saharon shelah- middle diamond

8/3/2019 Saharon Shelah- Middle Diamond

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MIDDLE DIAMOND

SH775

Saharon Shelah

The Hebrew University of JerusalemEinstein Institute of Mathematics

Edmond J. Safra Campus, Givat RamJerusalem 91904, Israel

Department of MathematicsHill Center-Busch Campus

Rutgers, The State University of New Jersey110 Frelinghuysen Road

Piscataway, NJ 08854-8019 USA

Abstract. Under certain cardinal arithmetic assumptions, we prove that for everylarge enough regular cardinal, for many regular < , many stationary subsets of concentrating on cofinality has the middle diamond. In particular, we have themiddle diamond on { < : cf() = }. This is a strong negation of uniformization.

Key words and phrases. Set theory, Normal ideals, Middle Diamond, Weak Diamond.

I would like to thank Alice Leonhardt for the beautiful typing.This research was partially supported by the Israel Science Foundation.Pub. 775First Typed at Rutgers - 6/13/01Latest Revision - 04/Oct/11

Typeset by AMS-TEX

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2 SAHARON SHELAH

Annotated Content

0 Introduction

[We state concise versions of the main results: provably in ZFC there aremany cases of the middle diamond, for many triples (, , ) where < are regular, 2 < we can give for any S a function f fromsome subset C of of order type unbounded in into such that anyf : extends f for stationarily many s. We also review somehistory.]

1 Middle Diamond: sufficient conditions

[We define families of relevant sequences

C = C : S and propertiesneeded for phrasing and proving our results (see Definition 1.5) and giveeasy implications; (later in 1.22, 1.25 we define other versions of middle dia-mond). Then (in 1.10) prove a version of the middle diamond for (, , , )guessing only the colour off C when = cf(2

), the number of possiblecolours, , is not too large and belongs to a not so small set of regularcardinals < (e.g., every large enough regular < if ). Thedemand on says 2 is not the number of -branches of a tree with < 2

nodes. We then prove that a requirement on (, ) is reasonable (see 1.11,1.9, 1.13). In 1.20 we get the version of middle diamond giving f C fromthe apparently weaker one guessing only the colour of f C. We quotevarious needed results and definitions in 1.3, 1.24.]

2 Upgrading to larger cardinals

[Here we try to get such results for many large enough regular . Wesucceed to get it for every large enough not strongly inaccessible cardinal.We prove upgrading: i.e., give sufficient conditions for deducing middlediamond on 2 from ones on 1 < 2 (see 2.1) and 2.3 (many disjoint ones),2.10, 2.11). We deduce (in 2.4) results of the form if is large enoughsatisfying a weak requirement, then it has middle diamond for quite many

, , using upgrading claims on the results of1 and generalization of [EK](see 2.5, 2.6, 2.7). We deal with such theorem more than necessary just forthe upgrading.]


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MIDDLE DIAMOND SH775 3

0 Introduction

Using the diamond (discovered by Jensen [J]) we can construct many objects;wonderful but there is a price - we must assume V = L or at least instancesof G.C.H. (see Gregory [Gr], Shelah [Sh 108]). In fact = + = 2 > { 2 which is not strongly inaccessible, for every large enough regular < , forevery < ,S has the -middle diamond (called later has the (,

+)-MD2-


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Note that having Ms is deduced in 1.20 from C has the -Md-property with large enough. Note that 0.1 has obvious applications on various constructions,e.g., construction of abelian groups G with Ext(G, K) = .

Note that in 0.1, 2.9 instead of using large enough < . . . , strong limitand quoting [Sh 460] we can specifically write what we use on (and on ). Alsoin 0.1, if we omit cf() > 0 (but still is uncountable) the case successor is notaffected, and in the case = when 2 divides , the statement instead everylarge enough can be replaced by for arbitrarily large regular < , or even:for arbitrarily large strong limit < , for every large enough regular < .A consequence is (see more on this principle in [Sh 667]).

0.3 Conclusion. 1) If ,,,S, M : S are as in 0.1 and < < , then

we can find = : S, is an increasing sequence of length of ordinals< with limit such that:

ifF : , then for stationarily many S we have i < F((2i)) =F((2i + 1)).

2) In fact it is enough to find a (,,)-Md-parameter C : S which has the

(, 2||+

)-MD-property (see Definition 1.6(1)).

Proof. 1) By (2).2) Let = {P : < ||

+}, pi a monadic predicate. Let M : S be as in0.1 for . Let F : and we let M be the following -model with universe : PM = { : = pr(0, 1), 0 < 1 < ||

+ and F( + 0) = F( + 1),

M + ||+ < Min(M\). Without loss of generality otp(|M|) = , S

||+/. Let ((2, i), (2i + 1)) = (i + 2 ,

i +

2) when

i is the i-th member of

M, i P.

Note that if a natural quite weak conjecture on pcf holds, then we can replace by (and similar changes) in most places but still need > 1 > , 2

1 > 20 2

in the stepping-up lemma; e.g. for every regular large enough , for some n, wehave middle diamond with 0 colours on S

n

- is helpful for abelian group theory.(Under weak conditions we get this for 1, see [Sh:F611].

Even without this conjecture it is quite hard to have the negation of this state-ment as it restricts 2 and cf(2) for every !


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6 SAHARON SHELAH

0.4 Definition. 1) Let S has square mean that is regular uncountable, Sa stationary subset of and: for some club E of and S, S E S there isa witness C = C : S

which means:

() C = C : S has the square property which means

(a) C is closed,

(b) S limit = sup(C),

(c) C C = C and

(d) S otp(C) < .

1A) We say S has a strict square if we add

(e) S E otp(C

) = cf().

2) We say S has a ( ) square if we replace (d) above by

(d) S otp(C) .

If we replace square by ( )-square this means we add otp(C) .

3) We say ( , )-square if above the demand C C = C is restrictedto the case cf() .4) S = { < : cf() = }.

On I[] see [Sh 420, 1].

0.5 Definition. 1) For regular uncountable let I[] be the family of sets S which have a witness (E, P) for S I[], which means

() E is a club of ,P = P : < ,P P(), |P| < , and for every E S there is an unbounded subset C of of order type < such that C C P.

2) Equivalently there is a pair (E, a), E a club of , a = a : < , a , a a = a and E S = sup(a) > otp(a) (or even = sup(a),otp(a) = cf() < .

0.6 Notation. 1) Let ,,,,, denote cardinals, a set of cardinals, let,,,,,,i,j, denote ordinals but 2 and 2 and is a limit ordi-nal if not said otherwise.

Clearly 1.10 and some results later continue Engelking Karlowic [EK] whichcontinue works on density of some topological spaces, see more history in [Sh 420].


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1 Middle Diamond: Sufficient conditions for cf(2)

1.1 Definition. For a regular uncountable , an ordinal 2 and cardinal 2

(1) we say S is (, )-small if it is F-(, )-small for some (, , )-colouringF, also called a -colouring of >, that is: a function F from > to where

(2) we say S is F (, )-small if (F is a -colouring of> and)

()F,S for every c

S for some the set { S : F( ) = c} is notstationary.

(3) Let Dmd,, = {A : \ A is (, )-small}, we call it the (, )-middle

diamond filter on . If = 2 we call this weak diamond filter and may writeDwd,.

(4) We omit if = , writing instead of (, ). We may omit from S isF -small, and ()

F,S when clear from the context.

1.2 Claim. 1) If = cf() > 0, 2 and then Dmd,, is a normal

filter on.2) Assume S , 2 , 2 for = 1, 2. Then S is (1, )-small iffS is (2, )-small

1.

3) In fact [2, 2


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8 SAHARON SHELAH

1.4 Remark. 1) On tr see [Sh 589], on see [Sh 430], on , [] seethere and in [Sh 460] but no real knowledge of these references is assumed here.2) Note that when = only F { : S} is relevant in Definition 1.1 because

for any

for some club E of we have E

. So we can restrictthe colourings to this set.

Below we may usually restrict ourselves to = . But if < = cf(), 2 = 2

even for = = 2, the set is small (as noted by Uri Abraham, see [DvSh 65]).

1.5 Definition. 1) We say C is a -Md-parameter if:

(a) is regular uncountable,

(b) S is a stationary subset of ,

(c) C = C : S and2 C = sup(C).

1A) We say C is a (,,)-Md-parameter if in addition ( S)( C)[cf() = |C | < ].2) We say that F is a (C, , )-colouring if: 2, 2, C = C : S is a-Md-parameter and F is a function from > to such that:

if S and 0, 1 and 0 C = 1 C then F(0) = F(1).2A) If = we may omit it and write (C, )-colouring.2B) In part (2) we can replace F by F = F : S where F :

(C) suchthat S F() = F( C). If = we may use F :

(C) (as we can ignore a non-stationary S S); so abusing notation we may writeF( C). Similarly if cf() = , : < is increasing continuous with limit we may restrict ourselves to (C)().3) Assume F is a (C, , )-colouring, where C is a -Md-parameter.We say c S (or c ) is an F-middle diamond sequence, or an F-Md-sequenceif:

() for every , the set { S : F( ) = c} is a stationary subset of .

We also may say that c is an (F, S)-Md-sequence or an (F, C)-Md-sequence; notethat can be reconstructed from F.3A) We say c S is a D F-Md-sequence or D (F, C)-Md-sequence if D is afilter on (no harm to assume that S belongs to D, or at least S D+) and

() for every we have

{ S : F( ) = c} = mod D.

2we can omit the requirement = sup(C) but then we trivially have middle diamond if =


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4) We say that C is a good (, 1)-Md-parameter, if C is a -Md-parameter andfor every < we have 1 > |{C : S & C}|. If1 = we may writea good -Md-parameter. Similarly for a (, 1, , )-good-Md-parameter and a

(,,)-good-Md-parameter (i.e. adding that C is a (,,)-Md-parameter.4A) We say that C is a weakly good -Md-parameter if it is a -Md-parameter andfor every < we have > |{C : S & nacc(C)}|. Similarly wedefine a weakly good (,,)-Md-parameter and a weakly good (, 1, , )-Md-parameter.4B) We say C is a good+-Md-parameter ifC = C : S is a -Md-parameterand C1 C2 C1 = C2 . Similarly we define a good

+ (,,)-Md-parameter.4C) We define a weakly good+ -Md-parameter or (,,)-Md-parameter anal-ogously, i.e., nacc(C1) nacc(C2) C1 = C2 .4D) For a set of regular cardinals we say that C = C : S is -closed ifeach C is; where a set C of ordinals is -closed if for every ordinal , cf() & = sup(C ) < sup(C) C.4E) We say (C, C

) : S is a good (,,)-Md parameter if C : S

is a (,,)-Md-parameter, C , sup(C) = and for every < the set

|{C : C, S}| has cardinality < .

1.6 Definition. 1) We say that C (or (, C)) has the (D, , )-Md-property if

(a) C is a -Md-parameter

(b) D is a filter on

(c) if F is a (C, , )-colouring, then there is a D F-Md-sequence, see Defini-tion 1.5, part (3A).

We may omit D if it is the club filter on .2) We say that C has the (D, )-Md-property as in part (1) only that F is a (C , , )-colouring. We may omit D if D is the club filter. We may omit if = 2.3) We say that S has the (, )-Md-property when some C = C : S has,and then we say the -Md-property if = .In Definition 1.5(2),(2A) we replace colouring by colouring2 if we have all : > (or : >(C) ); and let colouring1 be the original notion (soin Definition 1.5(3),(3A) we have new meaning when we use such Fs).4) We write Md2-property if we use colouring2 F and we write Md1-property forthe Md-property.

1.7 Remark. We may consider replacing C by (C, D), D a filter on C, that isF() would depend on ( C)/D only. For the time being it does not make areal difference.


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10 SAHARON SHELAH

1.8 Claim. 1) The restrictions in 1.5(2B) are O.K. That is in 1.5(2B), the firstsentence is obvious; for the second sentence (if = ) recall that for every , for some club E of , E ; and similarly in the third sentence.

2) In Definition 1.5(4), if 1 > then goodness holds trivially.3) If C = C : S is a weakly good (,,)-parameter, then for some club Eof there is C = C : S E which is a good

+ (,,)-parameter.4) If C = C : S has the (D, , )-Md-property, and C

= C : S, C

C, then also C has the (D, , )-Md-property.

Proof. 1),2) Easy.3) Note that the Cs are not required to be a club of , see [Sh 420, 1] butwe elaborate. Let A : < list without repetition {C : S and nacc(C)} and be such that A = A A {A : }. Let

E = { < : if < and S and nacc(C) then C {A : < }},clearly E is a club of.

Lastly, for S E let C = { < : for some nacc(C) we haveC = A}. Now C

: S E is as required.

4) Should be clear. 1.8

From the following definition the case Sep(, ) is used in the main claim below.

1.9 Definition. 1) Sep(,,0, , ) means that for some f:

(a) f = f : <

(b) f is a function from

(0) to (c) for every the set { (0): for every < we have f() = ()}

has cardinality < .

2) We may omit 0 if 0 = . We write Sep(, ) if for some = cf() 2 wehave Sep(,,,, ) and Sep(< , ) if for some = cf() 2 and < wehave Sep(,,,, ) .

1.10 The Main Claim. Assume

(a) = cf(2), D is a +-complete filter on extending the club filter

(b) = cf() < (c) C = C : S is a (,,)-good-Md-parameter, or just weakly good or

just (C, C) : S is a good (,,)-Md-parameter and S D

(d) 2


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Then C has the (D, 2, )-Md-property (recall that this means that the number ofcolours is not just 2).

We may wonder if clause (f) of the assumption is reasonable; the following Claimgives some sufficient conditions for clause (f) of 1.10 to hold.

1.11 Claim. Clause (f) of 1.10 holds, i.e., Sep(, ) holds, if at least one of thefollowing holds:

(a) =

(b) U() = + 2 ,

(c) UJ() = where for some we have J = []


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12 SAHARON SHELAH

By assumption (f) we have Sep(, ) which means (see Definition 1.9(2)) thatfor some = cf() 2 we have Sep(,,,, ).Let F = {f : < } where F exemplify Sep(,,,, ), see Definition 1.9(1)

and for

let Sol =: {

: for every < we have () = f()} whereSol stands for solutions, so by clause (c) of the Definition 1.9(1) of Sep it followsthat

() |Sol| < .

Let h be an increasing continuous function from into 2 with unbounded range; if is regular, h = id is O.K. Recall that = cf(2

) 2


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assume towards a contradiction that for each < there is a sequence (2)that exemplifies the failure of c to be an F-Md-sequence, so there is a E D suchthat

1 S E F( C) = f().

Define (2) by () = cd(() : < ). Now as is regular uncountableit follows by choice of h that E =: { < : for every < we have () < h()and if S, C and C

= C then C T


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14 SAHARON SHELAH

f(h()) if < & f(h()) < 1 and f

() = 0 otherwise. Let

()(1) and

let A = { ()(0): for every <

we have f() =()}. Let =:

so (1) (1), hence by the choice of f : < the set A = { (0):

for every < we have f() = ()} has cardinality < . If A then

(0) and h() (0), and for <

we have f() =() hence for

< ( ) we have either f(h()) < 1 & () =

() = f() = f(h()) orf(h())

1 & () =

() < 1 so f(h()) = (). Hence A

h() Aand as h is a one-to-one function we have |A | |A | <

and we are done.1.13

To show that 1.10 has reasonable assumptions we should still prove 1.11.

Proof of 1.11. Our claim gives sufficient conditions for Sep(, ), i.e. Sep(,,,, )for some = cf() 2. Clearly if < 2 we can waive regular as + canserve as well. Let 1 = .

Case 1: = , = , 0 [, ] and we shall prove Sep(,,0, 1, ). Let

F =

f :f is a function with domain (0) and

for some u [] and sequence = i : i <

with no repetition, i u(0), we have

( (0)[i f() = i] and

( (0)[(i


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choice of , f(i) = fu,(i) and by the definition of f

u, recalling i u = i we

have fu,(i) = i, so i = f(i) = (). This holds for every i < whereas ,

contradiction.

Case 2: 2 and U[] = .Let = (2)+ or just (2


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16 SAHARON SHELAH

Note that in case 2, we can replace 2 by: there is F of cardinality such that for every A [] for some f F we have = Rang(f A). Theproof is obvious. Similarly in case 3. 1.11

There are some obvious monotonicity properties regarding the Md-property.

1.14 Claim. 1) If (, C) has the -Md-property (see Definition 1.6(1)) and C =C : S and C

1 is an unbounded subset of C for S then (, C

1 : S)

has a -Md property. We can also increase S and decrease ; we can deal with the(, )-Md-property.2) Assume = cf() > 0, S { : < , cf() = } is stationary and < .Then some C : S is a (,,)-Md-parameter.

Proof. 1) Obvious.2) Easy (note: goodness is not required). 1.14

We may consider the following generalization.

1.15 Definition. 1) We say C is a -Md3-parameter if:

(a) is regular uncountable

(b) S is a stationary subset of (consisting of limit ordinals)

(c) C = C, C, C, : S, < cf()

(d) C = sup(C) and C = sup(C), otp(C) = and C = {C, : < cf()}

(e) < cf() sup( |{C, : S, (C)} and C1, = nacc(C). 1.18

1.19 Definition. 1) For a set A of ordinals and an ordinal let A = { : < or for some A we have [, + )}.2) The pair (, C) or just C has the (, )-MD0-property (or use MD instead ofMD0) if we can find f : S such that:

(a) f is a function from C = C

to for some

<

(b) if f is a function from to and < is non-zero then for stationarymany S we have f f and

= .

If we omit we mean = 2 so = 1 for every S.3) (D, ,)-MD2-property means: if is a relational vocabulary of cardinality < ,then we can find M : S such that:

(a) M is a -model with universe C = C

for some

<

(b) if M is a -model with universe and < then the set AM, = { S :M C = M and

= } is stationary and even belongs to D

+.

If we omit we mean = 2 so = 1 for every S. Similarly with D.4) We say (, S) has the (, )-MD-property if for some C = C : S thepair (, C) has the (, )-MD-property. We say that has the (,,)-propertyif (, S) has the (, )-MD-property.

The following claim shows the close affinity of the properties stated in theorem0.1 in the introduction and the -Md-property with which we deal in this sectionhere. This claim is needed to deduce Theorem 0.1 from 1.10.

1.20 Claim. 1) Assume (, C) has the -Md-property where C = C : S is a(,,)-Md-parameter. If < , =


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AM, = { S : = and (M cM(C

), c)cC

= (M, c)cC

}.

4) In part (2) we can replace good+ by good if we add

each C is closed under pr, where pr is a fixed pairing function on .

Proof. The proof consists of a translation of one version of middle diamond toanother (note: is a relational is a strong restriction).1) First fix < and deal with C : S, C

= C

\[0,

) recalling C = C . Let cd be a one-to-one function from > onto . Define F : S as follows:if f (C) then F

(f) = cd(f(

,) : < otp(C

)) where

,: < otp(C

)

list C in increasing order so F is a function from(C) into . As we assume (, C)

has the -Md-property, we can apply its definition, hence for F = F : Sthere is a diamond sequence c : S.

We now define f : C as follows: if 0 is a member of C and

[0, 0 + ) and < Min(C \(0 + 1)) and c = cd(, : < otp(C) and

, = cd(,,i : i < then f () = ,otp(C0),0 ; otherwise f

() = 0. So

f : C . Why is f

: S as required for C

: S (and

)? Assumethat f : , so we can define f : by f() = cd(f( + ) : < ),hence for stationarily many S we have c = F(f

C). Now we check thatf C is equal to f

defined above.

Let ,,i = f(,) and ,+i = f(, + i) for i < , < otp(C) so , =f(,) = cd(,,i : i <

), c = F(f C) = cd(, : < otp(C).

Hence by the definition of f we have: if = , C, [, + ) and < Min(C\(+1)) then f

() = ,, = f(,+()) = f(+()) =

as required.To have every < (and guess also f (C\C

) f [0,

)) we just haveto partition S to S : < such that each C S has the -Md-property whichwe can do by 1.23(3) below (or work a little more above, i.e., f() = cd(f(+) : < f() : < )).2), 3), 4) Left to the reader. 1.20

1.21 Claim. 1) Assume = cf() > = cf(), S S and

(a) C is a -Md-parameter for = 1, 2

(b) C1 = C1 : S has the (, )-Md-property

(c) A []


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20 SAHARON SHELAH

Then C2 has the (, )-Md-property.2) If C is a (,,+)-Md-parameter with the (, )-Md-property, = andC = { + i : C = 0 and i < }, then C

= C : Dom(C) is a

(,,+

)-Md-parameter with the -Md-property.

Proof. Easy by coding (as in the proof of 1.20).

1.22 Definition. Assume > are regular and C = C : S is a (,,)-Md-parameter with the (, )-Md-property. Let ID,,,,(C) = {A : C (S A)does not have the (, )-Md-property}. If = we may omit it.

1.23 Claim. LetC,,,, , be as above in Definition 1.22 andJ = ID,,,,(C).

1) J is an ideal on such that \S J, / J if 0.2) J is -complete and even normal if (if0 < it is still

+-complete).3) If S and S = mod J, then for any Fi = Fi : S, F

i :(C) ,

can be partitioned to subsets each having a Fi-Md-sequence.

Proof. 1) Let cd: be one-to-one onto and cd : be suchthat cd(cd(0, )) = . Clearly J is a subfamily ofP() and it is closed

under subsets, so we need just to prove that it is closed under union of two.So suppose A = A1 A2, A J for = 1, 2. As J is closed under subsets,without loss of generality A A2 = . Let the sequence F = F : S Aexemplify A J. Now we define F = F : S A as follows:

if S A and f (C) then: A F(f) = F (cd f), i.e., F(f) = F

(f

) where C f() = cd(f()).

2) Straight as the sequence F : S is not required to be nicely definable.3) Guessing is the same as guessing . 1.23

Well, are there good (,,+)-parameters? Some version of this is crucial forhaving interesting consequences of 1.10, but we shall have to quote (on I[] seeDefinition 0.5; see on it [Sh 420, 1]).


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1.24 Claim. 1) If S is a stationary subset of a regular cardinal and S I[],see Definition 0.5, and ( S)[cf() = ], then

() for some club E of , there is a good+ (S E,,+)-Md-parameter. (Ofcourse, we can replace the + by any ) (recall that we do not requiredthe Cs to be a closed subset of )

() for some club E of there is a weakly good+ (S E,,+)-Md-parameterconsisting4 of clubs (this is of interest when > 0)

() for some club E of there is a good (S E,,+)-Md-parameter C = C : S E such that

(i) C is weakly good+, i.e., nacc(C1)nacc(C2) C1 = C2

(ii) C is good, i.e., for each < we have > |{ C : S, C}|

(iii) C is -closed that is each C is -closed which means = sup(C

) & cf() C where = { : = cf() and < ||tr < }.

2) If = cf(), + < = cf(), then there is a stationary S I[] with ( S)[cf() = ].3) In part (1), moreover, we can find a good (S E , , +)-Md-parameter and a-complete filter D on such that: for every club E of , the set { S E :C E

E} belongs to D.

Proof. 1) By the definition ofI[] see Definition 0.4 and 1.5 above (see more in [Sh420, 1]) and for clause () the definition of ||tr .2) By [Sh 420, 1].3) By [Sh:g, III]. 1.24

1.25 Definition. Let C be a (,,)-Md-parameter

(1) For a family F of C-colourings and P 2, let idC,F,P beW : for some F F for every c P

for some the set

{ W S : F( C) = c}

is a non stationary subset of

4this means: the parameter C = C : S E, C a club of


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22 SAHARON SHELAH

(2) IfF is the family of all C-colouring we may omit it. If we write Def insteadofF this mean as in [Sh 576, 1] (not used so the reader can ignore this).

We can strengthen 1.5(2) as follows.

1.26 Definition. We say the -colouring F is (S,,)-Md-good if:

(a) S { < : cf() = } is stationary

(b) we can find E and C : S E such that

() E is a club of

() C is an unbounded subset of , |C| <

() if , , S E, and C = C then F() = F()

() for every < we have > |{C : S E and C}|

() S ||tr < or just S > |{C : C isunbounded in and for every C for some S we have C =C and C}|.

We can sum up (relying again on some quotations).

1.27 Conclusion. If = cf(2

) and we let =: { : = cf() and (


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(f) for , S as in clause (b) above 2, if > 0 C = C : S, C a

stationary subset of for each then we can find C = C : E Ssuch that C is a club of for each and C

C

: S has the

(, )-Md-property (really C

does not depend on C

)(g) in clause (f) if = , , 2 we can conclude also that C C

:

S has the (, )-MD2-property.

Proof. Clause (a) holds by [Sh 460], clause (b), (c) holds by 1.24, clause (d) by themajor claim 1.10 and clause (e) is a rephrasing of clause (d), and lastly for clause(f) use above case (iii) of clause (c) and use the obvious monotonicity (and C C

is unbounded in being stationary). 1.27

Note some open questions.

1.28 Question: 1) Assume is a weakly inaccessible which is not strong limit (i.e.a limit regular uncountable cardinal but not strong limit) and 2 : < is noteventually constant.Does have the weak diamond? Other consequences? In this case cf(2 2


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24 SAHARON SHELAH

(a) f = f : <

(b) f is a function from to

(c) f is increasing continuous in , i.e., < f = f

Dom(f

)

(d) for every < < the set { < : f () = ()} is not empty.

If we succeed, then we let f = f2

that is f =

{f : < 2} for < and

f : < is clearly as required.For clause (d) it is enough to consider the case = + 1, as the non trivial case

in the induction is = + 1. We can choose f+1 () : < such that it

holds by the assumption ().2) In the proof of 1.10, now |Sol| < 2 does not suffice. However, beforechoosing f : < we are given F and we can choose cd, cd : < and defineT, : < 2

, T


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In 1 we get the -middle diamond property for regular cardinals of the formcf(2) on most S S for many = cf() (and ). We would like to proveTheorem 0.1. For this we first give an upgrading, which is a stepping up claim(for MD) given as 2.1 below. Then we prove that it applies to 2 (for quite a few1), when 2 is a successor of regular, then when 2 is successor of singular andlastly when 2 is weakly inaccessible (not strong limit). We can upgrade each ofthe variants from 1, but we shall concentrate on the one from 1.5(5A) because ofClaim 1.20.The main result of this section is claim 2.9.

Question: What occurs to 2

strongly inaccessible, does it have to have some Md-property?

Possibly combining 1 and 2 we may eliminate the 2 > 21++ in 2.1, but

does not seem urgent now.

2.1 Upgrading Claim. 1) Assume

(a) 1 < 2 are regular cardinals such that 2 > 21+

+1 , 2 is a successor of regular.

Then for some stationary S2 S2 , there is C = C : S2, a good5 (2, , )-

Md-parameters which has the (D2 , , )-Md-property.2) We can in part (1) replace clause (c) by any of the following clauses

(c) there is a 1 square on some stationaryS S21

, (see Definition 0.4(1A);

this holds if 2 = +, 1 = )

(c) 2 has a (1, S1)-square (where S1 is from clause (b)) which means:

21,S1 1 < 2 are regular, S1 1 is stationary and there are S2 S21

stationary and C2 = C2 : S2 satisfying C2 = sup(C

2 ), C2

is closed, otp(C2 ) = 1 and if C21

C22 , {1, 2} S2 and =

sup(C21 ) and otp( C22

) S1, then C22

= C21 .

5the goodness is a bonus


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26 SAHARON SHELAH

(c) 2 has a (1, C1)-square C2 (where C1 is from clause (b)) which means:

21,C1,C2

() C2 = C2 : S2

() S2 S21

is stationary

() C2 is a closed unbounded subset of of order type 1

() if C21 C22

(so {1, 2} S2) and otp ( C21

) S1,and = sup(C21 ) then otp( C

21

) = otp( C22), call it and

{ C21 : otp(C21

) C1} = { C22

: otp(C22) C1}.

3) Assume {0, 2}

(a) 1 < 2 are regular uncountable

(b) some (1, , )-Md-parameter has the -MD-property

(c) at least one of the variants of clause (c) above holds.

Then some (2, , )-Md-parameter has the -MD-property.

2.2 Definition. We say that a (, )-colouring F = F : S for a -Md-parameter C = C : S is nice when:

if 1, 2 S and otp(C1) = otp(C2) and 1 (C1 ), 2 (C2 ) and1 C1 2 C2 otp(C1 1) = otp(C2 2) 1(1) = 2(2),then F1(1) = F2(2).

Proof. 1) If clause (c) holds then clause (c) of part (2) holds by [Sh 351, 4], so itis enough to prove part (2).2) Trivially clause (c) implies clause (c) and clause (c) implies clause (c); soassume (c); and let C2 = C2 : S2 be as there.

Let h = h : S2 be such that h is an increasing continuous function from

1 onto C2 recalling clause (), i.e., C2 is a closed unbounded subset of of ordertype 1. Let S = {h(i) : S2 and i S1} and if S, = h(i), S2 andi S1 then we let C

= {h(j) : j C

1i }; and let C

= C : S. Why is C

(hence C) well defined? We shall check that if S, = h(i), S2, i Sfor = 1, 2 then {h1(j) : j C

1i1

} = {h2(j) : j C1i2

}; this holds by clause() of the assumption (c). We should check the conditions for C is a good(2, , )-Md-parameter with the (D2 , , )-Md-property.


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The main point is: assume that F is a (C, , )-colouring, find a c as required.For each S2 we define a (C

1, , )-colouring F by F() = F( h1 ) that

is for i S1 and C1i we define F() as F() where C

h(i), where is

defined by Ch(i), we let () = (h1 ()). As6 2 > 21+||

22

(superfluous for nice colourings, e.g., for MD, {0, 2}), thenfor every large enough regular < , some good+ (,,+)-Md-parameter C hasthe (2, )-Md-property for every < .2) We can add: C has the -MD-property for = {0, 2}.3) We can find a sequence Si : i < of pairwise disjoint stationary subsets of S

such that for each i < there is Ci = Ci : S as in parts (1) and (2).

Proof. 1) Let 0 < be large enough such that [ {, 2} & < & 0 = cf() < ] ||[] < , (see Definition 1.3, exists by [Sh 460]).

By 1.10, for any regular [0, ) and < we have the result for cf(2) inplace of . If is a successor of regular we can apply 2.1(1), i.e., for clause ( c)

6only place this is used, ifF is nicely defined this is not necessary


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28 SAHARON SHELAH

there is a square as required by [Sh 351, 4]. In the case that is a successor of asingular cardinal, by 2.7 below, clause (c) of 2.1(2) holds for 2 = , 1 = cf(2

)so we can apply 2.1.

2) Follows by 1.20.3) In the proof of parts (1) + (2) above, instead of 2.1 we use 2.3. 2.4

Below we generalize Engelking Karlowic theorem [EK] (which says that if =, = 2 then there are f : for < such that every partial functionfrom to with domain of cardinality , is extended by f for some < ); wedo more than is necessary for the middle diamond.

2.5 Claim. 1) Assume

(a) = cov(, +, +, +)

(b) 2 (c) < 2

Then there is F such that:

() F = F : <

() each F is a function from to

() if u [] then we can find E such that

(i) E = Ei : i < is a sequence of equivalence relations on u

(ii) u {} = {/Ei : i < }

(iii) if f : u respects some Ei (i.e. Ei f() = f())then we can find a partition u : < of u such that < f u F for some < .

2) In part (1) we can strengthen clause () to () below if (d) for some {1, . . . , 5} below holds

() if u [] and f : u , then we can find u = ui : i < such that

(i) u = {ui : i < } and

(ii) for every i < for some < we have f ui F

(d)1 there is a sequence f = f : < of pairwise distinct functions from to such that: if u [] then we can find sequences u = ui : i < andBi : i < such that u = {ui : i < }, Bi [] {ui : i < } and(i < )( < )[f ui F)

then we can weaken the assumption in a parallel way, e.g.

(d)1,, there is f = f : < , f such that for any club E of we can find a

closed e E of order type + 1, and u = ui : i < , B = Bi : i < suchthat Bi [1] < 0, ui and u & cf() {ui : i < } andfor each i, f Bi : ui is with no repetitions (and in (d)3 it becomeseasy).

Remark. We can replace (as a domain) by []2, etc.

Proof. 1) Let A be a family of subsets of each of cardinality such thatany A [] is included in the union of many of them; it exists by assumption(a). By assumption (b) without loss of generality A A & B A B Aso we replace included by equal and also without loss of generality A1 A &A2 A A1 A2 A. Let pr be a pairing function on .

As < 2 there is a sequence f : < of pairwise distinct functionsfrom to {0, 1}.

Let

Y = {(A,B, g, ) :B A, g = gi : i <

gi B2 and i < j < gi = gj

and = i : i < satisfies {i : i < } A A}.


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30 SAHARON SHELAH

Clearly |Y| 2 = . For each x = (A, g, ) Y we define a functionFx : as follows:

Fx() is i if gi f and is 0 if there is no such i (clearly we cannot have

i0 = i1 < & gi0 f & gi1 f

).

Let F = F : < list {Fx : x Y}, clearly F satisfied demand () + (). Asfor demand () let u [] be given. For any 1 = 2 u let = (1, 2) < beminimal such that f1() = f

2

(), exists as f2 = f1

2 and let B = {(1, 2) :1 < 2 are from u}. By the choice ofA we can find pairwise disjoint B A for < such that B =

{B : < }. Lastly, we define E = E : < by

for 1, 2 u, 1E2 iff f1 B = f

2 B.

Assume that f : u respects E, then we can find non empty A A for <

such that Rang(f) =


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(a) Ai,j = Bi Aj

(b) gi,j = fi,j

Bi

(c) i,j

= f(i,j

).We can check that

() xi,j X and

()

{ui vj : i, j < } = u

() Fxi,j f (ui vj).

[Why? Let () ui vj so () = i,j() for some () < so g

i,j() = f

i,j

()

Bi by

clause (b) above hence () { : if < then g f()}. But f

Bi :

ui is a sequence without repetitions, so

< () i,j ui gi,j = f

i,j Bi

= fi,j

()

= gi,j()

(the middle inequality holds by the choice of (ui, Bi) : i < above, i.e., by clause(d)1).

So ixi,j (()) = () so Fxi,j (()) = i,j() = f(

i,j()), the last equality holds by

clause (c). So we are done.

Case 2: Clause (d)2 holds.Follows from Case 3.

Case 3: (d)3 holds.Let f be any function from to extending f . It suffices to show that

f : < satisfies clause (d)1. So let u [], so by (d)3 we can find a sequence

a = a : u of subsets of such that a / J and

1 = 2 u & i a1 a2 f(i) = f(i).

Let Bi = {i} []


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32 SAHARON SHELAH

Let 1 = Min{ : > }, so clearly 1 . As , 2 we have2 < 1, hence 1 = cf(1) is and ( < 1)(|| < 1). By (d)5 we have1 > 0. So by [Sh:g, VIII], ppJbd (1) (1)

= > so Clause (d)4 holds.

2.5


(a) = cov(, +, +, +) so >

(b) 2

(c) < = cf()

(d) = cf(), S S is stationary and S I[]

(e) T and [ < & T ||tr < ].

Then there is F such that

() F = F : <

() F is a function from to

() for a club of S, there is an increasing continuous sequence : < with limit such that:

() for every function f from { : T} to we can find a sequenceui : i < of subsets of T with union T such that (i < )( 0 and A is stationary in } and stillobtain an element of P.


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2.8 Remark. 1) If is (weakly) inaccessible we have weaker results: letting = { : = cf() and < ||tr < }, there is P : < ,P []

0, it folows by the weak hypothesis (see [Sh 410]), whosenegation is not known to be consistent and which has large consistency strength.3) If there is

i: i < is as above then in part (1) of 2.7 we can strengthen

clauses (), () in the conclusion. By demanding that wj : j < is increasingcontinuous. Why? Choosing f : < we add

if f1(i1) = f2(i2) then i1 = i2 & f1 i1 = f2 t2.so there is a sequence gi : i < , gi : and gi1(f1(i1)) = f1(i0)when i0 < i1 < , < and a function i : such that i(f(i)) = i.

Now in the end of the proof we replace = cd(j,f(j), ()) (so fixing j) by = cd().

Proof. There is B as in clause (), that is B I[] and B S is a stationarysubset of , see Claim 1.24(2). So let B be such a set; clearly omitting from Ba nonstationary subset of is O.K. as if B B, B\B is non-stationary and thedemands in clause () for B are exemplified by the sequence Ci, A

i , S

i ) : i <

then the same sequence exemplifies it for B.We can find a = a : < witnessing B I[] see Definition 0.5(2), so

(possibly omitting from B a nonstationary set) we have: B a =sup(a) & = otp(a) and [ a a = a] and [ \B a &otp(a) < ].

Let = + and M = M : < be an increasing continuous sequence of

elementary submodels of (H(), ,


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36 SAHARON SHELAH

(i


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For each j < let wj = { : T or T & j() = j} and apply clause (e)

of the assumptions to the sequence : < where = cd(j(), f(j), ()),

cd a coding function on so (by (e)) we can find A P satisfying clauses (i) and

(ii) of assumption (e). So by (i) of (e) there are i() < g(A) and j() < suchthat A() w

j().

For any pair (i, j) such that i < g(A), j < and Ai wj we shall define x = xi,j.

By (ii) of (e) we have { : Ai} P.We now define x = (hx1 , h

x2 , A

x, ix) as follows: we let ix = j, Ax = Ai,Dom(hx1) =

{f(j) : Ax}, hx1(f(j)) = , Dom(h2) = A

x, h2() = (). Easily x = xi,j M0 (as P M0, |P| , + 1 M0) and S

x, so we are done.

2) Let P = {A : A is a bounded subset of of cardinality }. Clearly |P | {|| : < } {|| : < } = , so clause (d) holds.

Now if = cf() < and = then tr = . [Why? For completeness if

T is a tree with set of nodes and levels, then very -branch of T includes a


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38 SAHARON SHELAH

Let (0, ) be strong limit; this is possible as is a strong limit of un-countable cofinality. Let 1 = 1[] < be > 0 and such that < 2

cov(||, , ,1) < 2 ; (such 1 exists by [Sh 460]).

Let Reg \1 and we shall prove the result for and every , < , sowithout loss of generality = . By Fodors Lemma on this is enough.Let = cf(2

). By 1.24(1) for every stationary S { < : cf() = } fromI[] (and there are such, see 1.24(2)) there are C

S = CS : S, D such thatCS is a weakly good+ (, ,

+)-Md-parameter, D is a -complete filter on containing the clubs of and by 1.10 + 1.11 S D

+, (e.g., D = D), the pair(CS, D) has the -Md-property.

For the case is (weakly) inaccessible we add each C is a club of . This ispermissible as we can use clause () of 1.24(1). Fix such S1 S and C = C : S1 = CS1 .

Note that we actually have

(a) if S S1 is stationary, C C = sup(C) for S

then C =C : S

is a weakly good+ (, , +)-Md-parameter and C has the(D, )-Md-property

(b) if = {Ai : i < i() < } then for some i, j < i(), C is a good

(, , +)-Md-parameter and C has the (D, )-Md-property where S =:

{ S1 : Aj & = sup(nacc(C) Ai)} and for S we let

C =: nacc(C) Aj .[Why? Let Si,j = { S1 : Aj and = sup(nacc(C) Ai)}. ClearlySi,j : i, j < i() is a sequence of subsets of S1 with union S1. By Claim1.23 for some i, j < i() the sequence C Si,j has the (D, )-Md-property.By monotonicity, i.e., Claim 1.14(1), the sequence nacc(C) Ai : Si,jhas the (D, )-Md-property.]

Now

Case 1: is the successor of regulars.We apply 2.1(1) with (, ) here standing for (2, 1) there; there is suitable

partial square (in ZFC by [Sh 351, 4]). Note that we do not lose any < because of the lifting to .

Case 2: is the successor of a singular.The proof is similar only now we use 2.7(3) to get the appropriate partial square

using to derive the C used as input for it (recalling that each sequence A Pis closed under the union of two!)

Case 3: is (weakly) inaccessible.


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Really the proof in this case covers the other cases (but 1 in this case contributesto increasing 0). This time we use 2.6 + 2.10 which is proved below.Let S2 be a stationary subset of S

from I[], it exists, see 1.24(2). Let F :

< 1 be as in 2.6 with (, 1, , ) here standing for (,,,) there. Letg be a function from 1 onto H(). Possibly replacing S2 by its intersectionwith some club of , there is a good -Md-parameter C

2 = C2 : S2. Lete = e : S, e a club of of order type . Clearly

() for some < 1 for any club E of for some increasing continuous sequencei : i of member of E the set B(i : i ) = {i S1 :g(F(i)) = {(i,, otp( C

2

), otp( ei)) : < i and ei}}is a stationary subset of

[Why? For every club E we can choose i E(i ) increasing continuous so

for some the set B(i : i ) is a stationary subset of . As the number ofpossible is 1 < = cf(), there is as above.]For this let

(i) S3 = { < : cf() = and g(F()) has the right form}

(ii) S3 let C3 = { e : (i,,j, otp( e)) g(F())}

(iii) let F : be defined by F() = if: = 1 for some (1, 2, 3, 4) g(F()).

We apply 2.10.

2),3) Similarly using Claim 2.3 when using Claim 2.4. 2.9

We now state another version of the upgrading of middle diamonds.


(a) 1 < 2 are regular cardinals

(b) S1 S1 is stationary, C1 = C1 : S1 is a (1, , )-Md-parameter

which has the (D1 , , )-Md-property, D1 is a filter on 1 to which S1belongs

(c) S2 S2 is stationary andC2 = C2 : S2 is a (2, , )-Md-parameter

(d) F : 2 1 satisfies: for every club E of 2 there is an increasing con-tinuous function h from 1 into 2 with sup(Rang(h)) acc(E) such that{ S1 : h() S2 and C

2h() {h() : C

1 } and F(h()) = } belongs

to D1

(e) 221

< 2.


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40 SAHARON SHELAH

Then C2 has the (, )-Md-property.2) Assume that in part (1) we omit assumption (e) and replace the Md-property byMD-property where {0, 2}.

Then C2

has the (, )-MD-property.3) In part (1) if we omit assumption (e) and strengthen clause (d) by demanding{ S1 : h() S2 and C

2h(0) = {h() : C

1 } and F(h()) = } belongs to D1 ,

then we can still conclude that: for every nice (C2, , )-colouring F (see below)there is an F-middle diamond sequence.

Remark. 1) Is clause (d) of the assumption reasonable? Later in [Sh:F611, 4.5] weprove that it holds under reasonable conditions.2) This + 2.11 are formalized elsewhere.

Proof of 2.10. Similar to 2.1.First, assume = 2 (so we are in part (2)) and let be a relational vocabulary

of cardinality < and by clause (b) of the assumption there is a sequence M1 =M1 : S1, M

1 a -model with universe C

1 such that

1 if M is a -model with universe 1, then the set of S1 for which M1 =

M C1 is = mod D1 .

Now for every S2 if F() S1 and {F() : C2 } C

1f() is an unbounded

subset of f(), then let M2 be the unique -model with universe C

2 such thatF C2 is an isomorphism from M

2 onto M

1F() {F() : C

2 }. If S1 does

not satisfy the requirement above let M2 be any -model with universe C2 .

Now it is easy to check that M2 = M2 : S2 is as required.Second, assume = 1 (and we are in part (1)) and let F2 be a (C2, , ) colouring.

For each (C1, , )-colouring F1 we ask:

Question on F1: Do we have, for every club E of 2 an increasing continuousfunction h : 1 2 such that

(i) sup(Rang(h)) E

(ii) the following set is a stationary subset of 1{ S1 : h() S2 and C2h() {h() : C

1 } and F(h()) = and

(1 (C1 ))(2

(C2h()))[( C1 )(1() = 2(h())) F1(1) =

F2(2)}.

If for some F1 the answer is yes, we proceed as in the proof for = 1. Otherwise,for each such F1 there is a club E[F1] exemplifying the negative answer. Let


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E =

{E[F1] : F1 is a (C1, , )-colouring}. As there are 221

such functionsF1 clearly E is a club of 2 and using (d) of the assumption, there is h : 1 2as there. Now we can define F1 by h, F2 naturally and get a contradiction.

The case = 0 and part (3) is left to the reader. 2.10

Trying to apply 2.10, clause (d) of the assumption looks the most serious. By thefollowing if each C1 is a stationary subset of (so > 0), it helps.

2.11 Claim. In 2.10, if 21 < 2, and we replace (d) by (d) + (f) below, then

we can conclude that for some C3 = C3 : S has the -MD-property whereC3 D

(f) D = D : S2, D is a filter on C

2 containing the co-bounded subsets

of C2 (or just D P(C2 )\{A C

2 : sup(A) < }

(d) F : 2 1 satisfies: for every club E of for some increasing continuousfunction h from 1 to 2 with sup(Rang(h)) acc(E) the set { S1 :h() S2 and {h() : C

1} C

2 D

} belong to D1 .

Proof. Let C = {e : e = e : S1, e is a subset of C2 }.

So |C| 21 < and for each e E we define Ce = Ce : Se2 by

Se = { S2 : Ce D} where C

e = { C

2 : otp(C

2 ) eF()}. If for some

e C, Ce is as required. 2.11


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42 SAHARON SHELAH

Glossary :

0 Introduction

Theorem 0.1: Many cases of MD-diamond.

Theorem 0.2: With division to .

Conclusion 0.3: Connection to the principle from [Sh 667].

Definition 0.4: Having (partial) squares.

1 Middle Diamond: sufficient conditions

Definition 1.1: Definition of S being F -small and the normal filter.

Claim 1.2: The filter is normal and does not matter.

Definition 1.3: Recalling pcf definitions.

Remark 1.4:

Definition 1.5: The main notions are defined:C is -Md-parameter, (,,)-Md-parameter,F is a (C , , )-coloring,C is a (D, F)-middle diamond sequence,C is good and C has the (D, ,)-Md-property or the(D, ,)-Md-property, also (good+-Md-property and Md.

Definition 1.6:

Claim 1.8: Easy implications.

Major Claim 1.10: Sufficient conditions for C having the (D, )-Md-properties(D a filter on , = cf(2)).

Claim 1.11: E.g., = implies Sep(, ), one of the assumptions of 1.10.

Remark 1.12: On variants.

Definition 1.9: Sep(,,,1, ), Sep(, ).

Claim 1.13: Monotonicity properties of Sep.

Claim 1.14: On some implications.

Definition 1.15: -MD-parameter.

Claim 1.16: Existence of good MD-parameters.

Claim 1.17: Like 1.10 for MD-parameters (1.14 - 1.17 are not central).

Claim 1.20: Getting, e.g., (D, , )-MD-property, i.e., M : S, |M| = Cwhich guess (as in Theorem 0.1) for having C with the(D, )-property; also md-properties MD-properties are defined.

Definition 1.22: -MD-property and the ideal ID,,,,(C)


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Claim 1.23: Non saturation ideal above (needed in 1.20)

Claim 1.24: Quotations on the existence of good Md-parameters (by quoting).

Definition 1.25: A related ideal.

Definition 1.26: F is (S,,)-good (1.25, 1.26 not central).

Conclusion 1.27: Phrasing some conclusions.

Question 1.28: weakly inaccessible, 2 : < is eventually constant.


Claim 2.1: For suitable 2 > 21 , we construct (2, , )-Md-parameters C which

has the (D, , )-property from (1, , )-Md-parameters whichhas the (D, , )-parameters.

Claim 2.3: Getting Ci = Ci : Si sets Si pairwise disjoint for i < in 2.1

Conclusion 2.4: If is strong limit, = cf() > 22

is successor of singularsthen for every large enough < , some good (,,+)-Md-parameterhas the (D, 2

, )-Md-property for every < .

Claim 2.5: A generalization of Engelking Karlowic. [no pf]

Claim 2.7: Dealing with successor of singulars. [ pf]

Claim 2.9: We advance the picture in 2.4 dealing with weakly inaccessible(i.e. regular limit) not strong limit cardinals; at a minor price - is of uncountable cofinality.

Claim 2.10: This is a stepping-up lemma used in 2.9 above.

Claim 2.11: Sufficient conditions for the lifting used in 2.10.


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44 SAHARON SHELAH

REFERENCES.

[AShS 221] Uri Abraham, Saharon Shelah, and R. M. Solovay. Squares with di-amonds and Souslin trees with special squares. Fundamenta Mathe-

maticae, 127:133162, 1987.

[DvSh 65] Keith J. Devlin and Saharon Shelah. A weak version of which followsfrom 20 < 21 . Israel Journal of Mathematics, 29:239247, 1978.

[DjSh 562] Mirna Dzamonja and Saharon Shelah. On squares, outside guessingof clubs and I


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[Sh 410] Saharon Shelah. More on Cardinal Arithmetic. Archive for Mathe-matical Logic, 32:399428, 1993. math.LO/0406550.

[Sh:g] Saharon Shelah. Cardinal Arithmetic, volume 29 of Oxford LogicGuides. Oxford University Press, 1994.

[Sh 430] Saharon Shelah. Further cardinal arithmetic. Israel Journal of Math-ematics, 95:61114, 1996. math.LO/9610226.

[Sh 462] Saharon Shelah. -entangled linear orders and narrowness of productsof Boolean algebras. Fundamenta Mathematicae, 153:199275, 1997.math.LO/9609216.

[Sh 506] Saharon Shelah. The pcf-theorem revisited. In The Mathematics ofPaul Erdos, II, volume 14 of Algorithms and Combinatorics, pages

420459. Springer, 1997. Graham, Nesetril, eds.. math.LO/9502233.[Sh:f] Saharon Shelah. Proper and improper forcing. Perspectives in Math-

ematical Logic. Springer, 1998.

[Sh 589] Saharon Shelah. Applications of PCF theory. Journal of SymbolicLogic, 65:16241674, 2000.

[Sh 460] Saharon Shelah. The Generalized Continuum Hypothesis revisited.Israel Journal of Mathematics, 116:285321, 2000. math.LO/9809200.

[Sh 576] Saharon Shelah. Categoricity of an abstract elementary class in twosuccessive cardinals. Israel Journal of Mathematics, 126:29128, 2001.math.LO/9805146.

[Sh 755] Saharon Shelah. Weak Diamond. Mathematica Japonica, 55:531538,2002. math.LO/0107207.

[Sh 667] Saharon Shelah. Successor of singulars: combinatorics and not col-lapsing cardinals in (< )-support iterations. Israel Journal ofMathematics, 134:127155, 2003. math.LO/9808140.

[Sh 829] Saharon Shelah. More on the Revised GCH and the Black Box. Annalsof Pure and Applied Logic, 140:133160, 2006. math.LO/0406482.

saharon shelah- middle diamond

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