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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 conality | can be extended to cover the
countable conality 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
conality.
We remind the reader that the square-brackets partition relation ![]


of
Erdos,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 conality;in general,we already know
stronger results for the case where  is the successor of a singular of uncountable
conality.The added diculties 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 Classication.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 conality (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.
Denition 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 suciently 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 denable
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:
Denition 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.
Denition 1.3.Let  be a singular cardinal of conality ,and let ~ = h
i
:i < i
be an increasing sequence of regular cardinals conal 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  dened 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 Denition 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 suciently 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 Denition 1.3.If i

<  and we dene
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 suciently large"and\9

"to mean\there are unboundedly
many".
Lemma 1.5.Let  = 
+
for  singular of conality ,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]:
Denition 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 conality (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 conality.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.
Denition 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 denition,there is precious little dierence between
calling he

: < i a C-sequence and calling it a -club system | the two names
exist for historical reasons.The dierence 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 dier-
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 denition 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.
Denition 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 dene Gap(;C),the gap in C determined by ,by
(2.1) Gap(;C) = (sup(C\);):
The next denition captures some standard ideas from proofs of club-guessing;
we have chosen more descriptive names (due to Kojman [7]) than those prevalent
in [9].
Denition 2.3.Suppose C and E are sets of ordinals with E\sup(C) closed in
sup(C).We dene
(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 dene
(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 dierence 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 conal 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 satised for all  <  and jC

j  .Suppose this is not the case,
and let h
i
:i < cf()i be an increasing sequence of cardinals conal 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 dene
(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 conal 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 satises
(3) of the above theorem | this is in concordance with notation from [9],and it
also ts in with the nice pairs dened 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 conality.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 conality,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 suces 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 dene 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 dene E
n+1
= acc(E
n
\E
0
n
\E
1
n
),dene 
n+1

to be the least such ,and
dene 
n+1

to be the least  corresponding to 
n+1

.
Now that E
n+1
has been dened,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 conality
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 dened for any  that needs attention at
this stage.
To nish the construction,we dene
(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 dene
(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
dened 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 conality,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 conality,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 conal subset of  (for conve-
nience,we dene 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

(n1);c

(n)].If we dene
(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 conality 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 satises the structural requirements of Theorem 3 if conditions (4)
and (5) hold,and say it satises the club-guessing requirements of Theorem 3 if
condition (6) holds.
The main thrust of our construction is to dene 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 dening

C
+1
and E
+1
from

C

and E

.The reader should also be warned that several auxiliary objects will be
dened 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  | dening 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 dene
(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 dene
(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 dene
(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 dene 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 dened,our S-club system

C
+1
satises 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 suciently large  < g:
The set C


dened above is closed in  by denition.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 satises 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 dene
(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 dened for all  <!
1
and hence there is a least n

<!
such that n(;) = n

for innitely 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 dene
(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 denition 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.
Denition 2.6.Let  = 
+
for  singular of countable conality,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 conal in 
(2) hf

C
(n):n <!i is a strictly increasing sequence of regular cardinals conal
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 conality.If f:!! enumerates a strictly
increasing sequence of regular cardinals that is conal 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 Denition 2.6,so for each  2 S we dene
(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 conal 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 denition.
Denition 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 conality,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 conality greater than ,and this follows
immediately from the denition 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 Denition 2.6,and

I = hI

: 2 Si is the
sequence of ideals dened 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 Todorcevic'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.
Denition 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 denition with concepts from the preceding
section.
Lemma 3.2.Let  = 
+
for  singular of countable conality,and let (

C;

I) be
a well-formed S-club system for some stationary S   consisting of ordinals of
countable conality.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 verication 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.
Denition 3.3.Let  = 
+
for  singular of conality @
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 Denition 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 denition.
Denition 3.5.Let e be a generalized C-sequence on some cardinal ,and let s
be a nite sequence of natural numbers.Given  <  < ,we dene 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 Todorcevic,at each stage of a minimal walk one has
a single ladder to use to make the next step.In our context,there are innitely
many ladders available,and the parameter s selects the one we use for our next
step.Even though there are innitely 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 denition.
Denition 3.6.We dene 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 Denition 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 conality,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 dene 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 dene
(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

\

) < 

:
Denition 3.8.Suppose  2 S,and  <  < .For each m <!,we let
s(;;m) 2
!
!be the sequence of length n

(;) dened 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 Denition 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 benets from a description in English.Given
 <  with  2 S,if we dene

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 eect 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 conality | 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 conality @
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 dened (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
innitely 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 suciently 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.
Denition 4.1.We dene a coloring c:[]
2
! as follows:
For  <  < ,let
(4.2) s

(;) = s
(;)
Next,dene
(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

fg
is constant with value 

.
Let hM

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

: < i;Ag,and
let E be the closed unbounded set dened 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 dened 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 denition 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 suciently 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 Denition 1.3.
For each  < ,let f
min

be the function with domain!dened 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
dene
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  (dened 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 denition 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  < 

satises (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 conality because stronger results are known for the un-
countable conality case (see [10],[2],and [3]).These results are also weaker than
those claimed for the countable conality 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 denitions:
Denition 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 conality,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
(fg) is id
p
(

C;

I)-positive
for each  < .Dene 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 hfg: 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 suces.
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 conality 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) satises 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 dene
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
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SUCCESSORS OF SINGULAR CARDINALS AND COLORING THEOREMS II 23
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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