Abstract. The main result is that for λ strong limit singular failing the continuum hypothesis (i.e. 2

155 (1998)

A polarized partition relation and failure of GCH at singular strong limit

by

Saharon S h e l a h (Jerusalem and New Brunswick, N.J.)

Abstract. The main result is that for λ strong limit singular failing the continuum hypothesis (i.e. 2

> λ

), a polarized partition theorem holds.

1. Introduction. In the present paper we show a polarized partition theorem for strong limit singular cardinals λ failing the continuum hypoth- esis. Let us recall the following definition.

Definition 1.1. For ordinal numbers α

, α

, β

, β

and a cardinal θ, the polarized partition symbol

 α

β



→

 α

β



θ

means that if d is a function from α

× β

into θ then for some A ⊆ α

of order type α

and B ⊆ β

of order type β

, the function d¹A×B is constant.

We address the following problem of Erd˝os and Hajnal:

(∗) if µ is strong limit singular of uncountable cofinality with θ < cf(µ),

does 

µ

µ



→

 µ µ



θ

?

The particular case of this question for µ = ℵ

and θ = 2 was posed by Erd˝os, Hajnal and Rado (under the assumption of GCH) in [EHR, Prob- lem 11, p. 183]). Hajnal said that the assumption of GCH in [EHR] was not crucial, and he added that the intention was to ask the question “in some, preferably nice, Set Theory”.

1991 Mathematics Subject Classification: Primary 03E05, 04A20, 04A30.

Research partially supported by “Basic Research Foundation” administered by The Israel Academy of Sciences and Humanities. Publication 586.

[153]

Baumgartner and Hajnal have proved that if µ is weakly compact then the answer to (∗) is “yes” (see [BH]), also if µ is strong limit of cofinality ℵ

. But for a weakly compact µ we do not know if for every α < µ

:

 µ

µ



→

 α µ



θ

.

The first time I heard the problem (around 1990) I noted that (∗) holds when µ is a singular limit of measurable cardinals. This result is presented in Theorem 2.2. It seemed likely that we could combine this with suitable collapses, to get “small” such µ (like ℵ

) but there was no success in this direction.

In September 1994, Hajnal reasked me the question putting great stress on it. Here we answer the problem (∗) using methods of [Sh:g]. But instead of the assumption of GCH (postulated in [EHR]) we assume 2

> µ

. The proof seems quite flexible but we did not find out what else it is good for.

This is a good example of the major theme of [Sh:g]:

Thesis 1.2. Whereas CH and GCH are good (helpful, strategic) assump- tions having many consequences, and, say, ¬CH is not, the negation of GCH at singular cardinals (i.e. for µ strong limit singular 2

> µ

, or the really strong hypothesis: cf(µ) < µ ⇒ pp(µ) > µ

) is a good (helpful, strategic) assumption.

Foreman pointed out that the result presented in Theorem 1.2 below is preserved by µ

-closed forcing notions. Therefore, if

V |=

 λ

λ



→

 λ λ



θ

then

V

|=

 λ

λ



→

 λ λ



θ

.

Consequently, the result is consistent with 2

= λ

& λ is small. (Note that although our final model may satisfy the Singular Cardinals Hypothesis, the intermediate model still violates SCH at λ, hence needs large cardinals, see [J].) For λ not small we can use Theorem 2.2.

Before we move to the main theorem, let us recall an open problem important for our methods:

Problem 1.3. (1) Let κ = cf(µ) > ℵ

, µ > 2

and λ = cf(λ) ∈ (µ, pp

(µ)). Can we find θ < µ and a ∈ [µ ∩ Reg]

such that λ ∈ pcf(a), a = S

i<κ

a

, a

bounded in µ and σ ∈ a

⇒ V

α<σ

|α|

< σ? For this it is

enough to show :

(2) If µ = cf(µ) > 2

but W

α<µ

|α|

≥ µ then we can find a ∈ [µ ∩ Reg]

such that λ ∈ pcf(a). (In fact, it suffices to prove it for the case θ = ℵ

.)

As shown in [Sh:g] we have

Theorem 1.4. If µ is strong limit singular of cofinality κ > ℵ

and 2

> λ = cf(λ) > µ then for some strictly increasing sequence hλ

: i < κi of regulars with limit µ, Q

i<κ

λ

/J

has true cofinality λ. If κ = ℵ

, this still holds for λ = µ

.

[More fully, by [Sh:g, II, §5], we know pp(µ) =

2

and by [Sh:g, VIII, 1.6(2)], we know pp

(µ) = pp

κ

(µ). Note that for κ = ℵ

we should replace J

by a possibly larger ideal, using [Sh 430, 1.1, 6.5] but there is no need here.]

Remark 1.5. Note that the problem is a pp = cov problem (see more in [Sh 430, §1]); so if κ = ℵ

and λ < µ

the conclusion of 1.4 holds; we allow J

to be increased, even “there are < µ

fixed points < λ

” suffices.

2. Main result

Theorem 2.1. Suppose µ is strong limit singular satisfying 2

> µ

. Then:

(1)

 µ

µ



→

 µ + 1 µ



θ

for any θ < cf(µ).

(2) If d is a function from µ

× µ to θ and θ < µ then for some sets A ⊆ µ

and B ⊆ µ we have otp(A) = µ + 1, otp(B) = µ and the restriction d¹A × B does not depend on the first coordinate.

P r o o f. (1) This follows from part (2) (since if d(α, β) = d

(β) for α ∈ A, β ∈ B, where d

: B → θ, and |B| = µ, θ < cf(µ) then there is B

⊆ B with |B

| = µ such that d

¹B is constant and hence d¹A × B

is constant as required).

(2) Let d : µ

× µ → θ. Let κ = cf(µ) and µ = hµ

: i < κi be a continuous strictly increasing sequence such that µ = P

i<κ

µ

, µ

> κ + θ.

We can find a sequence C = hC

: α < µ

i such that:

(A) C

⊆ α is closed, otp(C

) < µ, (B) β ∈ nacc(C

) ⇒ C

= C

∩ β,

(C) if C

has no last element then α = sup(C

) (so α is a limit ordinal) and any member of nacc(C

) is a successor ordinal,

(D) if σ = cf(σ) < µ then the set

S

:= {δ < µ

: cf(δ) = σ & δ = sup(C

) & otp(C

) = σ}

is stationary

(possible by [Sh 420, §1]); we could have added

(E) for every σ ∈ Reg ∩µ

and a club E of µ

, for stationary many δ ∈ S

, E separates any two successive members of C

.

Let c be a symmetric two-place function from µ

to κ such that for each i < κ and β < µ

the set

a

:= {α < β : c(α, β) ≤ i}

has cardinality ≤ µ

and α < β < γ ⇒ c(α, γ) ≤ max{c(α, β), c(β, γ)} and α ∈ C

& µ

≥ |C

| ⇒ c(α, β) ≤ i

(as in [Sh 108], easily constructed by induction on β).

Let λ = hλ

: i < κi be a strictly increasing sequence of regular cardinals with limit µ such that Q

i<κ

λ

/J

has true cofinality µ

(exists by 1.4 with λ = µ

≤ 2

). As we can replace λ by any subsequence of length κ, without loss of generality (∀i < κ)(λ

> 2

). Lastly, let χ = i

(µ)

and <

be a well ordering of H(χ)(:= {x : the transitive closure of x is of cardinality < χ}).

Now we choose by induction on α < µ

sequences M

= hM

: i < κi such that:

(i) M

≺ (H(χ), ∈, <

),

(ii) kM

k = 2

and

(M

) ⊆ M

and 2

+ 1 ⊆ M

, (iii) d, c, C, λ, µ, α ∈ M

, hM

: β < α, j < κi ∈ M

, S

β∈a^α_i

M

⊆ M

and hM

: j < ii ∈ M

, S

j<i

M

⊆ M

, (iv) hM

: β ∈ a

i belongs to M

.

There is no problem to carry out the construction. Note that actually clause (iv) follows from (i)–(iii), as a

is defined from c, α, i. Our demands imply that

[β ∈ a

⇒ M

≺ M

] and [j < i ⇒ M

≺ M

] and a

⊆ M

, hence α ⊆ S

i<κ

M

. For α < µ

let f

∈ Q

i<κ

λ

be defined by f

(i) = sup(λ

∩ M

). Note that f

(i) < λ

as λ

= cf(λ

) > 2

= kM

k. Also, if β < α then for every i ∈ [c(β, α), κ) we have β ∈ M

and hence M

∈ M

. Therefore, as also λ ∈ M

, we have f

∈ M

and f

(i) ∈ M

∩ λ

. Consequently,

(∀i ∈ [c(β, α), κ))(f

(i) < f

(i)) and thus f

<

f

. Since {f

: α < µ

} ⊆ Q

i<κ

λ

has cardinality µ

and Q

i<κ

λ

/J

is µ

-directed, there is f

∈ Q

i<κ

λ

such that (∗)

(∀α < µ

)(f

<

κ

f

).

Let, for α < µ

, g

∈

θ be defined by g

(i) = d(α, f

(i)). Since |

θ| < µ <

µ

= cf(µ

), there is a function g

∈

θ such that

(∗)

the set A

= {α < µ

: g

= g

} is unbounded in µ

. Now choose, by induction on ζ < µ

, models N

such that:

(a) N

≺ (H(χ), ∈, <

),

(b) the sequence hN

: ζ < µ

i is increasing continuous, (c) kN

k = µ and

(N

) ⊆ N

if ζ is not a limit ordinal, (d) hN

: ξ ≤ ζi ∈ N

,

(e) µ + 1 ⊆ N

, S

α<ζ, i<κ

M

⊆ N

and hM

: α < µ

, i < κi, hf

: α < µ

i, g

, A

and d belong to the first model N

.

Let E := {ζ < µ

: N

∩ µ

= ζ}. Clearly, E is a club of µ

, and thus we can find an increasing sequence hδ

: i < κi such that

(∗)

δ

∈ S

i

∩ acc(E) (⊆ µ

) (see clause (D) at the beginning of the proof).

For each i < κ choose a successor ordinal α

∈ nacc(C

) \ S

{δ

+ 1 : j < i}.

Take any α

∈ A

\ S

i<κ

δ

.

We choose by induction on i < κ an ordinal j

and sets A

, B

such that:

(α) j

< κ and µ

> λ

(so j

> i) and j

strictly increasing in i, (β) f

¹[j

, κ) < f

i+1

¹[j

, κ) < f

¹[j

, κ) < f

¹[j

, κ),

(γ) for each i

< i

we have c(δ

, α

) < j

, c(α

, α

) < j

, c(α

, α

)

< j

and c(δ

, α

) < j

, (δ) A

⊆ A

∩ (α

, δ

), (ε) otp(A

) = µ

, (ζ) A

∈ M

, (η) B

⊆ λ

, (θ) otp(B

) = λ

,

(ι) B

∈ M

i,ji

for ε < i, (κ) for every α ∈ S

ε≤i

A

∪ {α

} and ζ ≤ i and β ∈ B

∪ {f

(j

)} we have d(α, β) = g

(j

).

If we succeed then A = S

ε<κ

A

∪{α

} and B = S

ζ<κ

B

are as required.

During the induction at stage i concerning (ι), if ε + 1 = i then for some j < κ, B

∩M

i,j

has cardinality λ

, hence we can replace B

by a subset of the same cardinality which belongs to the model M

i,j

if j is large enough such that µ

> λ

; if ε + 1 < i then by the demand for ε + 1, we have W

j<κ

B

∈ M

i,j

. So assume that the sequence h(j

, A

, B

) : ε < ii has already been defined.

We can find j

(0) < κ satisfying requirements (α), (β), (γ) and (ι) and such that V

ε<i

λ

< µ

. Then for each ε < i we have δ

∈ a

i(0)

and

hence M

≺ M

i,ji(0)

(for ε < i). But A

∈ M

(by clause (ζ)) and B

∈ M

i,j_i(0)

(for ε < i), so {A

, B

: ε < i} ⊆ M

i,j_i(0)

. Since

κ>

(M

i,j_i(0)

) ⊆ M

i,j_i(0)

(see (ii)), the sequence h(A

, B

) : ε < ii belongs to M

i,ji(0)

. We know that for γ

< γ

in nacc(C

) we have c(γ

, γ

) ≤ i (remember clause (B) and the choice of c). As j

(0) > i and so µ

≥ µ

, the sequence

M

:= hM

: α ∈ nacc(C

)i

is ≺-increasing and M

¹α ∈ M

for α ∈ nacc(C

) and M

i,j_i(0)

ap- pears in it. Also, as δ

∈ acc(E), there is an increasing sequence hγ

: ξ < µ

i of members of nacc(C

) such that γ

= α

and (γ

, γ

) ∩ E 6= ∅, say β

∈ (γ

, γ

) ∩ E. Each element of nacc(C

) is a successor ordinal, so every γ

is a successor ordinal. Each model M

is closed under se- quences of length ≤ µ

, and hence hγ

: ζ < ξi ∈ M

(by choos- ing the right C and δ

’s we could have managed to have α

= min(C

), {γ

: ξ < µ

} = nacc(C

), without using this amount of closure).

For each ξ < µ

, we know that

(H(χ), ∈, <

) |= “(∃x ∈ A

)[x > γ

& (∀ε < i)(∀y ∈ B

)(d(x, y) = g

(j

))]”

because x = α

satisfies it. As all the parameters, i.e. A

, γ

, d, g

and hB

: ε < ii, belong to N

(remember clauses (e) and (c); note that B

∈ M

i,ji(0)

, α

< β

), there is an ordinal β

∈ (γ

, β

) ⊆ (γ

, γ

) satisfying the demands on x. Now, necessarily for some j

(1, ξ) ∈ (j

(0), κ) we have β

∈ M

. Hence for some j

< κ the set

A

:= {β

: ξ < µ

& j

(1, ξ) = j

}

has cardinality µ

. Clearly A

⊆ A

(as each β

∈ A

). Now, the sequence hM

: ξ < µ

i

hM

i is ≺-increasing, and hence A

⊆ M

. Since µ

> µ

= |A

| we have A

∈ M

. Note that at the moment we know that the set A

satisfies the demands (δ)–(ζ). By the choice of j

(0), as j

> j

(0), clearly M

≺ M

, and hence A

∈ M

. Similarly, hA

: ε ≤ ii ∈ M

, α

∈ M

and

sup(M

∩ λ

) = f

(j

) < f

(j

).

Consequently, S

ε≤i

A

∪ {α

} ⊆ M

(by the induction hypothesis or the above) and it belongs to M

. Since S

ε≤i

A

∪ {α

} ⊆ A

, clearly (H(χ), ∈, <

) |= “



∀x ∈ [

ε≤i

A

∪ {α

}



(d(x, f

(j

)) = g

(j

))”.

Note that [

ε≤i

A

∪ {α

}, g

(j

), d, λ

∈ M

and f

(j

) ∈ λ

\sup(M

∩ λ

).

Hence the set B

:=

n

y < λ

:



∀x ∈ [

ε≤i

A

∪ {α

}



(d(x, y) = g

(j

)) o

has to be unbounded in λ

. It is easy to check that j

, A

, B

satisfy clauses (α)–(κ).

Thus we have carried out the induction step, finishing the proof of the theorem.

Theorem 2.2. Suppose µ is a singular limit of measurable cardinals.

Then (1)

 µ

µ



→

 µ µ



θ

if θ = 2 or at least θ < cf(µ).

(2) Moreover , if α

< µ

and θ < cf(µ) then

 µ

µ



→

 α

µ



θ

.

(3) If θ < µ, α

< µ

and d is a function from µ

× µ to θ then for some A ⊆ µ

, otp(A) = α

, and B = S

i<cf(µ)

B

⊆ µ, d¹A × B

is constant for each i < cf(µ).

P r o o f. Clearly (3)⇒(2)⇒(1), so we shall prove part (3).

Let d : µ

× µ → θ. Let κ := cf(µ). Choose sequences hλ

: i < κi and hµ

: i < κi such that hµ

: i < κi is increasing continuous, µ = P

i<κ

µ

, µ

> κ + θ, each λ

is measurable and µ

< λ

< µ

(for i < κ). Let D

be a λ

-complete uniform ultrafilter on λ

. For α < µ

define g

∈

θ by g

(i) = γ iff {β < λ

: d(α, β) = γ} ∈ D (as θ < λ

it exists). The number of such functions is θ

< µ (as µ is necessarily strong limit), so for some g

∈

θ the set A := {α < µ

: g

= g

} is unbounded in µ

. For each i < κ we define an equivalence relation e

on µ

:

αe

β iff (∀γ < λ

)[d(α, γ) = d(β, γ)].

So the number of e

-equivalence classes is ≤

θ < µ. Hence we can find an increasing continuous sequence hα

: ζ < µ

i of ordinals < µ

such that:

(∗) for each i < κ and e

-equivalence class X, either X ∩ A ⊆ α

, or for every ζ < µ

, (α

, α

) ∩ X ∩ A has cardinality µ.

Let α

= S

i<κ

a

, |a

| = µ

, ha

: i < κi pairwise disjoint. Now, by induction on i < κ, we choose A

, B

such that:

(a) A

⊆ S

{(α

, α

) : ζ ∈ a

} ∩ A and each A

∩ (α

, α

) is a singleton,

(b) B

∈ D

,

(c) if α ∈ A

, β ∈ B

, j ≤ i then d(α, β) = g

(j).

Now, at stage i, h(A

, B

) : ε < ii are already chosen. Let us choose A

. For

each ζ ∈ a

choose β

∈ (α

, α

) ∩ A such that if i > 0 then for some

β

∈ A

, β

e

β

, and let A

= {β

: ζ ∈ a

}. Now clause (a) is immediate, and the relevant part of clause (c), i.e. j < i, is O.K. Next, as S

j≤i

A

⊆ A, the set

B

:= \

j≤i

\

β∈Aj

{γ < λ

: d(β, γ) = g

(i)}

is the intersection of ≤ µ

< λ

sets from D

and hence B

∈ D

. Clearly clause (b) and the remaining part of clause (c) (i.e. j = i) holds. So we can carry out the induction and hence finish the proof.

References

[EHR] P. E r d ˝o s, A. H a j n a l and R. R a d o, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93–196.

[BH] J. B a u m g a r t n e r and A. H a j n a l, Polarized partition relations, preprint, 1995.

[J] T. J e c h, Set Theory, Academic Press, New York, 1978.

[Sh:g] S. S h e l a h, Cardinal Arithmetic, Oxford Logic Guides 29, Oxford Univ. Press, 1994.

[Sh 430] —, Further cardinal arithmetic, Israel J. Math. 95 (1996), 61–114.

[Sh 420] —, Advances in cardinal arithmetic, in: Finite and Infinite Combinatorics in Sets and Logic, N. W. Sauer et al. (eds.), Kluwer Acad. Publ., 1993, 355–383.

[Sh 108] —, On successors of singular cardinals, in: Logic Colloquium ’78 (Mons, 1978), Stud. Logic Found. Math. 97, North-Holland, Amsterdam, 1979, 357–380.

Institute of Mathematics Department of Mathematics

The Hebrew University Rutgers University

91 904 Jerusalem, Israel New Brunswick, New Jersey 08903

E-mail: shelah@math.huji.ac.il U.S.A.

Received 4 November 1995;

in revised form 18 October 1996