1 then we easily finish: either forsome m

    nor(t)({s succT(t) : B(m) L(s) R(s) B(m + 1)}) T 7

    or we have m0 < m0 < m0 < m1 < m

    1 < m

    1 such that

    nor(t)({s succT(t) : B(m0) L(s) R(s) B(m1)}) T 7 and

    (s0, s1succT(t))(B(m0) L(s0) B(m

    0) & B(m

    1) R(s1) B(m

    1)).

    So suppose that nor(t)(A0) < T 1. In this case nor(t)(succT(t) \ A0) T 1. For each s succT(t) \ A0 there is m such that

    L(s) < B(m) < R(s).

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    Removing 4 extreme points from succT

    (t) \ A0

    (the first two and the lasttwo, counting according to the values of L) we get the required set A (withnor(t)(A) T 5).Now we would like to apply the induction hypothesis above each t succT()restricting ourselves to successors of t from the suitable set At (if applicable).A small difficulty is that we have decreased the norm of the creature abovethose t (possibly by 7, as the result of restricting to At). But now we apply theprocedure described in the case 2 below and we pass to Bt At such that

    nor(t)(bt) nor(t)(At) 7 T 14.

    Next we apply the inductive hypothesis above each s Bt, t succT() dom(nor). In this way we construct T as required (the point is that restricting

    to the sets Bt

    At

    causes that what happens above distinct t succT() isisolated in a sense).

    Case 2: k < |succT()|Then one of the following possibilities occurs (i = 0, 1):

    ()i nor()({t succT() : (m)(B(2m + i) L(t) R(t) 2)

    2. (l )(Tl > 15).

    For each Ti take the creature Ti Ti given by claim 2.4.2 for Ti and B. Look

    at the condition q = w,T0,T1, . . . Qk. Clearly q p and ifB(n) > max(w)

    thenq Qk (i k)(|w [B(n + i), B(n + i + 1))| < 2).

    This proves the claim.

    The next claim explains why we introduced the operation of taking the upperhalf of a creature as a part of the definition of (the order of) Qk

    Claim 2.4.4 Let p = w,T0,T1 . . . Qk, m . Suppose that is a Qk-

    name for an ordinal. Then there are n0 and a nice norm H on n0 such thatH(n0) m and

    ifT SH(Tuh0 , . . . ,Tuhn01), T

    > 0, w w L0()then there exist v cont(T) and T0,T

    1, . . . such that

    ,T0,T1, . . . ,Tn0 ,Tn0+1, . . .

    and w v,T0,T1, . . . decides the value of .

    Proof of Claim: This is essentially 2.14 of [Sh:207].Define the function H : []< by:

    H(u) 0 alwaysH(u) 1 if |u| > 1 and for each Ti T

    uhi , T

    i > 0 (for i u), for every w

    ,w w L0() there is v

    iu cont(T

    i) such that some pure extension

    of w v,Tl,Tl+1, . . . decides the value of (where l = max u + 1).

    H(u) n + 1 if for every u u either H(u) n or H(u \ u) n (for n > 0).

    As H is monotonic it is enough to find u such that H(u) m. The existenceof the u can be proved by induction on m, for all sequences w,T0,T1, . . ..Let us start with the case m = 1. Suppose that Ti T

    uhi , T

    i > 0 (for i ).

    For each i choose a creature Ti Ti such that

    cont(Ti

    ) = cont(Ti

    ) and Ti

    1

    2Ti

    (possible by the definition of the upper half of a creature). Then for eachw L0():

    w,T0,T1,T

    2, . . . Qk

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    and thus we find n(w) , v(w) []< such that

    v(w)

    n

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    (needed for iterations). So suppose that X is a Qk

    -name for an element of []

    and p = w,T0,T1, . . . Qk. We may assume that Tn n. We inductivelydefine integers

    0 = b0 < b1 < b2 < . . . and 0 = n0 < n1 < n2 < . . .

    and nice norms Hm on [nm, nm+1). Suppose we have defined bm, nm.Let m be a Qk-name for an integer such that

    Qk |[bm, m) X| > 2.

    Modifying the tail (above nm) of p we may assume that if nm n, w w Rn() and some pure extension of w,Tn+1,Tn+2, . . . decides the value of mthen w,Tn+1,Tn+2, . . . does it already (see 2.4.1). Applying 2.4.4 we find

    nm+1 > nm and a nice norm Hm on [nm, nm+1) such that

    Hm([nm, nm+1)) m + 1 and if T SHm(T

    uhnm

    , . . . ,Tuhnm+11),w w Lnm() then there is v cont(T

    ) such that some pureextension of

    w v,Tnm+1 ,Tnm+1+1, . . .

    decides the value of m and thus w v,Tnm+1 ,Tnm+1+1, . . . doesit.

    Let bm+1 be an integer larger than all possible values forced to m in the con-dition above.

    Now for each l we put

    Tl = S

    SHlk(Tuhnlk , . . . ,Tuhnlk+11), . . . , S H(l+1)k1(Tuhn(l+1)k1 , . . . ,Tuhn(l+1)k1)

    ,

    and thenq = w,T0,T

    1, . . ., B = {b0, b1, b2, . . .}.

    Check that q p and

    q Qk the set B Sklocalizes X

    (or see the end of the proof of 2) below).

    2) The construction of the condition q required in (Sk)Qk is similar to that in1). Here, however, we have to take care of all names for elements of [] fromthe model N (as well as names for ordinals to ensure the genericity).

    Let n : n enumerate all Qk-names from N for ordinals and let An : n list all names (from N) for infinite subsets of . Of course, both sequencesare not in N but all their initial (finite) segments are there.

    Now we inductively define sets Bn = {bn0 , bn1 , b

    n2 , . . .} []

    and conditionsqn = w,Tn0 ,T

    n1 ,T

    n0 , . . . Qk such that Bn, qn N:

    To start with we put B0 = , q0 = p = w,T0,T1, . . ..

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    Arriving at stage n > 0 we have defined Bn1

    , qn1

    N. We define Bn

    , qnapplying the following procedure inside the model N (so the result will be there;

    compare this procedure with that in part (1)):

    Let 0 be a Qk-name for an integer such that Qk (i < n)(|0 Ai| > 2). Wemodify qn1 (passing to a pure extension of it) and we assume that

    ()0,0,...,n1 if v Rn1i (), i and there exists a pure extension of

    w v,Tn1i+1 ,Tn1i+2 , . . . deciding the value of one of 0, 0, . . . , n1 then

    w v,Tn1i+1 ,Tn1i+2 , . . . decides it already

    (see 2.4.1). Next, by Claim 2.4.4, we find n0 and a nice norm H0 on n0 suchthat

    H0(n0) 1 and ifT

    SH0((Tn10 )

    uh

    , . . . , (Tn1n01)

    uh

    ), T

    > 0,w w Ln10 () then there exists v cont(T

    ) such that

    w v,Tn1n0 ,Tn1n0+1

    ,Tn1n0+2e . . . decides the value of 0.

    Let Tn0 = SH0((Tn10 )

    uh, . . . , (Tn1n01)uh) and let bn0 be greater than all possible

    values of 0 (i.e. the values forced in the condition on H0 above). Let 1 be aQk-name for an integer such that Qk (i < n)(| [b

    n0 , 1) Ai | > 2). We modify

    the tail of qn1 and we assume ()1,0,...,n1 (for i n0, qn1). Next wechoose n1 > n0 and a nice norm H1 on [n0, n1) such that

    H1([n0, n1)) 2 and ifT SH1((T

    n1n0

    )uh, . . . , (Tn1n11)uh), T >

    0, w w Ln1n0 () then there exists v cont(T) such that

    w v,Tn1n1

    ,Tn1n1+1, . . . decides the value of 1.

    Let Tn1 = SH1((Tn1n0

    )uh, (. . . ,Tn1n11)uh) and let bn1 be greater than all possibil-

    ities for 1 in the above property.We continue in this fashion and we determine integers n0 < n1 < n2 < . . .,bn0 < b

    n1 < b

    n2 < . . . and nice norms H0, H1, H2 . . . and we define creatures

    Tni = SHi((Tn1ni1

    )uh, . . . , ((Tn1ni1)uh). Finally we let Bn = {bn0 , b

    n1 , b

    n2 , . . .} and

    qn = w,Tn0 ,T

    n1 , . . . (actually we should have taken more care while getting

    ()i,0,...,n1 in constructing Tni : this is necessary for lim

    iTni = ).

    Suppose now that Y [] is a set Sk-localizing [] N. Then for eachn we have

    (i)(m < k)(| [Y(i + m), Y(i + m + 1)) Bn| 2).

    We inductively define increasing sequences in : n , ln : n of integersand a sequence Tn : n of creatures:

    Let i0 be such that

    (m < k)(| [Y(i0 + m), Y(i0 + m + 1)) B1| 2).

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    Let j0m

    (for m < k) be such that B1

    (j0m

    ) is the first element of

    [Y(i0 + m), Y(i0 + m + 1)) B1.

    Let T0 = S(T1j00+1

    ,T1j01+1

    , . . . ,T1j0k1

    +1). Note that the creatures T1j were ob-

    tained as results of the operation SHj (for some norms Hj) and hence theirroots (i.e. ) are > k-splitting points, thus no danger can appear in this proce-dure. Finally we choose l0 such that T

    0

    (T0, . . . ,Tl01).

    Assuming that we have defined in, ln, Tn, take in+1 > in + k such that

    (m < k)(| [Y(in+1 + m), Y(in+1 + m + 1)) Bn+2| 2)

    and if Bn+2(j) [Y(in+1), Y(in+1 + 1)) then Tn+2j

    (Tln , . . . ,Tm) (for

    some m > ln) and Tn+2j n + 1. Now, let jn+1m (for m < k) be such thatBn+2(j

    n+1m ) is the first element of [Y(in+1 + m), Y(in+1 + m + 1)) Bn+2.

    Finally, Tn+1 = S(Tn+2jn

    +10 +1

    , . . . ,Tn+2jn

    +1k1+1

    ) and ln+1 is such that

    Tn+1 (Tln , . . . ,Tln+11).

    The condition q is w,T0,T1, . . .. Clearly q p. To show that it is (N, Qk)-

    generic suppose that N is a Qk-name for an ordinal, say = n, and letq = w,T0,T

    1, . . . q be a condition deciding . Look at the construction

    of qn+1 = w,Tn+10 ,T

    n+11 , . . .. Because of ()0,0,...,n , if v M

    n+1i () and

    there is (in N) a pure extension of w v,Tn+1i+1 ,Tn+1i+2 , . . . deciding n then

    w v,Tn+1i+1 ,Tn+1i+2 , . . . decides it already.

    Let i be such that w Mn+1i (). Then there exists a pure extension ofthe condition w,Tn+1i+1 ,T

    n+1i+2 , . . . deciding n, e.g. that one which can be

    obtained from q (note that q and w,Tn+1i+1 ,Tn+1i+2 , . . . are compatible). By

    the elementarity of N there is such an extension in N. This implies thatw,Tn+1i+1 ,T

    n+1i+2 , . . . decides n (and the decision belongs to N). The condi-

    tions q and w,Tn+1i+1 ,Tn+1i+2 , . . . are compatible, so the values given by them to

    n are the same and we are done (with the genericity).Now we want to show that

    q Qk (X N[GQk ] [])((X, Y) Sk).

    Let A N be a Qk-name for an infinite subset of, say A = Am. We are going

    to prove thatq Qk (

    n )(i < k)(| [Y(n + i), Y(n + i + 1)) Am | 2).

    Let q = w,T0,T1, . . . q. Fix l > m. Look at T

    l for some m

    , m,l < m < m we have Tl

    (Tm , . . . ,Tm) and w

    i l and t Tl such that Tl above t comes from T

    n by decreasing norms

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    (like in the operation of taking the upper half but possibly with other valuessubtracted) and refining. Note that neccessarily |succT

    l(t)| = k and Tl above

    each successor of t refines some Tn+1jni+1 (for i < k) modulo decreasing norms by

    some values, which does not influence contributions. Next we find v cont(Tl)(actually v = v0 . . . vk1, vi is included in the contributions of the partabove that successor of t which refines Tnjn

    i+1) such that w

    v,Tl+1,Tl+2, . . .

    decides the values of the names jn0 +1, . . . , jnk1+1 (defined in the construction

    ofTn+1j s) and forces them to be bounded by bjn+10 +1, . . . , bjn+1

    k1+1, respectively.

    By the choice of j s this means that

    w v,Tl+1,Tl+2, . . . Qk (i < k)(j < n)(| [b

    njn

    +1i

    , bjn+1i

    +1) Aj | 2).

    Since m < l n and Y(in + i) Bn+1(jni ) < Bn+1(jni + 1) < Y(in + i + 1)(for all i < k) we get

    w v,Tl+1,Tl+2, . . . Qk (i < k)(| [Y(in + i), Y(in + i + 1)) Am | 2).

    As l > m was arbitrary we are done. The Main Claim and the theorem areproved.

    Remark: More examples of forcing notions distinguishing the localization prop-erties we have introduced in this section will be presented in [RoSh:470]. Theyare (like the forcing notion Qk) applications of the general schemata of thatpaper.

    3 Cardinal coefficients related to the localiza-

    tions

    In this section we discuss cardinal coefficients related to the localization prop-erties introduced earlier.

    Following Vojtas (cf [Voj]) with any relation R X Y we may associatetwo cardinal numbers (the unbounded and the dominating number for R):

    b(R) = min{|B| : (y Y)(x B)((x, y) / R)}

    d(R) = min{|D| : (x X)(y D)((x, y) R)}.

    For purposes of applications these cardinals are introduced for relations R X Y such that dom(R) = dom(cR) = X and rng(R) = rng(cR) = Y. Notethat for each such relation we have

    b(R) = d(cR1), d(R) = b(cR1), b(R1) = d(cR), d(R1) = b(cR).

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    All results of the previous sections provide information on dominating numbersd(R) for the considered relations. Let

    b = b() = min{|F| : F & (x )(y F)(n )(x(n) < y(n))}

    d = d() = min{|F| : F & (x )(y F)(n )(x(n) y(n))}

    non(K) = min{|X| : X 2 & X is not meager }.

    These are three of ten cardinal invariants forming the Cichon Diagram. For moreinformation on the cardinals related to measure and category see [BJSh:368] or[CiPa].

    Corollary 3.1

    1. (see 1.2) (k )(d(Rk) = c)

    2. (see 1.3) (k > 0)(d(cRk) = b)

    3. (see 1.4) (k )(d(cRk) = d)

    4. (see 1.6) (k )(d(Rk) = d(R0 ))

    5. (see 1.9) b d(R0) non(K)

    Corollary 3.2

    1. Con(d(R0) < d)

    2. Con(d < d(R0))

    Proof For the first model add 2 Cohen reals to a model of CH. Then inthe extension we will have d = 2 and d(R

    0) = 1 (see 1.10). The second model

    can be obtained by adding 2 random reals to a model of CH, which results ina model for d = 1 and d(R

    0) = 2 (see 1.10).

    Corollary 3.3 Let be an increasing function. Then:

    1. (see 2.1) b = d(S1) d(S2) . . . d(Sk) . . . d(S+),

    d(S+) d(S+), d(S+) d, and

    if is increasing fast enough (e.g. (n) > 22n) thend(S+) d(S+),

    2. (see 2.2) d(S+) non(K). .

    Corollary 3.4 Let be an increasing function. Then

    Con(d(S+) = 1 + non(K) = d = 2).

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    Proof Start with a model for CH and add to it first 2

    Cohen reals (what

    causes d = 2 but keeps d(S+) = 1) and next add 2 random reals (what

    preserves d = 2, d(S+) = 1 but causes non(K) = 2), see 2.3.

    Corollary 3.5 Letk > 0. Then

    Con(d(Sk) < d(Sk+1)).

    Proof Let Qk be the forcing notion from the proof of 2.4. It is proper (see2.4.1). To get the respective model it is enough to take the countable supportiteration of the length 2 of the forcing notions Qk over a model of CH. As Qkdoes not have the Sk+1-localization we easily get that in the resulting model wewill have d(Sk+1) = 2. The only problem is to show that the iteration has theSk-localization property (to conclude that in the extension d(Sk) = 1). Butthis is an application of 3 Chapter XVIII of [Sh:f]. We may think of Sk asa relation on (after canonical mapping). Keeping the notation of [Sh:f] weput:

    S S

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    [Bar2] T.Bartoszynski, Additivity of measure implies additivity of cat-egory, Transactions of the American Mathematical Society 281(1984):209213.

    [BJSh:368] T.Bartoszynski, H.Judah, S.Shelah, The Cichon Diagram, Journalof Symbolic Logic, 58 (1993):401423.

    [BsSh:242] A.Blass, S.Shelah, There may be simple P1 and P2 points andRudin-Keisler ordering may be downward directed , Annals of Pureand Applied Logic, 33 (1987):213-243.

    [CiPa] J.Cichon, J.Pawlikowski, On ideals of subsets of the plane and onCohen reals, Journal of Symbolic Logic, 51 (1986):560-569.

    [GoSh:448] M.Goldstern, S.Shelah, Many simple cardinal invariants, Archivefor Mathematical Logic, 32 (1993):203-221.

    [Kun] K.Kunen, Set Theory (An introduction to Independence Proofs),NorthHolland, Amsterdam, 1983.

    [Jec] T.Jech, Set Theory, Academic Press, New York 1978.

    [NeRo] L.Newelski, A.Roslanowski, The ideal determined by the unsymmet-ric game, Proceedings of the American Mathematical Society, 117(1993):823-831.

    [Sh:207] S.Shelah, On cardinal invariants of the continuum, Proceedings ofthe Conference in Set Theory, Boulder, June 1983; ed. J. Baum-

    gartner, D. Martin and S. Shelah, Contemporary Mathematics31 (1984):183-207.

    [RoSh:470] A.Roslanowski, S.Shelah, Norms on possibilities: a missing chapterof Proper and Improper Forcing, in preparation.

    [RoSh:475] A.Roslanowski, S.Shelah, More forcing notions imply diamond,Archive for Mathematical Logic, to appear.

    [Sh:b] S.Shelah, Proper Forcing, Lecture Notes in Mathematics 940,Springer 1982.

    [Sh:f ] S.Shelah, Proper and Improper Forcing, Springer, to appear.

    [Vo j] P.Vo jtas, Topological cardinal invariants and the Galois-Tukey cat-egory, Proceedings of the Conference on recent developments ofGeneral Topology and its applications, in memory of F.Hausdorff,Academic Verlag.