SUCCESSORS OF SINGULAR CARDINALS AND COLORING

THEOREMS II

TODD EISWORTH AND SAHARON SHELAH

Abstract.In this paper,we investigate the extent to which techniques used

in [10],[2],and [3] | developed to prove coloring theorems at successors of

singular cardinals of uncountable conality | can be extended to cover the

countable conality case.

1.Introduction

In this paper,we tackle some of the issues left unresolved by its predecessor [2]

and the related [3].In particular,we begin the project of extending the coloring

theorems found in those papers to a more general setting | a setting that will

allow us to draw conclusions concerning successors of singular cardinals of countable

conality.

We remind the reader that the square-brackets partition relation ![]

of

Erdos,Hajnal,and Rado [6] asserts that for every function F:[]

! (where

[]

denotes the family of subsets of of cardinality ),there is a set H of

cardinality such that

(1.1) ran(F [H]

) 6= ;

that is,the function F omits at least one value when we restrict it to [H]

.

This paper investigates the extent to which negations of square-brackets partition

relations hold at the successor of a singular cardinal.In particular,we examine

relatives of the combinatorial statement

(1.2) 9[]

2

;

where is the successor of a singular cardinal.Our main concern is the situation

where =

+

for singular of countable conality;in general,we already know

stronger results for the case where is the successor of a singular of uncountable

conality.The added diculties that arise in the work for this paper are due to

some issues involving club-guessing,and we prove some theorems in that area as

well.

We also remark that Chapter III of [9] (i.e.,[10]) claims something stronger than

our Theorem 5,but there is a problem in the proof given there.More precisely,the

Date:January 27,2009.

1991 Mathematics Subject Classication.03E02.

Key words and phrases.square-brackets partition relations,minimal walks,successor of sin-

gular cardinal.

The rst author acknowledges support from NSF grant DMS 0506063.Research of the sec-

ond author was supported by the United States-Israel Binational Science Foundation (Grant no.

2002323) The authors'collaboration was supported in part by NSF Grant DMS 0600940.This is

paper 819 in the publication list of the second author.

1

2 TODD EISWORTH AND SAHARON SHELAH

comments on page 163 dealing with extending the main theorem of that chapter to

the successor of a singular of countable conality (Lemma 4.2(4)) are not enough

to push the proof through.Theorems 4 and 5 provide a partial reclamation of this

earlier work of the second author.

We now take a moment to x our notation and lay out some results underpinning

our work.In particular,we need to discuss scales,elementary submodels,and their

interaction.

Denition 1.1.Let be a singular cardinal.Ascale for is a pair (~;

~

f) satisfying

(1) ~ = h

i

:i < cf()i is an increasing sequence of regular cardinals such that

sup

i<cf()

i

= and cf() <

0

.

(2)

~

f = hf

: <

+

i is a sequence of functions such that

(a) f

2

Q

i<cf()

i

.

(b) If < <

+

then f

<

f

,where the notation f <

g means that

fi < cf():g(i) f(i)g is bounded below cf().

(c) If f 2

Q

i<cf()

i

then there is an <

+

such that f <

f

.

Our conventions regarding elementary submodels are standard |we assume that

is a suciently large regular cardinal and let A denote the structure hH();2;<

i

where H() is the collection of sets hereditarily of cardinality less than ,and <

is

some well-order of H().The use of <

means that our structure A has denable

Skolem functions,and we obtain the set of Skolem terms for A by closing the

collection of Skolem functions under composition.With these Skolem terms in

hand,we can discuss Skolem hulls:

Denition 1.2.Let B H().Then Sk

A

(B) denotes the Skolem hull of B in the

structure A.More precisely,

Sk

A

(B) = ft(b

0

;:::;b

n

):t a Skolem term for A and b

0

;:::;b

n

2 Bg:

The set Sk

A

(B) is an elementary substructure of A,and it is the smallest such

structure containing every element of B.

We also make use of characteristic functions of elementary submodels.

Denition 1.3.Let be a singular cardinal of conality ,and let ~ = h

i

:i < i

be an increasing sequence of regular cardinals conal in .If M is an elementary

submodel of A such that

jMj < ,

h

i

:i < i 2 M,and

+1 M.

then the characteristic function of M on ~ (denoted Ch

~

M

) is the function with

domain dened by

Ch

~

M

(i):=

(

sup(M\

i

) if sup(M\

i

) <

i

,

0 otherwise.

If ~ is clear from context,then we suppress reference to it in the notation.

In the situation of Denition 1.3,it is clear that Ch

~

M

is an element of the product

Q

i<

i

,and furthermore,Ch

~

M

(i) = sup(M\

i

) for all suciently large i < .

The following result is essentially due to Baumgartner [1] | a proof can be found

in the introductory section of [4].

SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS II 3

Lemma 1.4.Let ,,~,and M be as in Denition 1.3.If i

< and we dene

N to be Sk

A

(M [

i

),then

(1.3) Ch

M

[i

+1;) = Ch

N

[i

+1;):

We need one more easy fact about scales;a proof can be found in [3].In the

statement of the lemma (and throughout the rest of this paper) we use the notation

\8

"to mean\for all suciently large"and\9

"to mean\there are unboundedly

many".

Lemma 1.5.Let =

+

for singular of conality ,and suppose (~;

~

f) is a

scale for .Then there is a closed unbounded C such that the following holds

for every 2 C:

(1.4) (8

i < )(8 <

i

)(8 <

i+1

)(9

< )[f

(i) > ^f

(i +1) > :]

We end this section with another bit of terminology due to Shelah [8]:

Denition 1.6.A -approximating sequence is a continuous 2-chain M= hM

i

:

i < i of elementary submodels of A such that

(1) 2 M

0

,

(2) jM

i

j < ,

(3) hM

j

:j ii 2 M

i+1

,and

(4) M

i

\ is a proper initial segment of .

If x 2 H(),then we say that Mis a -approximating sequence over x if x 2 M

0

.

Note that if M is a -approximating sequence and =

+

,then + 1 M

0

because of condition (4) and the fact that is an element of each M

i

.

2.Club-Guessing

In this section we investigate club-guessing.The coloring theorems presented

in [10],[3],and [2] make use of a particular type of club-guessing sequence.These

special club-guessing sequences are known to exist at successors of singular cardinals

of uncountable conality (we give a proof in this section,as the original proof

in [10] has some minor problems),but it is still open whether they must exist at

successors of singular cardinals of countable conality.For this case,the current

section provides club-guessing sequences satisfying weaker conditions,and then

in the sequel we demonstrate that these sequences can be used to obtain similar

coloring theorems.We will begin with some terminology.

Denition 2.1.Let be a cardinal.

(1) A C-sequence for is a family hC

: < i such that C

is closed and

unbounded in for each < .

(2) If S is a stationary subset of ,then an S-club system is a family hC

: 2

S

i such where

S

is a subset of S such that S n S

is non-stationary,and

C

is closed and unbounded in for each 2 S

.

As is clear by the above denition,there is precious little dierence between

calling he

: < i a C-sequence and calling it a -club system | the two names

exist for historical reasons.The dierence in terminology is worth preserving for

other reasons,however,because we will be using these objects in completely dif-

ferent ways |\C-sequences"are used exclusively for constructing minimal walks,

4 TODD EISWORTH AND SAHARON SHELAH

while\-club systems"are used only for club-guessing matters.Our use of dier-

ent terms makes it clear how the objects are to be used,and keeps our notation

consistent with the literature already in existence.

The use of the set S

in the preceding denition is for technical reasons |very

often,we will take an existing S-club system and modify in a way that makes sense

only for\almost all"elements of S,and we still would like to call the resulting

object an S-club system.

Denition 2.2.Suppose C is a closed unbounded subset of an ordinal .Then

(1) acc(C) = f 2 C: = sup(C\)g,and

(2) nacc(C) = C n acc(C).

If 2 nacc(C),then we dene Gap(;C),the gap in C determined by ,by

(2.1) Gap(;C) = (sup(C\);):

The next denition captures some standard ideas from proofs of club-guessing;

we have chosen more descriptive names (due to Kojman [7]) than those prevalent

in [9].

Denition 2.3.Suppose C and E are sets of ordinals with E\sup(C) closed in

sup(C).We dene

(2.2) Drop(C;E) = fsup(\E): 2 C n min(E) +1g:

Furthermore,if C and E are both subsets of some cardinal and he

: < i is a

C-sequence,then for each 2 nacc(C)\acc(E),we dene

(2.3) Fill(;C;E) = Drop(e

;E)\Gap(;C):

Our notation suppresses the dependence on the parameter he

: < i because

the precise choice of e

does not make a dierence at all;all that matters is that

Fill(;C;E) provides us with a simple way of generating a closed unbounded subset

of E\Gap(;C) for in nacc(C)\acc(E).

In our rst theorem,we characterize the existence of the special sorts of club-

guessing sequences that are crucial to proofs given in [3] and [2].

Theorem 1.Suppose =

+

for a singular cardinal,and let S be a stationary

subset of f < :cf() = cf()g.Then the following are equivalent:

(1) There is an S-club system hC

: 2 Si such that

(a) jC

j < for every 2 S,and

(b) for every closed unbounded E ,there are stationarily many such

that for all < ,

(2.4) f 2 nacc(C

)\E:cf() > g is unbounded in .

(2) There is an S-club system hC

: 2 Si such that

(a) supfjC

j: 2 Sg < ,and

(b) for every closed unbounded E ,there are stationarily many such

that for all < ,

(2.5) f 2 nacc(C

)\E:cf() > g is unbounded in .

(3) There is an S-club system hC

: 2 Si such that

SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS II 5

(a) otp(C

) = cf() for every 2 S,

(b) hcf(): 2 nacc(C

)i is strictly increasing and conal in ,and

(c) for every closed unbounded E ,there are stationarily many 2 S

with C

E.

Proof.Assume

C = hC

: 2 Si is as in (1).We claim that there is a <

such that for every closed unbounded E ,there are stationarily many such

that (2.4) is satised for all < and jC

j .Suppose this is not the case,

and let h

i

:i < cf()i be an increasing sequence of cardinals conal in .For

each i < cf(),there is a closed unbounded E

i

such that for all 2 S,either

jC

j >

i

or (2.4) fails for some < .The contradiction is immediate upon

consideration of the club E =

T

i<cf()

E

i

.

Having established the existence of such a ,we can modify

C by replacing those

C

of cardinality greater than by an arbitrary club (in ) of order-type cf(),and

this gives us an S-club system as in (2).

The journey from (2) to (3) is an application of standard club-guessing ideas.If

E is club in ,for the purpose of this proof,let us agree to say

C guesses E at

if (2.4) holds for all < .Our rst move is to establish that if

C is as in (2),then

there is a closed unbounded E

such that for every closed unbounded E ,

there are stationarily many 2 S where

C guesses acc(E

) at and such that

Drop(C

;E

) E.

Suppose this fails.Choose a regular cardinal such that

supfjC

j: 2 Sg < < :

By recursion on < we choose clubs E

of as follows:

Case = 0:E

0

=

Case limit:We let E

=

T

<

E

.

Case = +1:In this case,by our assumption we know that E

does not enjoy

the properties required of E

.Thus,there are closed unbounded sets E

0

and E

1

such that for all 2 E

0

\S,if

C guesses acc(E

) at ,then there is an

2 C

n (min(E

) +1) such that sup(E

\) =2 E

1

.We now dene

(2.6) E

= E

+1

= acc(E

)\E

0

\E

1

and the construction continues.

Nowlet E =

T

<

E

.It is clear that E is club in ,and so by our assumption we

can nd 2 S where

C guesses E.We note that 2 E,and therefore 2 E

0

for all

< .Furthermore,

C guesses acc(E

) at for all < because E acc(E

).Our

construction forces us to conclude that for each < ,there is an 2 C

nmin(E

)

such that sup(E

\) is not in E

1

(and therefore not in E

+1

either).

We now get a contradiction using a well-known argument | for each 2 C

greater than min(E),the sequence hsup(E

\): < i is decreasing,and therefore

eventually constant.Thus,there are

< and

< such that

< =)sup(E

\) =

:

6 TODD EISWORTH AND SAHARON SHELAH

Since jC

j < ,we know

:= supf

: 2 C

g is less than .We know

C guesses

acc(E

) at ,and so there is an 2 C

n (min(E

) +1) such that

(2.7) sup(E

\) =2 E

+1

:

But

,so

(2.8) sup(E

\) =

= sup(E

+1

\) 2 E

+1

;

and we have our contradiction.

To nish the proof,let us suppose that E

is the club whose existence was just

established.If

C guesses acc(E

) at ,then we can easily build a set D

such that

D

acc(E

)\C

,

D

is closed and unbounded in with otp(D

) = cf(),and

hcf(): 2 nacc(D

)i is strictly increasing and conal in .

Notice that D

Drop(C

;E

) for such | this is the reason for using acc(E

).

For all other 2 S,we can let D

be a subset of satisfying the last two conditions

above.It is now routine to verify that hD

: 2 Si is as required.Since it is clear

that (3) implies (1),the theorem has been established.

Let us agree to call an S-club system a nice club-guessing sequence if it satises

(3) of the above theorem | this is in concordance with notation from [9],and it

also ts in with the nice pairs dened in [3].We will say that S carries a nice

club-guessing sequence when such a sequence can be found.

Our next task is to demonstrate that nice club-guessing sequences exist when

we deal with successors of singular cardinals of uncountable conality.This result

actually follows from Claim 2.6 on page 127 of [9],but the proof of that claim has

some problems.The proof we give xes these oversights,and is actually quite a bit

simpler.

Theorem 2.If =

+

for a singular cardinal of uncountable conality,then

every stationary subset of f < :cf() = cf()g carries a nice club-guessing

sequence.

Proof.Let S be such a stationary set.By our previous work,it suces to produce

an S-club systemsatisfying (1) of Theorem1.Assume by way of contradiction that

no such S-club system exists.

Let hC

: 2 Si be an S-club system with otp(C

) = cf(),and let e be any

C-sequence on .

By recursion on n <!,we will dene objects hC

n

: <!i,h

n

: 2 Si,

h

n

: 2 Si,and E

n

such that

C

n

is closed and unbounded in ,

jC

n

j < ,

n

is a regular cardinal less than ,

n

< ,and

E

n

is closed and unbounded in .

SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS II 7

We let

C

n

denote hC

n

: 2 Si,and our initial set up has E

0

= ,

C

0

=

C,

0

= 0,

and

0

= 0.

Suppose we are given

C

n

.By our assumption,

C

n

does not satisfy the demands

of our theorem,and so there are clubs E

0

n

and E

1

n

such that

C

n

fails to guess E

0

n

on E

1

n

\S.This means for any 2 E

1

n

\S,there are < and a regular <

such that

(2.9) 2 nacc(C

n

)\E

1

0

n ( +1) =)cf() :

We now dene E

n+1

= acc(E

n

\E

0

n

\E

1

n

),dene

n+1

to be the least such ,and

dene

n+1

to be the least corresponding to

n+1

.

Now that E

n+1

has been dened,we declare an ordinal 2 S to be active at

stage n +1 if 2 acc(E

n+1

).For those 2 S that are inactive at stage n +1,we

do nothing | set C

n+1

= C

n

,

n+1

=

n

,and

n+1

=

n

.

For the remainder of this construction,we assume is active at stage n+1.Let

us say that ordinal < needs attention at stage n +1 if

(2.10) 2 nacc(C

n

)\acc(E

n+1

) n

n+1

+1:

Notice that any ordinal requiring attention at this stage is necessarily of conality

at most

n+1

.

Our construction of C

n+1

commences by setting

(2.11) D

n

= Drop(C

;E

n+1

):

This set D

n

is still closed and unbounded in since is active,and if needed

attention at this stage,then = sup(E

n+1

\) and therefore

(2.12) 2 nacc(D

n

)\acc(E

n+1

):

In particular,the set Fill(;D

n

;E

n+1

) is dened for any that needs attention at

this stage.

To nish the construction,we dene

(2.13) C

n+1

= D

n

[ fFill(;D

n

[E];E

n+1

): needs attention g:

The set C

n+1

is clearly unbounded in ,and it is closed since it was obtained from

D

n

by gluing closed sets into\gaps"in D

n

.It remains to see that jC

n+1

j < ,and

this follows by the estimate

(2.14)

C

n+1

jC

n

j +

n+1

jC

n

j:

Thus,the recursion can continue.

Let E =

T

n<!

E

n

,and choose 2 S\acc(E) such that divides the order-type

of \E.Since E acc(E

n

) for all n,it follows that is active at all stages of the

construction.Let us dene

(2.15)

= supf

n

:n <!g +1;

and

(2.16)

= supfjC

n

j:n <!g:

Since @

0

< cf() = cf(),we know

< and

< .Since 2 acc(E) and

divides otp(E\),

(2.17) jE\ n

j = ;

and an appeal to (2.16) tells us that we can choose an ordinal such that

8 TODD EISWORTH AND SAHARON SHELAH

2 E

< < ,and

=2

S

n<!

C

n

.

Our next move involves consideration of the sequence h

n

:n <!i of ordinals

dened as

(2.18)

n

= min(C

n

n ):

We will reach a contradiction by proving that this sequence of ordinals is strictly

decreasing.

Note that

n

is necessarily greater than by our choice of .This means that

n

is an element of nacc(C

n

).Moreover,

(2.19)

n+1

<

n

:

Two possibilities now arise | either

n

needs attention at stage n +1,or it does

not.We analyze each of these cases individually.

Case 1:

n

does not need attention at stage n +1

A glance at (2.10) establishes that

n

is not an element of acc(E

n+1

),and hence

if we set

n

= sup(

n

\E

n+1

),then

n

<

n

.Now 2 E E

n+1

,and therefore.

(2.20)

n

<

n

:

The ordinal

n

is in D

n

which is itself a subset of C

n+1

and so

(2.21)

n+1

n

<

n

:

Case 2:

n

needs attention at stage n +1

In this case,we have seen that Fill(

n

;D

n

;E

n+1

) is closed and unbounded in

n

and included in C

n+1

.Since must be strictly less than

n

,we see

(2.22) <

n+1

min(Fill(

n

;D

n

;E

n+1

) n ) <

n

and again we have

n+1

<

n

.

We now have the desired contradiction,as h

n

:n <!i allegedly forms a strictly

decreasing sequence of ordinals.

We now come to a very natural question that is still open.

Question 2.4.Suppose =

+

for singular of countable conality,and let S

be a stationary subset of f < :cf() =!g.Does S carry a nice club-guessing

sequence?

This question is particular relevant for this paper because a positive answer

would allow us to strengthen our results,as well as simplify the proof enormously

by using the techniques of [3].A positive answer follows easily from }(S),but we

leave the proof of this to the reader.The next theoremexplores the extent to which

we can obtain S-club systems with properties that approximate\niceness".

Theorem 3.Let =

+

for a singular cardinal of countable conality,and let

S be a stationary subset of f < :cf() = @

0

g.Further suppose that we have

sequences hc

: 2 Si and hf

: 2 Si such that

SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS II 9

(1) c

is an increasing function from!onto a conal subset of (for conve-

nience,we dene c

(1) to be 1)

(2) f

maps!to the set of regular cardinals less than ,and

(3) for every closed unbounded E ,there are stationarily many 2 S such

that c

(n) 2 E for all n <!.

Then there is an S-club system hC

: 2 Si such that

(4) c

(n) 2 C

for all n,

(5) jC

\(c

(n 1);c

(n)]j f

(n),and

(6) for every closed unbounded E ,there are stationarily many 2 S such

that

(2.23) (8n <!)(9 2 nacc(C

)\E) [c

(n 1) < < c

(n) and cf() > f

(n)]

We can get a picture of the case of most interest to us in the following manner.

First,notice that the functions hc

: 2 Si are essentially a\standard"club-

guessing sequence of the sort we know exist.Given 2 S,the sequence c

chops

into an!sequence of half-open intervals of the form (c

(n1);c

(n)].If we dene

(2.24) I

(n):= (c

(n 1);c

(n)];

then C

is constructed so that C

\I

(n) is of cardinality at most f

(n).The

club-guessing property tells us that for any closed unbounded E ,there are

stationarily many 2 S such that for each n <!,E\nacc(C

)\I

(n) contains an

ordinal of conality greater than f

(n).In particular,if the sequence hf

(n):n <!i

increases to for all 2 S,then for every closed unbounded E there are

stationarily many 2 S such that for any < ,

(2.25) f 2 E\nacc(C

):cf() > g is unbounded in :

This almost gives us the assumptions needed to apply Theorem 1;the problem,

however,is that our hypotheses admit the possibility that C

is of cardinality ,

and this takes us out of the purview of Theorem 1.

Proof.Our starting point for this proof is the bare-bones sketch of a similar proof

given for Claim 2.8 on page 131 of [9].By way of contradiction,assume that there

is no such family hC

: 2 Si.The proof will require us to construct many S-club

systems in an attempt to produce the desired object;let us agree to say that an

S-club system satises the structural requirements of Theorem 3 if conditions (4)

and (5) hold,and say it satises the club-guessing requirements of Theorem 3 if

condition (6) holds.

The main thrust of our construction is to dene objects E

and

C

= hC

: 2 Si

by induction on <!

1

.The sets E

will be closed unbounded in ,while each

C

will be an S-club system satisfying the structural requirements of Theorem 3.Our

convention is that stage refers to the process of dening

C

+1

and E

+1

from

C

and E

.The reader should also be warned that several auxiliary objects will be

dened along the way.

Construction

Initial set-up

We set E

0

= and C

0

= fc

(n):n <!g for each 2 S.

10 TODD EISWORTH AND SAHARON SHELAH

Stage | dening E

+1

and

C

+1

We assume that

C

is an S-club system satisfying the structural requirements

of Theorem 3,and E

is a closed unbounded subset of .We have assumed that

Theorem 3 fails,and so there are closed unbounded subsets E

0

and E

1

of such

that for each 2 E

0

\S,there is an n <!such that

(2.26) 2 nacc(C

)\E

1

\I

(n) =)cf() f

(n):

We dene

(2.27) E

+1

:= acc(E

\E

0

\E

1

):

Let us agree to say that an ordinal 2 S is active at stage if C

0

acc(E

+1

),

and note that the set of such is stationary.If 2 S is inactive at stage ,then

we do nothing and let C

+1

= C

.

If is active at stage ,then we know 2 E

0

and so there is a least n(;) <!

such that

(2.28) 2 nacc(C

)\E

1

\I

(n(;)) =)cf() f

(n(;)):

The construction of C

+1

will modify C

only on the interval I

(n(;)),that is,

we ensure that

(2.29) C

+1

\( n I

(n(;))) = C

\( n I

(n(;))):

Our next step is to dene

(2.30) D

= Drop(C

\I

(n(;));E

+1

\I

(n(;))):

Note that D

is a closed unbounded subset of c

(n(;)) of cardinality at most

f

(n(;)).

We still have some distance to traverse before arriving at C

+1

| one should

think of D

as being the rst approximation to how C

+1

will look on the interval

I

(n(;)).To nish,let us say that an element of D

needs attention if

2 acc(E

+1

)\nacc(D

),and

cf() f

(n(;)).

If needs attention,then Fill(;C

\I

(n(;));E

+1

\I

(n(;)) is closed and

unbounded in Gap(;C

) and of cardinality cf() f

(n(;)).We dene

(2.31) A

= D

[fFill(;C

\I

(n(;));E

+1

\I

(n(;)): needs attention g:

Since the needed instances of\Fill"are always a closed subsets lying in a\gap"of

D

,the set A

is still closed and unbounded in c

(n(;)).Also,simple cardinality

estimates tell us

(2.32)

A

f

(n(;)):

We now dene C

+1

piecewise |as indicated in 2.29,we do nothing outside of the

interval I

(n(;)),while we set

(2.33) C

+1

\I

(n(;)) = A

:

So dened,our S-club system

C

+1

satises the structural requirements of Theo-

rem 3 and the construction continues.

C

and E

for limit

SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS II 11

We begin by setting E

=

T

<

E

.Next,for each 2 S we let C

be the

closure in of

(2.34) f: 2 C

for all suciently large < g:

The set C

dened above is closed in by denition.Since it contains C

0

,it is also

unbounded.Finally,

(2.35) C

\I

(n)

[

<

C

\I

(n):

Since is countable and f

(n) is a cardinal,it follows that

(2.36)

C

\I

(n)

f

(n)

for all n,and therefore hC

: 2 Si satises the structural requirements of Theo-

rem 3.

End Construction

Having constructed

C

and E

for all <!

1

,we turn now to obtaining a

contradiction.Let us dene

(2.37) E

:=

\

<!

1

E

:

It is clear that E

is club in ,and so there is a 2 S such that

(2.38) C

0

f < : divides otp(E

\)g:

Let us x such a ,and note that

(2.39) jE

\I

(n)j = for all n <!:

For each <!

1

,we know from (2.38) that is active at each stage <!

1

.

In particular,n(;) is dened for all <!

1

and hence there is a least n

<!

such that n(;) = n

for innitely many .Let h

n

:n <!i list the rst!such

ordinals,and let

= supf

n

:n <!g.

Choose an ordinal

2 E

\I

(n

)n

S

<

C

|this is possible because of (2.39),

as

(2.40)

[

<

C

\I

(n

)

@

0

f

(n

) < :

Finally dene

(2.41)

n

:= min(C

n

n

)

for each n <!.Notice that our choice of

guarantees that

is strictly less than

n

for all n.

Claim 2.5.For each n,we have

n+1

<

n

.

Proof.Fix n.It is clear from our construction that

(2.42) min(C

n

+1

n

) = min(C

n+1

n

) =

n+1

because

2 I

(n

) and n(;) 6= n

if

n

< <

n+1

.

We now track what happens to

n

during stage

n

by splitting into two cases.

Case 1:

n

=2 acc(E

n

+1

).

12 TODD EISWORTH AND SAHARON SHELAH

In this case,we note that since

2 E

n

+1

we have

(2.43)

sup(

n

\E

n

+1

) <

n

:

Since

=2 C

n

+1

while

(2.44) sup(

n

\E

n

+1

) 2 D

n

+1

C

n

+1

;

it follows that

<

n+1

<

n

and we are done.

Case 2:

n

2 acc(E

n

+1

).

Since

<

n

,the denition of

n

tells us that

n

must be in nacc(C

n

).Also,

both and

n

are in E

n

+1

,so in particular 2 E

0

n

and

n

2 E

1

n

.This tells us

cf(

n

) f

(n(;

n

)).

By our case hypothesis,

n

= sup(E

n

+1

\

n

) and so

n

2 D

n

and

(2.45)

n

= min(D

n

n

) >

:

We conclude

(2.46)

n

2 nacc(D

n

);

and so

n

needs attention during the construction of C

n

+1

.In particular,

(2.47) Fill

n

;C

\I

n(;)

;E

+1

\I

n(;)

C

n

+1

and so

(2.48) C

n

+1

\(

;

n

) 6=;:

We conclude

(2.49)

<

n+1

= min(C

n

+1

n

) = min(C

n+1

n

) <

n

as required.

Using the preceding claim,we get a strictly decreasing set of ordinals.This is

absurd,and Theorem 3 is established.

Club-guessing systems structured like those provided by Theorem 3 will occupy

our attention for the rest of this paper,so we will give them a name.

Denition 2.6.Let =

+

for singular of countable conality,and let S be

a stationary subset of f < :cf() = @

0

g.An S-club system hC

: 2 Si is

well-formed if there is a function f

C

:!! and functions c

:!! for each

2 S such that such that

(1) c

is strictly increasing with range conal in

(2) hf

C

(n):n <!i is a strictly increasing sequence of regular cardinals conal

in

(3) for each n,jC

\(c

(n 1);c

(n)]j f

C

(n)

(4) for each n,if 2 nacc(C

\(c

(n 1);c

(n)] then cf() > f

C

(n)

(5) if E is closed and unbounded in ,then there are stationarily many 2 S

such that

(2.50) E\nacc(C

)\(c

(n 1);c

(n)] 6=;for all n <!:

SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS II 13

If there is a well-formed S-club system,then we say that S carries a well-formed

club-guessing sequence.We continue to use the notation I

(n) to indicate the in-

terval (c

(n 1);c

(n)] (where our convention is that c(1) = 1),and refer to

this sequence of intervals as the interval structure of C

.The function f

C

is said

to measure

C.

Proposition 2.7.Let S be a stationary subset of f < :cf() = @

0

g where

=

+

with singular of countable conality.If f:!! enumerates a strictly

increasing sequence of regular cardinals that is conal in ,then S carries a well-

formed club-guessing sequence that is measured by f.

Proof.For each 2 S,we set f

= f and apply Theorem 3 to any relevant choice

for hc

: 2 Si.The S-club system hC

: 2 Si that arises need not satisfy

condition (4) of Denition 2.6,so for each 2 S we dene

(2.51) D

= f 2 nacc(C

):if 2 I

(n),then cf() > f

(n)g;

and let D

equal the closure of D

in .The proof that hD

: 2 Si is as required

is routine and left to the reader.

We remark that any S-club system hC

: 2 Si providing a positive answer

to Question 2.4 is also essentially well-formed | given any increasing function f

mapping!onto a set of regular cardinals conal in ,it is straightforward to\thin

out"the C

to get a well-formed S-club system

D measured by f.

We move nowto some terminology concerning club-guessing ideals taken from[9].

We start with a basic denition.

Denition 2.8.Let

C = hC

: 2 Si be an S-club system for S a stationary

subset of some cardinal ,and suppose

I = hI

: 2 Si is a sequence such that I

is an ideal on C

for each 2 S.The ideal id

p

(

C;

I) consists of all sets A such

that for some closed unbounded E ,

(2.52) 2 S\E =)E\A\C

2 I

:

Proposition 2.9.Suppose =

+

for singular of countable conality,and let

C be a well-formed S-club system for some stationary S f < :cf() = @

0

g.

Let I

be the ideal on C

generated by sets of the form

(2.53) f 2 C

: 2 acc(C

) or cf( ) < or < g

for < and < .Then id

p

(

C;

I) is a proper ideal.

Proof.We need to verify that =2 id

p

(

C;

I).If we unpack the meaning of this,we

see that we need that for every closed unbounded E ,there is a 2 S such

that E\C

=2 I

.This means that for each < and < ,there needs to be

a 2 E\nacc(C

) greater than with conality greater than ,and this follows

immediately from the denition of well-formed.

With the preceding proposition in mind,if we say that (

C;

I) is a well-formed

S-club system,we mean that

C is as in Denition 2.6,and

I = hI

: 2 Si is the

sequence of ideals dened as in Proposition 2.9.The ideals id

p

(

C;

I) for well-formed

(

C;

I) lie at the heart of the coloring theorems presented in the sequel.

14 TODD EISWORTH AND SAHARON SHELAH

3.Parameterized Walks

In this section,we develop a generalization of Todorcevic's technique of mini-

mal walks [11,12,13].The notation is a bit cumbersome,but this seems to be

unavoidable given the complexity of the ideas we are trying to voice.

Denition 3.1.Let be a cardinal.A generalized C-sequence is a family

he

n

: < ;n <!i

such that for each < and n <!,

e

n

is closed unbounded in ,and

e

n

e

n+1

.

The next lemma connects the above denition with concepts from the preceding

section.

Lemma 3.2.Let =

+

for singular of countable conality,and let (

C;

I) be

a well-formed S-club system for some stationary S consisting of ordinals of

countable conality.There is a generalized C-sequence he

n

: < ;n <!i such

that

je

n

j cf() +f

C

(n) +@

1

,and

2 S\e

n

=)C

\I

(n) e

n

.

Proof.We will obtain e

n

as the closure (in ) of a union of approximations e

n

[]

for <!

1

.We start by letting e

be closed unbounded in of order-type cf()

for each < .The construction proceeds as follows:

e

0

[0] = e

e

n

[ +1] = closure in of e

n

[] [

[

2S\e

n

[]

C

\I

(n)

e

n+1

[0] = e

n

e

n

[] = closure in of

[

<

e

n

[ ] for limit

e

n

= closure in of

[

<!

1

e

n

[]:

The verication that he

n

: < ;n <!i has the required properties is routine.

The relationship between the generalized C-sequence obtained above and the

given well-formed S-club system (

C;

I) is important enough that it ought to have

a name.

Denition 3.3.Let =

+

for singular of conality @

0

,and suppose (

C;

I)

is a well-formed S-club system for some stationary S f < :cf() = @

0

g.A

generalized C-sequence e is said to swallow (

C;

I) if

(1) je

n

j cf() +f

C

(n) +@

1

,and

(2) 2 S\e

n

=)C

\I

(n) e

n

.

The most important property enjoyed by these cumbersome generalized C-sequences

is isolated by the following lemma.

SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS II 15

Lemma 3.4.Suppose e swallows the well-formed S-club system (

C;

I).If is in

S\e

m

for some m<!,then

(3.1) (8

n <!) [nacc(C

)\I

(n) nacc(e

n

)]:

Proof.Choose n

<!so large that m < n

and cf() f

C

(n

).If n

n <!

and 2 nacc(C

)\I

(n),then 2 e

n

by Denition 3.3,and cannot be in acc(e

n

)

because

(3.2) je

n

j cf() +f

C

(n) +@

1

< cf( ):

Up until this point in the section,we have been developing the context in which

our generalized minimal walks will take place,and now we turn to their denition.

Denition 3.5.Let e be a generalized C-sequence on some cardinal ,and let s

be a nite sequence of natural numbers.Given < < ,we dene St(;;s;`)

|\step`on the s-walk from to (along e)"| by induction on`<!.

St(;;s;0) = ;

and

St(;;s;`+1) =

8

>

>

>

>

>

>

<

>

>

>

>

>

>

:

if = St(;;s;`)

min(e

0

St(;;s;`)

n ) if St(;;s;`) > and` lg(s)

min(e

s(`)

St(;;s;`)

n ) otherwise.

Finally,let

n(;;s) = least`such that = St(;;s;`).

In the C-sequences used by Todorcevic,at each stage of a minimal walk one has

a single ladder to use to make the next step.In our context,there are innitely

many ladders available,and the parameter s selects the one we use for our next

step.Even though there are innitely many ladders available,nevertheless there are

only nitely many possible destinations,for given < ,the sequence he

n

:n <!i

increases with n and therefore the sequence hmin(e

n

n):n <!i is decreasing and

hence eventually constant.This brings us to our next denition.

Denition 3.6.We dene St

(;;`) |\step`of the settled walk from to

(along e)"| by the following recursion:

St

(;;0) = ;

and

St

(;;`+1) =

8

>

>

<

>

>

:

if = St

(;;`),

lim

n!1

min(e

n

St

(;;`)

n )

otherwise.

We let n

(;) denote the least n for which St

(;;n) = .

16 TODD EISWORTH AND SAHARON SHELAH

The settled walks described above avoid the use of parameters s;unfortunately,

we seem to need the greater generality furnished by Denition 3.5 in our proof of

the main result of this paper.The following straightforward lemma connects the

two concepts.

Lemma 3.7.There is an m

<!such that if s 2

<!

!,lg(s) n

(;),and

s(i) m

for all i < lg(s),then

St(;;s;`) = St

(;;`) for all`< n

(;):

We say that m

settles the walk from to (along e),and let m

(;) denote the

least such m

.

Our discussion now returns to a familiar context | let =

+

for singular

of countable conality,and let S be a stationary subset of f < :cf() = @

0

g.

Further suppose (

C;

I) is a well-formed S-club system swallowed by the generalized

C-sequence e.In the course of this discussion,we will dene several auxiliary

functions.

Suppose 2 S and < < ,and let m

= m

(;) be as in Lemma 3.7.For

`< n

(;) 1,we know =2 e

m

St

(;;`)

and so if we dene

(3.3)

=

(;) = supfmax(e

m

St

(;;`)

\):`< n

(;) 1g;

then

must be less than .

Let = (;) denote the ordinal St

(;;n

(;)1);our choice of m

ensures

that is in S\e

m

.An appeal to Lemma 3.4 tells us there must exist a least

m= m(;) <!such that

(1) m m

,

(2) nacc(C

)\I

(m) nacc(e

m

) for all m m,and

(3) if m m and

2 nacc(C

)\I

(m),then

(3.4)

< sup(e

m

\

) <

:

Denition 3.8.Suppose 2 S,and < < .For each m <!,we let

s(;;m) 2

!

!be the sequence of length n

(;) dened by

s(;;m)[`] =

(

m

(;) if`< n

(;) 1,

m if`= n

(;) 1:

Proposition 3.9.Suppose 2 S, < < ,and m m(;).For any

2

nacc(C

)\I

(m),if sup(e

m

(;)

\

) < <

,then

(3.5) St(;;s(;;m);`) = St

(;;`) for all`< n

(;);

and

(3.6) St(;;s(;;m);n

(;)) =

:

Proof.Assume and s:= s(;;m) are as hypothesized,and suppose

St(;;s;`) = St

(;;`)

SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS II 17

with`+1 < n

(;).Then

St(;;s;`+1) = min(e

s(`)

St(;;s;`)

n )

= min(e

m

St

(;;`)

n )

= min(e

m

St

(;;`)

n ) (as >

(;))

= St

(;;`+1):

In particular,we know

St(;;s;n

(;) 1) = St

(;;n

(;) 1) = (;):

We now use Denition 3.5 to compute

St(;;s;n

(;)) = min(e

s(n

(;)1)

St

(;;s;n

(;)1)

n )

= min(e

m

(;)

n )

=

;

where the last equality holds because

2 e

m

(;)

and

sup(e

m

(;)

\

) < <

:

The preceding argument certainly benets from a description in English.Given

< with 2 S,if we dene

as in (3.3),then the usual sort of minimal walks

argument guarantees that for any in the interval (

;),the\m

{walk"(i.e.,the

walk obtained by always stepping in the m

th ladder) from to will agree with

the m

{walk from to until the last step before the latter arrives at .Varying

the ladder used for the next step (i.e.,changing the particular value of m) gives

us a way of gaining control over one more step,provided we have a little more

information on the ordinal .

Notice that even though we assume m m

,we cannot simply replace s(;;m)

with a sequence of the same length that is constant with value m | doing this

change has no eect on our steps in the initial portion of the walk,but it might

increase the value of

so that it exceeds the particular

we were aiming for,and

then the argument no longer works (although something could be said if we were

working with of uncountable conality | see the forthcoming paper [5]).Thus,

we seem to be stuck with sequences s that are not constant if we want our proof to

go through.

4.The main theorem

Throughout this section,we will be operating in the following general context:

=

+

for singular of conality @

0

S is a stationary subset of f < :cf() = @

0

g

(

C;

I) is a well-formed S-club system

e = he

n

:n <!; < i is a generalized C-sequence that swallows (

C;

I)

(~;

~

f) is a scale for with

0

> @

0

.

:[]

2

!!is the function dened (for < ) by

(4.1) (;) = maxfi <!:f

(i) f

(i)g:

18 TODD EISWORTH AND SAHARON SHELAH

hs

i

:i <!i is an enumeration of

<!

!in which each element appears

innitely often

x = f;;S;(

C;

I);e;(~;

~

f);hs

i

:i <!ig (so x codes all of the parameters

listed previously)

A is a structure of the formhH();2;<

i for some suciently large regular

cardinal and well-ordering <

of H().

We apologize to the reader for the preceding bare list of assumptions |writing

all of the above out each time results in a dramatic loss of clarity.

Denition 4.1.We dene a coloring c:[]

2

! as follows:

For < < ,let

(4.2) s

(;) = s

(;)

Next,dene

(4.3) k(;) = least` n(;) such that (;St(;;s

(;);`)) 6= (;).

Finally,let

(4.4) c(;) = St(;;s

(;);k(;)):

The computation of c(;) seems more reasonable when written out in English

|we start by computing (;) and use this to select the element s

of

<!

!that

will guide our walk.We then walk from to using s

,and we stop when we

reach a point where\ changes".This stopping point is the value of c(;).The

same basic idea is exploited in [3];the current version is complicated by our need

for the parameter s

.

Theorem 4.If ht

: < i is a pairwise disjoint sequence of nite subsets of

and A is an unbounded subset of ,then for id

p

(

C;

I)-almost all

< ,we can

nd < and 2 A such that

(4.5) c(;) =

for all 2 t

:

Proof.By way of contradiction,suppose ht

: < i and A form a counterex-

ample (without loss of generality, < min(t

)).Then there is an id

p

(

C;

I)-positive

set B such that for each

2 B,there are no < and 2 Asuch that c t

fg

is constant with value

.

Let hM

: < i be a -approximating sequence over fx;ht

: < i;Ag,and

let E be the closed unbounded set dened by

E:= f < : = M

\g:

By our assumptions,we can choose 2 E\S such that

(4.6) E\B\C

=2 I

:

Finally,let be some element of A greater than .

The discussion preceding Proposition 3.9 applies to and ,so we can safely

speak of m(;) and the other functions dened there.Since E\B\C

=2 I

,

we know that E\B must contain members of nacc(C

)\I

(n) for arbitrarily

large n.Thus,we can nd

2 E\B such that

2 nacc(C

)\I

(m) for some

m m(;).In particular,

(4.7)

2 nacc(e

m

(;)

)

SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS II 19

by the denition m(;).

Let s

= s(;;m) for this particular value of m.We know

(4.8) sup(e

m

(;)

\

) <

;

and Proposition 3.9 can now be brought into play | if lies in the interval deter-

mined by (4.8),then we know that the s

-walk from to will pass through

and in addition,we know exactly what the walk looks like up to that point.

Since

2 E and ht

: < i 2 M

0

,we note

(4.9) <

=)t

:

We assumed < min(t

),and so we conclude

(4.10) sup(e

m

(;)

\

) < <

=)t

(sup(e

m

(;)

\

);

):

We now prove the following claim.

Claim 4.2.For all suciently large i <!,there are unboundedly many <

such that

(4.11) (;St(;;s

;`)) = i for all`< n

(;) and 2 t

,

while

(4.12) (;

) > i for all 2 t

:

Proof.Let M be the Skolem hull (in A) of fx;ht

: < i;A;

g.Since M is

countable and the

i

are uncountable,it follows that

(4.13) Ch

M

(i) = sup(M\

i

) for all i <!;

where Ch

M

is the characteristic function of M from Denition 1.3.

For each < ,let f

min

be the function with domain!dened as

(4.14) f

min

(i) = minff

(i): 2 t

g:

It is easy to see that (~;hf

min

: < i) is a scale for ,and this scale is also an

element of M

.Since

is an element of every closed unbounded subset of that

is an element of M

,we can appeal to Lemma 1.5 and conclude that there is an

i

0

<!such that whenever i

0

i <!,

(4.15) (8 <

i

)(8 <

i+1

)(9

<

)[f

min

(i) > ^f

min

(i +1) > :]

Next,note that M is an element of M

,as the required Skolem hull can be

computed in M

using the model M

+1

.This means that the function Ch

M

is in

M

and therefore

(4.16) Ch

M

<

f

:

Thus,we can nd i

1

<!such that

(4.17) Ch

M

[i

1

;!) < f

St(;;s

;`)

[i

1

;!) for all`< n

(;):

Finally,choose i

2

so large that

(4.18) cf(

) <

i

2

;

and let i

= maxfi

0

;i

1

;i

2

g.

20 TODD EISWORTH AND SAHARON SHELAH

We claim now that (4.11) and (4.12) holds for any i i

.Given such an i,we

dene

N = Sk

A

(M [

i

)

= supff

St(;;s

;`)

(i):`< n

(;)g;and

= f

(i +1):

We know (4.15) holds in the model N,and since both and (dened above)

are in the model N (as

i

N and f

2 N),it follows that

(4.19) N j= (9

<

)[f

min

(i) > ^f

min

(i +1) > ]

The denition of N together with (4.18) imply that N\

is unbounded in

,

and so we can conclude that the set of 2 N\

for which

(4.20) f

min

(i) > and f

min

(i +1) >

is unbounded in

.

Suppose now that <

satises (4.20).If in addition sup(e

m

(;)

\

) < ,

then given 2 t

,we know

(4.21) sup(e

m

(;)

\

) < <

:

An appeal to Proposition 3.9 tells us

(4.22) St(;;s

;`) = St

(;;`) for all`< n

(;);

and

(4.23) St(;;s

;n

(;)) =

:

Now it should be clear that (;

) i +1 because of our choice of .Given

`< n

(;),we know

(4.24) f

St(;;s

;`)

(i) = f

St

(;;`)

(i) < f

min(i)

f

(i):

On the other hand,given j > i we know (from Lemma 1.4) that

(4.25) Ch

M

(j) = Ch

N

(j) = sup(N\

j

);

and since 2 N (as 2 t

2 N and t

is nite),it follows from (4.17) that

(4.26) f

(j) Ch

N

(j) = Ch

M

(j) < f

St

(;;`)

(j) = f

St(;;s

;`)

(j)

for all`< n

(;).The statement (4.11) now follows immediately and with it the

claim.

We are now in a position to obtain a contradiction.First,use the preceding

claim to x an

i such that such that

(4.27) s

i

= s

;

and for which there are unboundedly many

satisfying both (4.11) and (4.12).

In particular,we can x an <

in A satisfying (4.11) and (4.12) such that

(4.28) sup(e

m

(;)

\

) < <

;

we now prove

(4.29) c(;) =

for all 2 t

;

and this will yield the desired contradiction.

SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS II 21

Given 2 t

,from (4.11),we conclude (;) =

i,and hence

(4.30) s

(;) = s

:

For`< n

(;),we know

(4.31) (;St(;;s(;);`)) = (;St

(;;`)) =

i = (;);

while

(4.32) (;St(;;s(;);n

(;))) = (;

) >

i:

Thus

(4.33) k(;) = n

(;);

and

(4.34) c(;) = St(;;s

(;);k(;)) = St(;;s

;n

(;)) =

;

as required.

The contradiction is immediate as no such and are supposed to exist for our

choice of

.

5.Conclusions

We now use Theorem 4 to draw some conclusions concerning negative square-

brackets partition relations and their connection with saturation-type properties of

club-guessing ideals.These results are framed in terms of successors of singular

cardinals of countable conality because stronger results are known for the un-

countable conality case (see [10],[2],and [3]).These results are also weaker than

those claimed for the countable conality case in Section 4 of [10] |as mentioned

before,there is a problem in the proof of Lemma 4.2(4) on page 162;the present

paper provides a partial rescue.

Let us recall the following denitions:

Denition 5.1.Let I be an ideal on some set A,and let and be cardinals,

with regular.

(1) The ideal I is weakly -saturated if A cannot be partitioned into disjoint

I-positive sets,i.e.,there is no function :A! such that

1

(i) =2 I

for all i < .

(2) The ideal I is -indecomposable if

S

i<

A

i

2 I whenever hA

i

:i < i is an

increasing sequence of sets from I.

Theorem 5.Suppose =

+

for singular of countable conality,and let .

If there is a well-formed pair (

C;

I) for which the ideal id

p

(

C;

I) fails to be weakly

-saturated,then there is a coloring c

:[]

2

! such that for any two unbounded

subsets A and B of and any & < ,there are 2 A and 2 B with < and

(5.1) c

(;) = &:

In particular, 9[]

2

.

22 TODD EISWORTH AND SAHARON SHELAH

Proof.Suppose there is a function :! such that

1

(fg) is id

p

(

C;

I)-positive

for each < .Dene the function c

:[]

2

! by

(5.2) c

(;) = (c(;)):

Given A and B unbounded in and & < ,since

1

(f&g) is id

p

(

C;

I)-positive we

can apply Theorem 4 (with hfg: 2 Ai in place of ht

: < i) to nd 2 A

and 2 B such that

(5.3) c(;) 2

1

(f&g);

and this suces.

We state the following corollary in such a way that it covers all successors of

singular cardinals,though we remind the reader that stronger results are known

(see [3]) in the situation where the conality of is uncountable.

Corollary 5.2.Let be a singular cardinal.If

+

![

+

]

2

+

,then there is an

ideal I on

+

such that

(1) I is a proper ideal extending the non-stationary ideal on

+

,

(2) I is cf()-complete

(3) I is -indecomposable for all uncountable regular with cf() < < ,

and

(4) I is weakly -saturated for some < .

Proof.Let S be any stationary subset of f <

+

:cf() = cf()g,and let (

C;

I) be

a well-formed (or nice in the case where cf() > @

0

) S-club system.An elementary

argument tells us that

+

![

+

]

2

must hold,and therefore the ideal id

p

(

C;

I) is

weakly -saturated | this follows from Theorem 5 in the case where cf() = @

0

,

and Theorem 3 of [3] if cf() > @

0

.It is also routine to check (see Observation

3.2(1) on page 139 of [9]) that id

p

(

C;

I) satises conditions (1)-(3).

Now if id

p

(

C;

I) happens to be weakly cf()-saturated (a situation which might

not even be consistent |see Section 6 of [3]) then we are done.Otherwise,we can

nd a family fA

i

:i < cf()g of disjoint id

p

(

C;

I)-positive sets.Since id

p

(

C;

I) is

weakly -saturated,there must exist an i < cf() and a < such that A

i

cannot

be partitioned into disjoint id

p

(

C;

I)-positive sets.If we dene

I:= id

p

(

C;

I) A

i

:= fB

+

:A

i

\B 2 id

p

(

C;

I)g;

then I has all of the required properties.

References

[1] James E.Baumgarter.A new class of order-types.Ann.Pure Appl.Logic,54(3):195{227,

1991.

[2] T.Eisworth and S.Shelah.Successors of singular cardinals and coloring theorems I.Arch.

Math.Logic,44(5):597{618,2005.

[3] Todd Eisworth.A note on strong negative partition relations.Fund.Math.,202:97{123,2009.

[4] Todd Eisworth Successors of singular cardinals.(chapter in the forthcoming Handbook of Set

Theory).

[5] Todd Eisworth Club-guessing,stationary re ection,and coloring theorems.(in preparation).

[6] P.Erd}os,A.Hajnal,and R.Rado.Partition relations for cardinal numbers.Acta Math.Acad.

Sci.Hungar.,16:93{196,1965.

[7] Menachem Kojman.The ABC of PCF.unpublished manuscript.

SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS II 23

[8] Saharon Shelah.On successors of singular cardinals.In Logic Colloquium'78 (Mons,1978),

volume 97 of Stud.Logic Foundations Math.,pages 357{380.North-Holland,Amsterdam,

1979.

[9] Saharon Shelah.Cardinal Arithmetic,volume 29 of Oxford Logic Guides.Oxford University

Press,1994.

[10] Saharon Shelah.There are Jonsson algebras in many inaccessible cardinals.In Cardinal Arith-

metic,volume 29 of Oxford Logic Guides,Chapter III.Oxford University Press,1994.

[11] Stevo Todorcevic.Coherent sequences.Chapter in the forthcoming Handbook of Set Theory.

[12] Stevo Todorcevic.Partitioning pairs of countable ordinals.Acta Math.,159(3-4):261{294,

1987.

[13] Stevo Todorcevic.Walks on ordinals and their characteristics,volume 263 of Progress in

Mathematics.Birkhauser,2007.

Department of Mathematics,Ohio University,Athens,OH 45701

E-mail address:eisworth@math.ohiou.edu

Institute of Mathematics,The Hebrew University of Jerusalem,Jerusalem,Israel,

Department of Mathematics,Rutgers University,New Brunswick,NJ

E-mail address:shelah@math.huji.ac.il

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