Combinatorics for the Dominating and Unsplitting Numbers

Combinatorics for the Dominating and Unsplitting NumbersAuthor(s): Jason AubreySource: The Journal of Symbolic Logic, Vol. 69, No. 2 (Jun., 2004), pp. 482-498Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/30041739 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 69, Number 2, June 2004

COMBINATORICS FOR THE DOMINATING AND UNSPLITTING NUMBERS

JASON AUBREY

Abstract. In this paper we introduce a new property of families of functions on the Baire space, called

pseudo-dominating, and apply the properties of these families to the study of cardinal characteristics of

the continuum. We show that the minimum cardinality of a pseudo-dominating family is min{t, 0 }. We

derive two corollaries from the proof: r 2 min{, u} and min{, r} = min{, r~,}. We show that if

a dominating family is partitioned into fewer that s pieces, then one of the pieces is pseudo-dominating. We finally show that u < g implies that every unbounded family of functions is pseudo-dominating, and

that the Filter Dichotomy principle is equivalent to every unbounded family of functions being finitely

pseudo-dominating.

S1. Introduction. The purpose of this paper is to introduce a useful class of families of functions on co. Studying the properties of these families provides some new insight into the relationship between some cardinal characteristics of the continuum. We call these families of functions pseudo-dominating, and as the name suggests, these families are closely related to dominating families. These families will be shown to have connections with many other cardinal characteristics. In this section, we give the definitions and relevant properties of the cardinal characteristics and other concepts which we will refer to in the main body of the paper.

DEFINITION 1. A family F C "oo is unbounded if and only if for every f E 0ow there exists some g E 7 such that g f* f, where g 0* f means that f(n) > g(n) for all but finitely many n. The minimum cardinality of an unbounded family 2c C0 W is called the boundedness number, and is denoted b.

DEFINITION 2. A family 9 C "wo is dominating if and only if for each f E Wo(

there is a g C 9 with f <* g. The dominating number, a, is the smallest cardinality of any dominating family.

It is clear that b < Z. It is known to be consistent with the axioms of set theory that b < 0. This is the case, for example, in the model obtained by adding N2 Cohen reals to a model of set theory.

We record here an easy and well known fact which will be referred to later. This observation and its proof are folklore.

LEMMA 1.1. If a dominating family is partitioned into fewer than b pieces, then one of the pieces is dominating.

Received August 27, 2003; final version February 4, 2004.

© 2004, Association for Symbolic Logic 0022-4812/04/6902-0011/$2.70

482



COMBINATORICS FOR THE DOMINATING AND UNSPLITTING NUMBERS 483

It will be necessary to consider families of functions which are very close to being dominating. These definitions were first made in [4].

DEFINITION 3. A family of functions f C_

"w is k-dominating if the collection ' = {g : (3F = k)(Vn e wo)g(n) = max{f(n) : f e F}} is dominating. A family of functions & is finitely dominating if the collection S= {g: (3F c F)(IF| < w)(Vn E co)g(n) = max{f(n) : f e F}} is dominating.

We define the following cardinal characteristic for the purpose of stating the subsequent theorem. This cardinal characteristic, called the groupwise density number, was introduced in [5] and studied extensively in [3].

DEFINITION 4. A family ' C [wc]" is groupwise dense if 1. if Ye and X C* Y, then X E ', and 2. if H = {I, : ne w } is an interval partition of co, then there is an infinite

A C cw such that UnEA In E .

The groupwise density number, g, is the minimum number of groupwise dense families with empty intersection.

The groupwise density number is provably less than or equal to 0, and it is strictly less than 0 in the model of set theory obtained by adjoining N2 Cohen reals to a model of set theory. However, g is incomparable with b. It is strictly larger than b in the model obtained by adding N2 superperfect reals to a model of the continuum hypothesis, and it is strictly smaller than b in the model obtained by adding N2 Hechler reals to a model of the continuum hypothesis.

A slightly weaker version of the following theorem is implicit in the work of Mildenberger [10]. It was explicitly noticed and improved to the present version by Blass [4]. It should be compared with Lemma 1.1 and Theorem 5.1.

THEOREM 1.2 (Blass [4]). If a dominating family 9 is partitioned into fewer than g sets, then one of the pieces is 2-dominating.

Ultrafilters are a main tool used in this paper.

DEFINITION 5. A filter on cw is a family S- C_

9(o) satisfying the following conditions:

1. IfX e Y and X C Y, then Ye Y, 2. For any finite F

_ -, I") FE .

A filter is called proper if it does not contain the empty set. A proper filter is called non-trivial if it contains all cofinite sets. An ultrafilter on co is a filter such that for every X C o either X or its complement belongs to 5. A base for a filter Y is a subfamily 4 of 7 with the property that for any X e J there is some Ye . such that Y C X. A pseudobase for a filter . is a family

c_ [co]" with the property

that for any X e " there is some Y e 6 such that Y c X.

Every filter we consider will be a non-trivial proper filter on co, and so we will often just say "filter," when we mean "non-trivial proper filter on 0c."

We note here a close connection between the dominating number and ultrafilters. First, for any ultrafilter / on wco one can define an associated ordering on "w as follows: For f,g E "o, f

s<_ g if and only if {n : f(n)

_ g(n)} e '. Then



484 JASON AUBREY

cf(?-prod w) is defined as the minimum cardinality n of a subset Y c 0w such

that for every f E Wow there exists a g E 7 such that f _<

g. Let X(1) denote the smallest cardinality of a base for W. Then a result of Nyikos [11] states that cf(1-prod w) x X(W)

_ i. This fact will be used later, as will the observation that

cf('-prod w) is always a regular cardinal. We also define here some notation we will be using later. Given a function

h : wo - wco and a set X C w, h(X) will denote, as usual, the image of X under the function h. However, if . is a family of subsets of ow, then h(.) will denote the collection {X : h-' (X) E }. This last notion will be of particular importance when working with filters and ultrafilters because if # is a filter (ultrafilter) and h : w - w is finite-to-one, then h(-) is also a filter (ultrafilter). Properties of filters are in some cases preserved under this map. In particular, we will need the observation that given an ultrafilter ? and a finite-to-one function h : w -+ o, cf(1-prod w) = cf(h (/)-prod w).

We will also consider the following types of ultrafilters:

DEFINITION 6. An ultrafilter / is a P-point if, for every countable descending sequence X1 2* X2

_* X3 -. with each

X,0 E W, there exists a set X E 6 with

X C* X,~ for each n E w.. Ketonen [8] established the following result which we will refer to later. THEOREM 1.3. Every ultrafilter generated by fewer than t sets is a P-point. We will also be concerned with the following cardinal characteristic:

DEFINITION 7. The ultrafilter number u is the minimum cardinality of a base for a non-principal ultrafilter on w.

Solomon [12] proved that b _

u, and b < u holds in the model obtained by adding N2 Cohen reals to a model of set theory.

Ultrafilters are closely related two other cardinal characteristics which we will use. To define them, we first define a notion of splitting and the associated cardinal characteristic, the splitting number: X

_ w splits an infinite set Y w c if both

Y n X and Y - X are infinite.

DEFINITION 8. A family ' of infinite subsets of co is a splitting family if for every Y E [cw]" there is some X E ' which splits Y. The splitting number is s = min{ '] : 7 is a splitting family.}.

DEFINITION 9. A family 2 of infinite subsets of co is unsplittable if no single set splits all members of F. It is a-unsplittable if no countably many sets suffice to split all members of 2. The unsplitting number t is the smallest cardinality of any unsplittable family. The a-unsplitting number tr, is the smallest cardinality of any a-unsplittable family.

The unsplitting number is also referred to in the literature as the refining number or the reaping number. Clearly t < tr.

It is also easy to see that r < u: Suppose that ' is an ultrafilter, and that ,

is a base for W'. If X C were to split every set in 9, then neither X nor co \ X could be in ?. Solomon [12] demonstrated how one can define an unbounded family of functions from an unsplittable family of sets, so we have b < i. It is also the case that s < Z (see [2]). Each of the last three inequalities can consistently be strict.




The unsplitting number r is incomparable with t and g. We will use the following facts concerning these concepts. First, we observe that any unsplittable family . which is also a filter is in fact an ultrafilter. To see this, we only need to show that for every X

_ w either X E 7 or w \ X E Y. Since Y is unsplittable, for every

X c ow there exists a Y E Y such that Y C* X or Y _*

co \ X. Since # is closed upwards and contains all co-finite sets, this implies that X E - or cw \ X E 5r. The unsplitting number has an even stronger connection to ultrafilters.

THEOREM 1.4. [1] The minimum cardinality ofa pseudobase for an ultrafilter is i. The cardinal tq is also of importance in the theory of ultrafilters. The following

theorem is folklore: LEMMA 1.5. If W is a P-point, then any base for e has cardinality at least rt. In the next section we define the main concepts employed in this paper and

establish some properties of these concepts. While giving the definition of b we also gave the definition of the relation <*. Below we will restrict this relation to subsets of cw. For functions f, g E OcW and subset Q c wo, when we say f 0* g on Q we mean that Q \ {n : f(n) < g(n)} is finite.

S2. Pseudo-domination.

DEFINITION 10. A family ' C wo is n-pseudo-dominating (written n-yi-dominating) if for every g E Wo there exists an interval partition H of cw such that for every union of H-blocks, say Q, there exist n functions f l ..., f, E ~7 such that max{fi,..., fn} * g on Q or max{fl,..., fn} * g on w\Q. We call a family y-dominating if it is 1-y/-dominating. Finally, we call a family F finitely ry-dominating if for every g E "0w there exists an interval partition H of co such that for every union Q of H-blocks there exists a finite set F C & such that the function f defined by f(n) = max{h(n) : h E F} satisfies f 2* g on Q on ow \ Q.

In proving that a family of functions is n-i-dominating we will often employ the following notion and the subsequent theorem.

DEFINITION 11. A set C_

co is n-big if and only if for every upward closed

" C [wCO] and every function g : co --, co such that Y contains {nlf(n) _

g(n)} for every f E ' there exists an increasing finite-to-one function h : c - co such that the set {h(X1 n X2 n ... n X,) : X1, X2

.... X, E Y} is unsplittable. The notion

of "finitely big" is defined similarly.

LEMMA 2.1. A family is n-big if and only if it is n-ly-dominating. PROOF. Suppose that F is n-big, and let g E WW. Let Y be the upward closure of

the collection of sets {{j : g(j) > f(j)} : f E }. Since ' is n-big there exists an increasing finite-to-one h : co - co such that {h(XlnX2n... . nX,) : X1, X2 ..., Xn E Y} is unsplittable. Define an interval partition H by letting li = h-'(j). Let Q be any union of H blocks. Let R c co be such that h-' (R) = Q. Then there exist X1, X2, ... . Xn E " such that

h(X~ nX2 n. -nXn) c* R or

h (Xi X02 ['-1.. -nXn) c* co\R.



486 JASON AUBREY

But then

Xinx2 n... x * h-'(R)= Q or

X1 n X2 N-.-n X, C* 0\h-'(R) -0\Q

Find fl,f2, .. fn E such that {j : fk(j) _

g(j)} _

Xk for 1 < k < n. The first conclusion implies that g 0* max{f l, f2, ... fn } on w\Q and the second implies that g 0* max{f l, f2 ... ,n on Q. Thus ' is n-u-dominating.

Now suppose that ' is n-yi-dominating. If ' is n-dominating then given a function g and # as above, the family {X1 n X2 n ... n X, : X1, X2,... Xn 7} contains a finite set, and so given any h : wO - w, {h(X1 n X2 N

- ) n Xn) X, X2,..... Xn E } is unsplittable. Suppose that F is not n-dominating and suppose, without loss of generality, that our g was chosen so that it is not dominated by the maximum of any n functions from F. Let H be an interval partition of w such that for any union Q of H blocks there exist f l, f2

.... fn, e such that

max{f l, f2 . .

, fn } g on Q or w\Q. Let h : w -* wc be such that h(I/) = n for each I, E H. Let R C w, Q = h-'(R) and let fl, f2 ..., fn E c be such that

{j:g(j) . max{fl(j), f2(j),. , fn(j)}} _*

h-1(R)= Q, or

{j: g(j) > max{fl(j), f2(j). ..,

fn(j)}} C* w\h-'(R) = w\Q.

Then

h({j:g(j) . max{fl(j), f2(j) ..,

fn(j)}}) c* R, or

h({j:g(j) max{fl(I),f2z(j)...fn(j)}}) C* w\R.

The set {j : g(j) _

max{fl (j), fz(j) ... f, (j)}} is an intersection ofn sets from 7. So {h(Xi n

"

- n X,) : Xl

..., Xn E Y} is unsplittable. - We note here that a very slight modification of this proof shows that finitely

pseudo-dominating is equivalent to finitely big.

DEFINITION 12. 0,

is the minimum cardinality of a k-y-dominating family and

0* is the minimum cardinality of a finitely y-dominating family.

To provide some orientation regarding these concepts we establish the following lemmas.

LEMMA 2.2. If T C W" is a n-dominating family of functions, then it is n-yy- dominating.

PROOF. Suppose that C ( "wo is n-dominating. The set 2' = {max{fl, f2,....

fn, } : f l, f2, . f , n 2} is then dominating. Therefore 2' is a/-dominating. It then follows that 7 is n-y/-dominating. -

LEMMA 2.3. For all n E o, b* = Z* < Z. PROOF. From Lemma 2.2 an n-dominating family is n-y-dominating, giving the

last inequality. The definitions clearly imply that a n-y-dominating family is finitely ey-dominating. Closing an n-y-dominating family under maximums of n many functions turns an n-,-dominating family into a 1-y-dominating family with the




same cardinality as the original family. Similarly, closing a finitely y-dominating family under finite maximums turns it into a 1-yi-dominating family with the same cardinality. These last observations imply that the minimum cardinality of an

n-,- dominating family and the minimum cardinality of a finitely ag-dominating family are the same; i.e., 0* = 0

In light of this lemma we will only refer to 0* when proving theorems about the size of y-dominating families.

S3. a-pseudo-domination. The next definition introduces a slight variation on ry-dominating families. These concepts will be seen to be more closely related than is at first apparent.

DEFINITION 13. We say that a family 2' C "wco is n-a-pseudo-dominating (written

n-a-t-dominating) if for every g E "w there exists an interval partition H of wo such that for every countable collection { Qk k, of unions of H-blocks, there exist n functions fi,...,.. f, E such that for each k e cw max{fl,..., f,}

_* g on

Qk or max{fi,....

f,} >* g on cw\Qk. We call a family a-y-dominating if it is I-ra-y-dominating. Finally, we call a family 7 finitely

a-,-dominating if for every

g e Wo there exists an interval partition H of w such that for every countable

collection { Qk k0w

of unions of H-blocks there exists a finite set F C F such that for each k E cw the function f defined by f(l) = max{h(l) : h e F} satisfies f * g on Qk or on c \ Qk.

As is the case with ii-dominating families, we have the following definition which will be used in applications of r-yq-dominating families.

DEFINITION 14. A set C wco is n-a-big if and only if for every upward closed Y C [o]w and every function g : co --+ co such that " contains {nlf(n) < g(n)} for every f e 2 there exists an increasing finite-to-one function h :co - co such that the set {h(X n X2 n X,) :

Xl, X2 ..., Xn E .} is a-unsplittable.

We will develop the theory of a-qy-dominating families in parallel with the theory of

,-dominating families. For the most part the proofs are very similar. In general

we will simply state the results and indicate how the proofs differ. LEMMA 3.1. A family ' is n-a-big if and only if it is n-a-ay-dominating. PROOF. Suppose that 2 is n-a-big, and let g L Wo. Let Y be the upward closure

of the collection of sets { {j : g(j) > f(j)} : f E '}. Since 2" is n-a-big there exists afinite-to-oneh :co -+ wsuch that {h(XinX2n. . nX,) : X1, X2, ... X, E } is a-unsplittable. Define an interval partition H by letting Ik = h-l(k). Let { Qk

}k0, be a countable collection of unions of H blocks. Let Rk C co be such

that h-'(Rk) = Qk. We then proceed as before, showing that for each k E co, g 0* max{fi, f2 .

.. f} lon o \Qk or g <* max{fl,

f2 ....

f, o)on

Qk. Thus 2 is a-yi-dominating.

The other direction is similarly a slight variation of the corresponding direction from Lemma 2.1. -1

DEFINITION 15. 0n,,

is the minimum cardinality of a n-ar-y/-dominating family and 0: is the minimum cardinality of a finitely r-y/-dominating family.

LEMMA 3.2. If ' C 'wco is a n-dominating family of functions, then it is n-a-iy- dominating.



488 JASON AUBREY

PROOF. As in the proof of Lemma 2.2, if f is n-dominating, then the collection F' = {max{fl, f2 : fl, f2 .

... fn E '} is dominating. Then 7" is

a-yi-dominating, and so 7 is n-a-y-dominating. - LEMMA 3.3. For all n E 0, * < 0

,, = Z

<_ .

PROOF. From Lemma 3.2 an n-dominating family is n-a-y-dominating, giving the last inequality. The definitions clearly imply that a n-a-yi-dominating family is finitely a-y-dominating. Closing an n-a-yi-dominating family under maximums of n many functions turns an n-a-y-dominating family into a 1-a-yI-dominating family with the same cardinality as the original family. Similarly, closing a finitely a-yi-dominating family under finite maximums turns it into a 1-a-ty-dominating family with the same cardinality. These last observations imply that the minimum cardinality of an n-a-g-dominating family and the minimum cardinality of a finitely a-y-dominating family are the same; i.e., 0Z = ),.

We now turn to the main theorem we will use to apply these concepts.

S4. Main theorems. The following theorem will be of central importance in applications of pseudo-dominating families to cardinal characteristics. It is a gener- alization of a result proved by Blass and Mildenberger [6].

THEOREM 4.1. Let r and ' be subsets of [w]", and let I = {In = [an,an+l) n E } be an intervalpartition of co. Then either

1. for every pair of infinite disjoint sets D and D' which are each a union of intervals from I and such that no interval in D is adjacent to one in D', every set in r has infinite intersection with both D and D'; or

2. there exists a pair of disjoint sets D and D' which are each a union of intervals from H, such that no interval in D is adjacent to one in D', and such that every set in 7 has infinite intersection with both D and D'; or

3. there exists a set X E Y and an increasingfinite-to-onefunction f such that the set {f(X n Y): Y e 7'} is unsplittable.

PROOF. As above, we let Y, ' c [wo]', and let H = {I,~ = [as, an+l) : ne w } be an interval partition of co. There are three cases to consider.

CASE 1. Suppose that for every X E .7, X meets all but finitely many intervals of H. Then let D and D' be a pair of infinite disjoint sets D and D' which are each a union of intervals from H and such that no interval in D is adjacent to one in D'. (For example, one could form D and D' in this manner: D = Ukem I4k and D' = Ukew 14k+2.) We need to establish that both D and D' have infinite intersection with every set in 7.

Recall that every X e .7 meets all but finitely many intervals of H. Now D and D' both contain infinitely many intervals of H. This implies that X must intersect all but finitely many intervals in D and all but finitely many intervals in D'. In particular, every X E .7 intersects both D and D' infinitely. This takes care of Case 1.

The next two cases require some preparation. In both cases we will assume that we are not in case 1, and so we fix an X E 7 which has empty intersection with infinitely many intervals of H.

We will define an interval partition tP with the following property. The first interval of ' will be the union of all H intervals up to and including the first H




interval which X does not intersect, say Io. The second interval of ' will be the union of all H blocks from 1,0+1 up to and including the next interval of H which X fails to intersect. We continue in this manner to define '. We do this formally as follows: Let L = {nk k E cw} C co be such that nk E L if and only if X misses

Ink, and suppose no < nil < .*-.

Define Ro = Uro0 Ij, and given m E co let

Rm+l =

Ujrm:1 Ij Let f : w -+ co be constant on exactly the intervals from the partition '. CASE 2. Recall that we have fixed a set X e Y which misses infinitely many

blocks of H. Suppose that there exists an infinite and coinfinite set Q C co such that for every Y 7

f the sets X n Y n f-'(Q) and X n Y n f -' (wo\ Q) are infinite. Then define D = f-'(Q) - Uk=0o Ink and D' = f-'(co\Q) - Uko'= Ink. These sets are clearly disjoint. Notice that f-'(Q) is a union of intervals and f-1(w \ Q) is the complementary union of ' intervals. Therefore the intersection of D and D' is empty.

Also, Uk-0

Ik is a union of those H intervals which fail to intersect X. Recall that each ' interval is a union of H intervals, the last of which is disjoint from X. Therefore D = f-'(Q) - Uk=O Ink and D' = f-'(cw\Q) - Uk=0 IN'k are unions of H blocks, and if In is in D, neither I,+1 nor In-1 can be in D'. In other words, no interval in D is adjacent to one in D'.

Finally, suppose that Y E 7. We are assuming that for every Y E ' the sets X nY n f -'(Q) and X n Y n f-l(w\Q) are infinite. Now

OO< OO

DO Y= (f-'1(Q)- U In'k) Y f-'(Q)- U Ink)n(Yn X). k=O k=O

But f-'1 (Q) n (Y n X) is infinite, and Uk0 Ink n (Y n X) is empty. Therefore, 00

DN Y -(f-l(Q)- UInk)n Y k=O

is infinite. The same argument shows that 00

D' nY = (f-'(wo \ Q) - U nk)n

Y k=O

is infinite. Therefore every set in ' has infinite intersection with both D and D'. This establishes the second case. We now turn to the final case. CASE 3. Suppose that for every infinite and coinfinite Q co there exists a Y E '

such that at least one of the sets Xn Yn f -' (Q) or Xn Yn f -' (w\Q) is finite. Now ifXn Ynf -' (Q) is finite, then Qnf (Xn Y) is finite. Similarly, ifXN Ynf -i (w\Q) is finite, then cw\Q n f(X n Y) is finite. Thus {f(X n Y)| Y e '} is not split by Q. But Q is arbitrary so we have that {f(X n Y)| Y E '} is unsplittable. Notice that the function f, defined before the discussion of cases two and three, is increasing and finite-to-one, as claimed in the theorem. -1

This theorem has the following corollary. COROLLARY 4.2. Let - and & be subsets of [o]", and let H = {I, = [an, an+l)l

n E co} be an interval partition of co. Then either case 1 or case 2 of Theorem 4.1 holds or 3 there exists a set X E 7 and an increasing finite-to-one function f such that the set { f (X n Yi n Y2n . . . 0 Yk) : Y1, Y2 . , Yk E } is unsplittable.



490 JASON AUBREY

PROOF. Given 7, 7, and H as above, apply the Theorem 4.1 to 7, 7' = { Y1 n

Y2 ... n Yk : Y1, Y2 ..., Yk E '}, and H. If case 1 of the theorem obtains with Y, then case 1 of the corollary clearly also obtains with -. If case 2 of the theorem obtains with "', then case 2 of the corollary obtains because the D and D' given by the theorem will also have infinite intersection with the sets in &. If case 3 of the theorem obtains with . and &', then case 3 of the corollary obtains with J and

We have the analogous facts for use with a-y/-dominating families.

THEOREM 4.3. Let Y and & be subsets of [Wo]", and let H = I = [an,an+l)l n co} be an intervalpartition ofo w. Then either

1. for every pair ofinfinite disjoint sets D and D' which are each a union of intervals from H and such that no interval in D is adjacent to one in D', every set in 7 has infinite intersection with both D and D'; or

2. there exists a countable collection of pairs of disjoint sets Dn and D( which are each a union ofintervalsfrom H, such that no interval in Dn is adjacent to one in D(, and such that for every X E & there is some n E ow such that X has infinite intersection with both Dn and D(; or

3. there exists a set X E 7 and an increasingfinite-to-onefunction f such that the set (f (X n Y): Y E '} is a-unsplittable.

PROOF. CASE 1. As in case 1 of the proof of Theorem 4.1 we suppose that for every X E 7, X meets all but finitely many intervals of H. Then D and D' are constructed exactly as in the proof of case 1 of Theorem 4.1.

As in the proof of cases 2 and 3 of Theorem 4.1, for the proofs of cases 2 and 3 here, we will assume that we are not in case 1. So we fix an X E 7 which has empty intersection with infinitely many intervals of H.

As in Theorem 4.1, we define a new interval partition ' as follows. Let L =

{nklk E c} ow be such that nk E L if and only if X misses I,,, and suppose

n~ j n ie o_ in0

,_ no < nl < ... Define Ro =

U,=o I, and given m w let

Rm+l =

U,-"=0+ Ij. Set

S= {Rk :kE cow}. Let f : w

-- w be constant on exactly the intervals from the partition '.

CASE 2. Recall that we have fixed a set X E Y which misses infinitely many blocks of H. Suppose that there exists a countable collection { Q,, }e such that for every Y E ' there exists some n E w such that the sets X n Y n f-l(Q,) and X n Y n f-' (cw\Q) are infinite.

Define, for each n E 0, Dn = f-'(Qn) - U.=olnk

and D,

= f-'(w\Qn) -

UJk=0 Ink,. Notice that for each n, f-' (Q,) is a union of'P intervals and f(w-1\ Q,)

is the complementary union of ' intervals. Therefore, for each n, the intersection of D, and D( is empty. We similarly follow the proof of case 2 of Theorem 4.1 to show that for each n, both D, and D( are unions of H blocks, and no interval in

D, is adjacent to one in D,. Finally, suppose that Y c . We are assuming that for every Y E ' there exists

some n E w such that the sets X n Y n f-'(Q,) and X n Y n f-l(o\Q,) are infinite. Find such an n for this Y. The corresponding sets D, and D( have infinite intersection with Y; this is shown as in the proof of case 2 of Theorem 4.1.

CASE 3. In this case we assume that for every countable collection { Q, } of infinite subsets of cw, there exists a Y e 7 such that for every n E w at least one of the sets




X n Y n f-'1(Q<n) or X n Y n f-'(w\Qn) is finite. Proceeding as in the proof of case 3 of Theorem 4.1, it follows that {f(X n Y)| Y c } is a-unsplittable. The function f is increasing and finite-to-one. -1

COROLLARY 4.4. Let .-

and Y be subsets of[co]w, and let H = {In = [a,, an+l) n E }o) be an interval partition of w. Then either case 1 or case 2 of Theorem 4.3 holds or 3 there exists a set X E " and an increasing finite-to-one function f such that the set {f(X n Yi n Y2 0

. . n Yk) : Y1, Y2 ....

Yk E 7} is r-unsplittable. PROOF. This Corollary is proved from Theorem 4.3 exactly as Corollary 4.2 is

proved from Theorem 4.1. -

S5. Applications to dominating families. We now present our first application of the idea of Theorem 4.1. From Section 1 recall Lemma 1.1 which concerned partitions of dominating families into fewer than b pieces, and recall Theorem 1.2 which concerned partitions of dominating families into fewer than g pieces. The following theorem provides another such result.

THEOREM 5.1. Suppose that ~ " %o is a dominating family, and ,e < s. If we partition 9 as

au<K

then one of the pieces is 2-qy-dominating. PROOF. Suppose that no S is 2-y-dominating. Therefore no o is 2-big. For

each a let Y0 and go witness that o is not 2-big. For each go define an interval partition He = {I, = [if , il) :n E cow} with the property that if x < if then ga(x) <

i+1. Notice that 7" contains the collection of sets {{n: f(n) _

ga(n)} : f E a). We next apply Theorem 4.1, applied with 7, in the roles of both - and '. Case 3 is ruled out since Y- and go witness the assumption that o is not 2-big, and case 2 is a weakening of case 1; so case 2 must hold. Thus we can find disjoint sets Da and D', each a union of intervals from Hi, with the property that no interval of

H0 contained in Da is adjacent to one in D', and with the property that every set

in Y- meets both Da and DL infinitely. We define sets EL and EL as follows: Define EL to be the union of all In, and

In,l such that In, C Da, and define EL as the union of all I," and

I,n+ such that

Ia C D . Notice first that since Da and D' do not contain adjacent intervals from Ha, it follows that EL and EL are disjoint. Now let X E [w]W, and suppose that Y0 contains a set Hx which differs only finitely from {n : next(n, X) < ga(n)}. We claim that X n EL and X n EL are infinite. To see that X n EL is infinite, notice first that since Hx E 70 it follows that Hx has infinite intersection with Da. Now suppose that the number k E w is an element of that intersection. If k E {n : next(n,X) 0 ga(n)) then next(k,X) e [k,ga(k)] n X. In this case, by definition of the interval partition Ha, it follows that k and next(k, X) are either in the same Ha interval or that next(k, X) is in the Ha interval after the one containing k. By definition of Ea, it follows that next(k, X) E Ea, and so in EL n X. Since there are infinitely many elements of the intersection Da n Hx, it follows that there



492 JASON AUBREY

are infinitely many elements of the intersection DQ m {n : next(n, X) _

go (n)}, and in turn infinitely many elements of X n EQ.

Similarly, X n Ee is infinite. Since EQ and E' are disjoint, it follows that X n EQ and X \ EQ are both infinite. Therefore, we have shown that if Hx c -, then EQ splits X.

However, note that for every X E [o]W, there exists some f E 9 such that next(-, X)

_* f. Now, there exists an a such that f e 0 and so {n : f(n) <

gQ(n)} E 3-. Also, {n : next(n,X) _

go(n)} D* {n : f(n) _

gQ(n)}. So Y- contains a set which differs only finitely from {n : next(n, X) < gQ (n)}. Therefore, every X E [o]W is split by some EQ, and so {EQ : a < s} constitutes a splitting family. This is a contradiction because n < s.

Therefore, some .

is 2-big, and so 2-yi-dominating. - COROLLARY 5.2. If an n-dominating family is partitioned into < s pieces, then some

union of n pieces is 2n-y-dominating. PROOF. Let 2' be a n-dominating family of functions and let F for a <

partition ' into a < s pieces. For each set F consisting of n ordinals < let

VF = (max{fl, f2.

... fn} fl,

f2...f e UyCF 'Y}' and let y =

{max{f1, f2 ....

fn : flf, f2, ...

n E '}. Then y is a dominating family, and the sets yF partition ( into < s pieces. Therefore, according to Theorem 5.1, one of the pieces, say yF, is 2-big.

Choose a function h, and for each subset G C UyEF 7 of size n consider the sets SG = {j : max{f f : G} < h(j)}. Since 'F is 2-big, it follows that there exists some finite to one function r : O * wo such that - = {r(SG, n SG2) GI, G2 C UyeF ~7y} is unsplittable. But = {r({j : fl(j) h(j)}n{j : fz(j) h(j)} n --. n {j : f2n(j) 0 h(j)}) : f l, f2, f2n E UyEF 7 }. Hence UeF 07 is 2n-y-dominating.

As another corollary to Theorem 5.1, we obtain a new proof of a result of Heike Mildenberger's [10].

THEOREM 5.3. cf(D) s PROOF. Suppose that 9 is a dominating family of size Z. Suppose n < s, and

suppose that we have partitioned .

as

S- U. . Qr<tK

We intend to show that the sets , cannot all have cardinalities less than D. Assume that they all do have cardinality < D. By the previous proposition, some o is 2-big. The closure of 90 under finite maximums has cardinality strictly less than 0, and therefore it is not dominating. Therefore, we may assume that Sa is closed under finite maximums. Now we can fix a function g : ow -+ c which is not 0* any function from PQ.

Let 3 be the filter generated by all co-finite sets plus the sets of the form {nlf (n) < g(n)} for all functions fE C9. As o is 2-big, there exists a finite-to-one function h : wo -, co such that {h(X n Y) : X, Y c Y} is unsplittable. Since 3T is a filter, {h(X n Y) : X, Y e = {h(X) : X e 7}. Clearly, {h(X) : X E K } = h(-). Since {h(Xn Y) : X, Y e 7} is unsplittable, it follows that h (7) is also unsplittable, and so is an ultrafilter.




In particular, we have an ultrafilter generated by < 0 sets. The result of Nyikos mentioned in the introduction implies that cf(h(J)-prod w) = 0, and this implies that 0 is regular. Since 0 2 s (see [2]), the proof is complete. -H

S6. Applications to unsplittable families. THEOREM 6.1. If 9 is an unsplittable family of sets, then the collection offunctions = {next(-, X) : X E } is a 2-,-dominating family.

PROOF. Suppose that 3 c [o]" is an unsplittable family and that the collection of corresponding functions next(-, X) for X E Q, called 2, is not 2-big. Then there are a function g E Wow and an upward closed set - with the following properties: Y7 contains all of the sets {n : next(n, X)

_ g(n) } for X E Q, and for every finite

to one function h E Wco the set

(h(Y n z) : YZ Z e }

can be split by a single set. Partition wo into finite intervals I, = [an, an+l) such that g(x) < an+l for all

x < an. Theorem 4.1 applies to give disjoint sets D and D', each a union of intervals I,, such that (a) no interval in D is adjacent to one in D' and (b) every set in 7 has infinite intersection with both D and D'.

Let E be the union of all the I,'s and I,+ 's such that I, C D. Define E' similarly from D', and note that E and E' are disjoint. We claim that if Y c then Y n E is infinite.

By hypothesis - contains {n : next(n, Y) < g(n)} for every Y E 9. So each {k E wo : next(k, Y) < g(k)} contains infinitely many k E D since D has infinite intersection with every set in Y. For each such k, next(k, Y) E [k, g(k)] n Y. By our choice of the intervals I,, this element of Y is either in the same interval as k or in its right neighbor. In either case, it is in E because k e D. Thus we have infinitely many elements of Y n E, as claimed.

Similarly, for each Y E 9, Y n E' is infinite and therefore so is Y\E, since E and E' are disjoint. Thus E splits every element of 9, a contradiction, since 91 was supposed to be unsplittable. -

THEOREM 6.2. If 9 is a a-unsplittablefamily ofsets, then the collection offunctions = {next(-, X) : X E ~9 } is a 2-o-y -dominatingfamily.

PROOF. The proof is a slight modification of the proof of Theorem 6.1, and uses Theorem 4.3 instead of Theorem 4.1. H

COROLLARY 6.3. If r < 0 then r = u = re,. PROOF. Suppose that r < 0, and let 9 be an unsplittable family of cardinality

r. By Theorem 6.1, the family 2F = {next(-, X) : X e ~9} is 2-big. Since t < 0, 2 is not finitely dominating. Let g : w o w be a function not dominated by the maximum of any finite collection of functions from 2'. Then the collection 7 = {{n : f(n) < g(n)} : f E 2'} has the finite intersection property. Since 2 is 2-big, there is a function h : wo -* ow be such that {h(X n Y) : X, Ye . } is unsplittable. This collection also has the finite intersection property and so generates an ultrafilter with a base of cardinality |1J =

-. Since u is the minimum cardinality

of a base for an ultrafilter and t < u, it follows that r = u. Notice also, that the ultrafilter generated by {h(X n Y) : X, Y c } is generated by < 0 sets, and so by



494 JASON AUBREY

Theorem 1.3, this ultrafilter is a P-point. Lemma 1.5 shows that this base therefore has cardinality at least rt, so we have r > to. Since t < to always holds, we have tc - ta --1

COROLLARY 6.4. 1. t > min{u, 0}. 2. t

_ min{ta, Z}.

PROOF. We first prove 1. If t < min{u, Zi}, then t < d. But then Corollary 6.3 implies that t = u, so we have a contradiction.

The proof of 2 is the same. -1 COROLLARY 6.5. 1. Z* = min{u, 0} = min{t, 0}. 2. = 0*. PROOF. To prove 1, we note that Lemma 2.2 implies that 0* < 0, and Theorem 6.1

implies that 0* K t. So 0* _

min{t, 0} _

min{u, 0}. Suppose 0* < min{u, 0}. Let 2 be a 2-big family of functions of cardinality 0*. Since 1'I < ?, ' is not finitely dominating, and so we can fix a function g : o - C which is not dominated by the maximum of any finitely many functions from 2'. Let =

{{n : f(n) _

g(n)} : f E '}. We can then find a function h : cw - w such that

- = {h(X n Y) : X, Y E -} is unsplittable. But we then have an unsplittable

family with the strong finite intersection property (by our definition of g). So .7

generates an ultrafilter, and therefore must have cardinality at least u. So we have a contradiction. Thus Z* = min{u, 0} = min{t, 0}.

To prove 2, we note that by Corollary 6 case 2 we know that min{r, b} =

min{t(r, 0}. Thus by Corollary 7 case 1, 0* = min{te, 0}. - In [7] Brendle has defined variants oft and

to which he calls the finitely unsplitting

number, and the finitely a-unsplitting number. These variants will be seen to have a close connection to 2-y-dominating families. Let {I, : n E ow} be a partition of co into finite sets. In the discussion below, a set X c co is said to split a partition {I, : n e co } if both {n : I, c X } and {n : In X = 0} are infinite. Then the finitely unsplitting number and the finitely a-unsplitting number are defined as follows:

ft = min{t 1f : ' consists of partitions of co into finite sets, and no

single X C ow splits every element of 7} ft, = min{I0'I : "

consists of partitions of co into finite sets, and no countable 2 C [w]m' splits every element of }.

In the same paper, it is proved that ft = min{t, 0}, and ft, = min{t,, 0}. There- fore ft = 0*. Brendle asked whether ft < frt, is consistent. The results above provide a negative answer to that question. Corollary 6.4 gives t > min{rc,, 0}, and it follows from this that ft = min{t, ,} = min{t,, i } = ft,.

S7. Pseudo-domination if u < g. We now explore pseudo-dominating families in the case when u < g. This can be seen as a strong negation of the continuum hypothesis, due to Blass and Laflamme's result that under this hypothesis there is only one near coherence class of ultrafilters [5]. Our result here depends on Laflamme's classification of subfamilies of co

./ o, the monotonically increasing functions on

co [9]. Laflamme's result classifies subfamilies of co / co into equivalence classes given by the following relation.




DEFINITION 16. [9] Let 7 and " be subsets ofw 7 a. Define 7 _ _

if

(3r E cw 7 cw)(Vf E 9)(3g E y)f 0* go r.

We say that 7 and " are equivalent if both '

_ y and y(

_ .

THEOREM 7.1. [9] Assume u < g. Then every family _

co w co not bounded by a constant is equivalent to one of the following:

1. w / w (The dominating case), 2. {f E cv / cv : (30n E o))f(n)

_ n} (The high case),

3. {f E c/ ,7

ov : {n : f(n) _

n} E W } = .(W) for some (or every ultrafilter ?) (The ultrafilter case),

4. {f E co / c: {n : f (n) _

n} E #-} = 9.(-) for the cofinite filter Y (The

low case), 5. {f E ov o : (3c E c)(Vn)(f(n) < c)} (The boundedcase). The point of this section is to establish the following theorem. THEOREM 7.2. Under the hypothesis u < g, every unbounded subfamily ofco 7 cv

is q-dominating. This will be proved in the sequence of lemmas below. LEMMA 7.3. Suppose that ', y " co / c, that F is q/-dominating, and that

- - 1. Then " is y/-dominating.

PROOF. Suppose that F is a ly-dominating subset of ov ,7

c and that y is a family of monotonically increasing functions above ' under Laflamme's relation; that is,

(3r E Wco) (~f E ~) (3h E ~)h o r 2* .f

Fix such a function r : co --

co, and without loss of generality, suppose that r is strictly increasing. For any g E ov / c define

. : cv - o by (n) = g(r(n + 1)).

Fix such a g and let H = {I, = [i,, in+l) : n E c} be an interval partition which witnesses the fact that 7 is qi-dominating for g. Let Jo = [0, r(il)) and for n > 0, let Jn = [r(in), r(in+l)). Then define T = {Jn : n E o}. Let Q be a union of T blocks, and let Q' be the corresponding collection of H blocks. That is, let Q' be such that In c Q' iff Jn C Q.

Then we can find a function f ~ such that

f * g on Q' or on co \ Q'.

Using the relationship between W' and y we can find h E ye such that h o r >* f. Then

h o r , on Q' or on w \ Q'.

Suppose this occurs on Q', and fix some large n E Q. Find m such that r(im) 0 n

< r(im+l). Then there exists some k E [im, im+l) such that r(k) < n < r(k + 1). Then because n is large (enough) we have

h(n) . h(r(k)) > g(k) = g(r(k + 1)) > g(n)

So we've shown that h 2* g on Q. Similarly, if

h or >* ,

on c \ Q' then h >* g on c \ Q.



496 JASON AUBREY

According to this lemma, we only need to show that the unbounded families enumerated in the theorem are

,-dominating. Families in the bounded and low

case are clearly bounded by the identity function g(n) = n. So we only need to consider the dominating, high and ultrafilter cases. The dominating family is clearly y/-dominating, since it contains all increasing functions. The next lemma shows that families in the ultrafilter case are ly-dominating.

LEMMA 7.4. Families in the ultrafilter case are q-dominating. PROOF. Suppose that 1 is a non-principal ultrafilter on cw and that 2 = {f e

w j : {n : f(n) < n} E ?'}. Let g E ow and let F = (In = [ia, in+1l) : n E w} be an interval partition of co with the property that if x < in then g(x) < in+1.

Define an interval partition H from F as follows. The ultrafilter ?' contains either the union of the even numbered intervals or the union of the odd numbered intervals of F. Suppose first that ? contains the union of the odd numbered intervals. In this case let Jn = I2n U I2n+l, and H = {J : n E co}. The point of this construction is that the union of the right halves of each H interval is contained in '. Suppose next that { contains the union of the even numbered intervals of F. In this case let Jo = Io and for n > 0 let J, = 12n-1 U I2n, and H = {J : n e w}. In this case, except for Jo, the union of the right halves of every Jn is contained in '. Since what happens with our functions on a finite set (namely the first H interval) will not affect the argument, we may assume without loss of generality that we are in the first case above.

Let Q be a coinfinite union of H blocks. Either Q or w \ Q is in ?; without loss of generality, suppose that Q cW. We will construct a function in {f E co / : {n : f(n) < n} E W} which eventually dominates g on co \ Q. On an interval Ji =

Izl U 121+1 in Q, we define f to be the identity function on 121+1

and constantly equal to the left endpoint of 21+1, i21+1, on 121. On an interval Jn = I2n U 2n+1 in co \ Q we define f as follows: Find the next H interval Ji in Q; on J, let f be constantly equal to the left endpoint of J1 's right F-subinterval.

Now, f is the identity function on a set in W, namely the union of the right F subintervals of all H intervals in Q. We need to show that f eventually dominates g on wco \ Q. Let Jn be a n interval in co \ Q, and let J, be the next H interval in Q. Notice that by the definition of H, the images under g of the points in J, at most reach into the left F subinterval of Ji and so are less than or equal to that subinterval's right endpoint; this endpoint plus 1 is the value of f on Jn, and so f dominates g on Jn. Since Jn was an arbitrary H interval in o \ Q, it follows that f dominates g on wo \ Q.

Notice, however, that this lemma, along with Lemma 7.3, also implies that families in the high case are ry-dominating. To see this, it suffices to observe that for every ultrafilter , , {f E co / co : {n : f(n) < n} E }

C_ {f E co 7 co : (3"n E o)

f(n) < n}. Recall that the principle of Filter Dichotomy states every filter on co is either

nearly coherent to the cofinite filter (and is said to be "feeble") or is nearly coherent to an ultrafilter. It is well known that u < g implies Filter Dichotomy, but it is not known whether the implication can be reversed.

THEOREM 7.5. Filter Dichotomy holds if and only if every unbounded family of functions on co is finitely pseudo-dominating.




PROOF. First assume that Filter Dichotomy holds, and suppose that .-

is an unbounded family of functions on co. Fix a function g : w - co, and consider the collection 7 = {{n : f(n)

_ g(n)} : f E Y}. Suppose there are finitely

many functions (say k) in - whose pointwise maximum dominates g. Then the collection of k-ary intersections of sets from 7 contains a finite set. In this case, the collection of k-ary intersections is unsplittable. Assume that there is no finite collection of functions from Y whose maximum dominates g. By this assumption, the family F has the strong finite intersection property, and so it generates a filter. Identify K with the filter it generates. Filter Dichotomy implies that F is either feeble or nearly coherent to an ultrafilter.

The filter F cannot be feeble since that would imply Y is bounded (according to Talagrand [13]).

Next suppose that Z is nearly coherent to an ultrafilter. Then there is some increasing finite-to-one function h : co -+ wC such that h(%) is an ultrafilter. But then h(F) is unsplittable, and so the collection {h(X1 n X2 n

."" n XI) : l E

co and for each 1 < i < / Xi E } is unsplittable. Thus F satisfies the definition of finitely big, and is therefore finitely y-dominating.

Next suppose that every unbounded family of functions is finitely y-dominating. Let 7 be a filter, and consider the collection of functions . = {next(-, X) : X E

7}. If this family of functions is bounded, then the filter 7f is feeble. So suppose the family Y is unbounded. By our assumption, this family is finitely v-dominating. Hence it is finitely big. Let g(n) = n and consider the collection {{n : next(n, X) < g(n)} : X E 7}. Notice that for each X e 7, {n : next(n, X) 0 g(n)} = X so {{n : next(n,X) < g(n)} : X E 7} is just W7. Then by our assumption, there exists an increasing finite-to-one function h : - cw so that h () is an ultrafilter. That is, W is either feeble or nearly an ultrafilter. Thus, Filter Dichotomy holds. -

Thus we have two closely related notions which fall between u < g and Filter Dichotomy. Two interesting questions are 1) whether the condition "all unbounded families of functions are v-dominating" implies u < g and 2) what are the possible relationships between ,-dominating families and finitely v-dominating families? Are these notions the same, or are there models in which, for example, all unbounded families are finitely y-dominating but not all unbounded families are i-dominating?

Acknowledgements. The results presented here are part of my doctoral disserta- tion, which was written under the direction of Professor Andreas Blass. I would like to thank Professor Blass for his guidance, and I would like to thank the referee for many helpful comments.

REFERENCES

[1] BOHUsLAV BALCAR and PETR SIMON, On minimal it-character of points in extremally disconnected compact spaces, Topology and its Applications, vol. 41 (1991), pp. 133-145.

[2] ANDREAS BLAss, Combinatorial cardinal characteristics of the continuum, Handbook of Set Theory (Matthew Foreman, Akihiro Kanimori, and Menachem Magidor, editors), Kluwer, to appear.

[3] , Groupwise density and related cardinals, Archive for Mathematical Logic, vol. 30 (1990), no. 1, pp. 1-11.

[4] , Nearly adequate sets, Logic and Algebra, Contemporary Mathematics, vol. 302, 2002, pp. 33-48.



498 JASON AUBREY

[5] ANDREAS BLASS and CLAUDE LAFLAMME, Consistency results about filters and the number of in- equivalent growth types, this JOURNAL, vol. 54 (1989), no. 2, pp. 50-56.

[6] ANDREAS BLASS and HEIKE MILDENBERGER, On the cofinality ofultrapowers, this JOURNAL, vol. 64 (1999), no. 2, pp. 727-736.

[7] JORG BRENDLE, Around splitting and reaping, Commentationes Mathematicae Universitatis Caroli- nae, vol. 39 (1998), no. 2, pp. 269-279.

[8] JusslN KETONE, On the existence of P-points in the Stone-Cech compactification of the integers, Fundamenta Mathematicae, vol. 92 (1976), pp. 92-94.

[9] CLAUDE LAFLAMME, Equivalence of families of functions on the natural numbers, Transactions of the American Mathematical Society, vol. 40 (1992), pp. 307-319.

[10] HEIKE MILDENBERGER, Groupwise densefamilies, Archivefor MathematicalLogic, vol. 330 (2001), pp. 93-112.

[11] PETER NYIKOS, Special ultrafilters and cofinal subsets of"w, to appear. [12] R. C. SOLOMON, Families of sets and functions, Czechoslovak Mathematical Journal, vol. 27

(1977), pp. 556-559. [13] MICHEL TALAGRAND, Compacts de fonctions mesurables etfiltres non mesurables, Studia Mathe-

matica, vol. 67 (1980), no. 1, pp. 13-43.

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UNIVERSITY OF NORTHERN IOWA

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Combinatorics for the Dominating and Unsplitting Numbers