2 8 8 r e v i s i o n : 1 9 9 3 0 8 2 7 m o d i f i e d : 1 9 9 3 - 0 8 2 7 COLLOQUIA M AT H E M AT I C A SOCIETATIS J ´ ANOS BOLYA I 60. SETS, GRAPHS AND NUMBERS, BUDAPEST (HUNGAR Y), 1991 Strong Partition Realations Below the Po wer Set: Consist ency Was Sier pinski Righ t? II. S. SHELAH ∗ We continue here [Sh276] (see the introduction there) but we do not relay on it. The motivation was a conjecture of Galvin stating that 2 ω ≥ ω 2 + ω 2 → [ω 1 ] n h(n) is consistent for a sui table h : ω → ω. In sect ion 5 we disprove this and giv e similar neg ati ve result s. In section 3 we prove the consistency of the conjecture replacing ω 2 by 2 ω , which is quite large, starting with an Erd˝ os cardinal. In section 1 we present iter ation lemmas which needs when we replace ω by a larger λ and in section 4 we generalize a theorem of Halpern and Lauchli replacing ω by a larger λ. 0. Preli minar ies Let < ∗ χ be a well ordering ofH (χ), where H (χ) = {x : the transitive closure ofx has ca rd inality< χ}, agre ein g with the usual well -ordering of the ∗ Public ation no 288, summer 86. The author would like to thank the United States – Israel Binationa l Science F ounda tion for partia lly support ing this researc h.
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8/3/2019 Saharon Shelah- Strong Partition Realations Below the Power Set: Consistency Was Sierpinski Right? II
We continue here [Sh276] (see the introduction there) but we do notrelay on it. The motivation was a conjecture of Galvin stating that 2 ω ≥ ω2+ ω2 → [ω1 ]nh (n ) is consistent for a suitable h : ω → ω. In section 5we disprove this and give similar negative results. In section 3 we provethe consistency of the conjecture replacing ω2 by 2ω , which is quite large,starting with an Erd˝ os cardinal. In section 1 we present iteration lemmaswhich needs when we replace ω by a larger λ and in section 4 we generalizea theorem of Halpern and Lauchli replacing ω by a larger λ.
0. Preliminaries
Let < ∗χ be a well ordering of H(χ ), where H(χ ) = {x : the transitive closureof x has cardinality < χ }, agreeing with the usual well-ordering of the
∗
Publication no 288, summer 86. The author would like to thank the United States– Israel Binational Science Foundation for partially supporting this research.
8/3/2019 Saharon Shelah- Strong Partition Realations Below the Power Set: Consistency Was Sierpinski Right? II
2ordinals. P (and Q, R) will denote forcing notions, i.e. partial orders witha minimal element ∅= ∅P .
A forcing notion P is λ-closed if every increasing sequence of membersof P , of length less than λ, has an upper bound.
If P ∈H(χ ), then for a sequence p = pi : i < γ of members of P letα˜
= α˜ p
def = sup{ j˜
: {β j : j < j˜} has an upper bound in P } and dene the
canonical upper bound of p, &¯ p as follows:
(a) the least upper bound of { pi : i < α˜
} in P if there exists such anelement,
(b) the < ∗χ -rst upper bound of p if (a) can’t be applied but there is such,
(c) p0 if (a) and (b) fail, γ > 0,
(d) ∅P if γ = 0.
Let p0& p1 be the canonical upper bound of p : < 2 .
Take [ a]κ = {b⊆ a : |b| = κ} and [a]<κ = θ<κ [a]θ .For sets of ordinals, A and B , dene H OP
A,B as the maximal orderpreserving bijection between initial segments of A and B , i.e, it is thefunction with domain {α ∈A : otp( α ∩ A) < otp( B )}, and H OP
A,B (α ) = β if and only if α ∈A, β ∈B and otp( α ∩ A) = otp( β ∩ B ).
Denition 0.1 λ →+ (α)<ωµ holds provided whenever F is a function from
[λ]<ω to µ, C ⊆ λ is a club then there is A ⊆ C of order type α such that[w1 , w2 ∈[A]<ω , |w1 | = |w2 | ⇒ F (w1) = F (w2 )].
Denition 0.2 λ → [α ]nκ,θ if for every function F from [λ]
n
to κ there isA ⊆ λ of order type α such that {F (w) : w ∈[A]n } has power ≤ θ.
Denition 0.3 A forcing notion P satises the Knaster condition (has property K) if for any { pi : i < ω 1} ⊂ P there is an uncountable A ⊂ ω1
such that the conditions pi and pj are compatible whenever i, j ∈A.
1. Introduction
Concerning 1.1–1.3 see Shelah [Sh80], Shelah and Stanley [ShSt154, 154a].
8/3/2019 Saharon Shelah- Strong Partition Realations Below the Power Set: Consistency Was Sierpinski Right? II
8has cardinality < σ (2) .2) Pr ht (µ,n,σ (1), σ(2)) is dened similarly with “similar” instead of
“strongly similar”.
3) Pr x µ,< κ, σ1 : < κ σ2 : < κ , Pr f x (µ,n,σ (1), σ(2)) , Pr f
x (µ, <
ℵ0 , σ1, σ
2) are dened in the same way.
There are many obvious implications.
Fact 2.5. 1) For every T ∈Per( µ > 2) there is a T 1 ⊆ T , T 1 ∈Per u (µ> 2).2) In dening Pr f
x (µ,n,σ ) we can demand T ⊆ T 0 for any T 0 ∈Per f (µ> 2),similarly for Pr f
x (µ, < κ, σ ).3) The obvious monotonicity holds.
Claim 2.6. 1) Suppose µ is regular, σ ≥ ℵ0 and Pr f eht (µ, n, < σ ). Then
Pr f aht (µ, n, < σ ) holds.
2) If µ is weakly compact and Prf aht (µ, n, < σ ), σ < µ , then Pr
f ht (µ, n, < σ )holds.
3) If µ is Ramsey and Pr f aht (µ, < ℵ0 , < σ ), σ < µ , then Pr f
ht (µ, < ℵ0 , < σ ).4) If µ = ω, in the “nice” version, the orders < ∗α : α < µ disappear.
Proof. : Check it.The following theorem is a quite strong positive result for µ = ω.
Halpern Lauchli proved 2.7(1), Laver proved 2.7(2) (and hence (3)), Pincuspointed out that Halpern Lauchli’s proof can be modied to get 2.7(2), andthen Pr f
eht (ω,n,< σ ) and (by it) Pr f ht (ω,n,< σ ) are easy.
Theorem 2.7. 1) If d ∈Colnσ (ω> 2), σ < ℵ0 , then there are T 0 , . . . , T n − 1 ∈Per f (ω> 2) and k0 < k 1 < . . . < k < . . . and s < σ such that for every
< ω : if µ0 ∈ T 0 , µ1 ∈ T 1 , . . . , ν n − 1 ∈ T n − 1 ,m<n
lg(ν m ) = k , then
d(ν 0 , . . . , ν n − 1) = s.2) We can demand in (1) that
SP( T ) = {k0 , k1 , . . . }
3) Pr f htn (ω,n,σ ) for σ < ℵ0 .
4) Prf
htn ω, < ℵ0 , σ1n : n < ω , σ
2n : n < ω if σ
1n < ℵ0 and σ
2n : n < ωdiverge to innity.
8/3/2019 Saharon Shelah- Strong Partition Realations Below the Power Set: Consistency Was Sierpinski Right? II
9Denition 2.8. Let d be a function with domain ⊇ [A]n , A be a set of ordinals, F be a one-to-one function from A to α (∗) 2, < ∗α be a well ordering of α 2 for α ≤ α(∗) such that F (α) < ∗α F (β ) ⇐⇒ α < β , and σ be a cardinal.1) We say d is (F, σ )-canonical on A if for any α 1 < · · · < α n ∈A,
d(β 1 , . . . , β n ) : F (β 1), . . . , F (β n ) similar to
F (α 1), . . . , F (α n ) ≤ σ.
2) We dene “almost (F, σ )-canonical” similarly using strongly similar instead of “similar”.
3. Consistency of a strong partition below the continuum
This section is dedicated to the proof of
Theorem 3.1. Suppose λ is the rst Erd˝ os cardinal, i.e. the rst such thatλ → (ω1)<ω
2 . Then, if A is a Cohen subset of λ , in V [A] for some ℵ1–c.c.forcing notion P of cardinality λ, P “MAℵ1 (Knaster ) + 2 ℵ0 = λ” and:1.) P “ λ → [ℵ1]n
h (n ) ” for suitable h : ω → ω (explicitly dened below).
2.) In V P for any colorings dn of λ, where dn is n-place, and for any diver-gent σn : n < ω (see below), there is a W ⊆ λ, |W | = ℵ1 and a functionF : W → ω 2 such that: dn is (F, σ n ) − canonical on W for each n.(See denition 2.8 above.)
Remark 3.2. h(n) is n! times the number of u ∈ [ω 2]n satisfying (if η1 , η2 , η3 , η4 ∈ u are distinct then sp(η1 , η2), sp(η3 , η4) are distinct) up tostrong similarity for any nice < ∗α : α < ω .
2) A sequence σn : n < ω is divergent if ∀m ∃k ∀n ≥ k σn ≥ m.
Notation 3.3. For a sequence a = α i , e∗i : i < α , we call b⊆ α closed if (i) i ∈b⇒ a i ⊆ b
(ii) if i < α, e ∗i = 1 and sup( b ∩ i) = i then i ∈b.
Denition 3.4. LetK
be the family of ¯Q = P i , Q j , a j , e
∗
j : j < α, i ≤ αsuch that
8/3/2019 Saharon Shelah- Strong Partition Realations Below the Power Set: Consistency Was Sierpinski Right? II
11(α ) b = bw : w ∈W is an indexed set of Q-closed subsets of lg(Q),(β ) W |= “ w1 ≤ w2” ⇒ bw 1 ⊆ bw 2 ,(γ ) ζ ∈bw 1 ∩bw 2 , w1 ≤ w, w2 ≤ w then (∃u ∈W )ζ ∈bu ∧u ≤ w1∧u ≤ w2 .
We assume b codes (W, ≤ ).2) For b∈IN W (Q), let
Q[b]def = { pw : w ∈W : pw ∈P cn
bw , [W |= w1 ≤ w2 ⇒ pw 2 bw 1 = pw 1 ]}
with ordering Q[b] |= p1 ≤ p2 iff w∈W p1w ≤ p2
w .3) Let K 1 be the family of Q ∈K such that for every β ≤ lg(Q) and (Q β )-closed b, P β and P β /P cn
b satisfy the Knaster condition.
Fact 3.8. Suppose Q ∈K 1 , (W, ≤ ) is a nite partial order, b ∈ IN W (Q)and p∈Q[b].1) If w ∈W , pw ≤ q∈P cn
bwthen there is r ∈Q[b], q ≤ r w , p ≤ r , in fact
r u (γ ) =
pu (γ ) if γ ∈Dom pu \ Dom q pu (γ ) & q(γ ) if γ ∈bu ∩ Dom q and for some v ∈W ,
v ≤ u, v ≤ w and γ ∈bv
pu (γ ) if γ ∈bu ∩ dom q but the previous case fails
2) Suppose (W 1 , ≤ ) is a submodel of (W 2 , ≤ ), both nite partial orders,bl∈IN W l (Q), b1
w = b2w for w ∈W 1 .
(α ) If q∈Q[b2 ] then qw : w ∈W 1 ∈Q[b1].(β ) If p∈Q[b1] then there is q ∈Q[b2], q W 1 = p, in fact qw (γ ) is pu (γ ) if u ∈W 1 , γ ∈bu , u ≤ w, provided that(∗∗) if w1 , w2 ∈W 1 , w ∈W 2 , w1 ≤ w, w2 ≤ w and ζ ∈bw 1 ∩ bw 2 then for some v ∈W 1 , ζ ∈bv , v ≤ w1 , v ≤ w2 .(this guarantees that if there are several u’s as above we shall get the same value).3) If Q ∈K 1 then Q[b] satises the Knaster condition. If ∅is the minimal element of W (i.e. u ∈W ⇒ W |= ∅ ≤u) then Q[b]/P cn
b∅ also satises the Knaster condition and so ◦Q[b], when we identify p ∈P cn
b with p : w ∈W .
Proof. 1) It is easy to check that each r u (γ ) is in P cnbu
. So, in order to prover ∈Q[b], we assume W |= u1 ≤ u2 and has to prove that r u 2 bu 1 = r u 1 . Letζ ∈bu
1.
First case: ζ ∈Dom( pu 1 ) ∪Dom q.
8/3/2019 Saharon Shelah- Strong Partition Realations Below the Power Set: Consistency Was Sierpinski Right? II
12So ζ ∈ Dom( r u 1 ) (by the denition of r u 1 ) and ζ ∈ Dom pu 2 (as
p ∈ Q[b]) hence ζ ∈ (Dom pu 2 ) ∪ (Dom q) hence ζ ∈ Dom( r u 2 ) by thechoice of r u 2 , so we have nished.
Second case: ζ ∈Dom pu 1 \ Dom q.As ¯ p∈Q[b] we have pu 1 (ζ ) = pu 2 (ζ ), and by their denition, r u 1 (ζ ) =
pu 1 (ζ ), r u 2 (ζ ) = pu 2 (ζ ).Third case: ζ ∈Dom q and (∃v ∈W ) (ζ ∈bv ∧v ≤ u1 ∧v ≤ w). By the
denition of r u 1 (ζ ), we have r u 1 (ζ ) = pu 1 (ζ )&q(ζ ), also the same v witnessesr u 2 (ζ ) = pu 2 (ζ )&q(ζ ), (as ζ ∈bv∧v ≤ u1∧v ≤ w ⇒ ζ ∈bv∧v ≤ u2∧v ≤ w)and of course pu 1 (ζ ) = pu 2 (ζ ) (as p∈Q[b]).
Fourth case: ζ ∈Dom q and ¬(∃v ∈W ) (ζ ∈bv ∧v ≤ u1 ∧v ≤ w).By the denition of r u 1 (ζ ) we have r u 1 (ζ ) = pu 1 (ζ ). It is enough to prove
that r u 2 (ζ ) = pu 2 (ζ ) as we know that pu 1 (ζ ) = pu 2 (ζ ) (because p ∈ Q[b],u1 ≤ u2). If not, then for some v0 ∈W , ζ ∈bv0 ∧ v0 ≤ u2 ∧ v0 ≤ w. Butb∈INW (Q), hence (see Def. 3.7(1) condition ( γ ) applied with ζ, w1 , w2 , wthere standing for ζ, v0 , u1 , u2 here) we know that for some v ∈W , ζ ∈v ∧ v ≤ v0 ∧ v ≤ u1 . As (W, ≤ ) is a partial order, v ≤ v0 and v0 ≤ w, wecan conclude v ≤ w. So v contradicts our being in the fourth case. So wehave nished the fourth case.
Hence we have nished proving r ∈Q[b]. We also have to prove q ≤ r w ,but for ζ ∈ Dom q we have ζ ∈ bw (as q ∈ P cn
w is on assumption) andr w (ζ ) = q(ζ ) because r w (ζ ) is dened by the second case of the denitionas (∃v ∈W ) (ζ ∈bw ∧v ≤ w ∧v ≤ w), i.e. v = w.
Lastly we have to prove that p ≤ r (in Q[b]). So let u ∈W , ζ ∈Dom pu
and we have to prove r u ζ P ζ “ pu (ζ ) ≤ P ζ r u (ζ )”. As r u (ζ ) is pu (ζ ) or
pu (ζ )&q(ζ ) this is obvoius.
2) Immediate.3) We prove this by induction on |W |.For |W | = 0 this is totally trivial.For |W | = 1 , 2 this is assumed.For |W | > 2 x ¯ pi
∈Q[b] for i < ω 1 . Choose a maximal element v ∈W andlet c = {bw : W |= w < v }. Clearly c is closed for Q.
We know that P cnc , P cn
bv/P cn
c are Knaster by the induction hypothesis.We also know that pi
v c∈P cnc for i < ω 1 , hence for some r ∈P cn
c ,
r ” A˜
def = i < ω 1 : piv c∈G
˜ P cnc
is uncountable”
8/3/2019 Saharon Shelah- Strong Partition Realations Below the Power Set: Consistency Was Sierpinski Right? II
For i ∈A2 let, r i be dened using 3.8(1) (with pi , piv &qi ). Let W 1 =
W \ { v}, b = bw : w ∈W 1 .By the induction hypothesis applied to W 1 , b , r i W 1 , for i ∈ A2
there is an uncountable A3⊆ A2 and for i < j in A3 , there is r i,j
∈Q[b ],r i W 1 ≤ r i,j , and r j W 1 ≤ r i,j . Now dene r i,j
c ∈P cnc as follows: its domain
is dom ri,jw : W |= w < v , r
i,jc (dom r
i,jw ) = r
i,jw whenever W |= w < v .Why is this a denition? As if W |= w1 ≤ v ∧w2 ≤ v, ζ ∈bw 1 ∧ ζ ∈bw 2
then for some u ∈W , u ≤ w1 ∧u ≤ w2 and ζ ∈u. It is easy to check thatr i,j
c ∈P cnc . Now r i,j
c P cnc “ pi
bv, pj
bvare compatible in P cn
bv/P cn
c ”.
So there is r ∈P cnbv
such that r i,jc ≤ r , pi
bv≤ r , pj
bv≤ r . As in part (1) of
3.8 we can combine r and r i,j to a common upper bound of pi , pj in Q[b].
Claim 3.9. If e = 0 , 1 and δ is a limit ordinal, and P i , Q˜ i
, α i , e∗i (i < δ ) are such that for each α < δ , Qα = P i , Q
˜ j, α j , e∗j : i ≤ α, j < α belongs to
K , then for a unique P δ , Q = P i , Q˜ j
, α j , e∗j
: i ≤ δ, j < δ belongs to K .
Proof. We dene P δ by (d) of Denition 3.4. The least easy problem is toverify the Knaster conditions (for Q ∈K 1). The proof is like the preservationof the c.c.c. under iteration for limit stages.
Convention 3.9A. By 3.9 we shall not distinguish strictly between P i , Q˜ j
,α j , e∗j : i ≤ δ, j < δ and P i , Q
˜ i, α i , e∗i : i < δ .
Claim 3.10. If Q ∈K , α = lg( Q), a ⊂ α is closed for Q, |a | ≤ ℵ1 , Q˜ 1
is a P cn
a -name of a forcing notion satisfying (in V P α ) the Knaster condition,
its underlying set is a subset of [ω1]< ℵ0
then there is a unique ¯Q
1
∈
K ,lg(Q1) = α + 1 , Q1
α = Q˜
, Q α = Q.
8/3/2019 Saharon Shelah- Strong Partition Realations Below the Power Set: Consistency Was Sierpinski Right? II
A Stage: We force by K 1<λ = Q ∈K 1 : lg(Q) < λ, Q ∈H (λ) ordered by
being an initial segment (which is equivalent to forcing a Cohen subset of λ). The generic object is essentially Q∗∈K 1λ , lg(Q∗) = λ, and then we force
by P λ = lim Q∗. Clearly K <λ is a λ-complete forcing notion of cardinality
λ, and P λ satises the c.c.c. Clearly it suffices to prove part (2) of 3.1.
Suppose d˜ n is a name of a function from [ λ]n to k
˜ n for n < ω , σ˜ n < ω ,
σn : n < ω diverges (i.e. ∀m ∃k ∀n ≥ k σn ≥ m) and for some Q0∈K 1
<λ .
Q0K 1
<λ“there is p ∈P
˜ λ [ p P λ d˜ n : n < ω is a
counterexample to (2) of 3.1”].
In V we can dene Qζ : ζ < λ , Qζ∈K 1<λ , ζ < ξ ⇒ Qζ = Qξ lg(Qζ ),
in¯Q
ζ +1
, e∗
lg( Q ζ ) = 1,¯Q
ζ +1
forces (inK 1
<λ ) a value to p and the P ˜ λ -namesd˜ n ζ , σ
˜ n , k˜ n for n < ω , i.e. the values here are still P λ -names. Let Q∗
be the limit of the Qξ -s. So Q∗∈K 1 , lg(Q∗) = λ, Q∗ = P ∗i , Q˜∗
j, α∗j , e∗j :
i ≤ λ, j < λ , and the P ∗λ -names d˜ n , σ
˜ n , k˜ n are dened such that in V P ∗λ ,
d˜ n , σ
˜ n , k˜ n contradict (2) (as any P ∗λ -name of a bounded subset of λ is a
P ∗lg( Q ξ ) -name for some ξ < λ ).
B Stage: Let χ = κ+ and < ∗χ be a well-ordering of H(χ ). Now we can applyλ → (ω1 )<ω
2 to get δ,B,N s (for s ∈[B ]< ℵ0 ) and h s,t (for s, t ∈[B ]< ℵ0 , |s | =|t |) such that:
(a) B ⊆ λ, otp( B ) = ω1 , sup B = δ,(b) N s (H (χ ),∈, < ∗χ ), Q∗∈N s , d
˜ , σ˜ n , k
˜ n : n < ω ∈N s ,
(c) N s ∩ N t = N s ∩t ,
(d) N s ∩ B = s,
(e) if s = t ∩ α, t ∈[B ]< ℵ0 then N s ∩ λ is an initial segment of N t ,
(f) h s,t is an isomorphism from N t onto N s (when dened)
(g) h t,s = h − 1s,t
(h) p0 ∈N s , p0 P λ “ d˜ n , σ
˜ n , k˜ n : n < is a counterexample”,
(i) ω1 ⊆N s , |N s | = ℵ1 and if γ ∈N s , cf γ > ℵ1 then cf(sup( γ ∩ N s )) =ω1 .
8/3/2019 Saharon Shelah- Strong Partition Realations Below the Power Set: Consistency Was Sierpinski Right? II
18For p, q∈Θ, p ≤ q iff n p ≤ n q and for every s ∈fs T p we have ps ≤ qs .
F Stage: Let g˜
: ω → ω, g˜∈N s , g
˜grows fast enough relative σn : n < ω .
We dene a game Gm. A play of the game lasts after ω moves, in the n th
move player I chooses pn∈Θn and a function hn satisfying the restrictions
below and then player II chooses ¯qn ∈Θn , such that pn ≤ qn (so T pn = T qn ).Player I loses the play if sometimes he has no legal move; if he never loses,he wins. The restrictions player I has to satisfy are:
(a) for m < n , qm ≤ pn , pns forces a value to g
˜(n + 1),
(b) hn is a function from [ B T p n]≤ g (n ) to ω,
(c) if m < n ⇒ hn , hm are compatible,(d) If m < n , < g(m), s ∈[B T p n ] , then pn
s d˜
(s) = hn (s),(e) Let s1 , s 2 ∈ Dom hn . Then hn (s1) = hn (s2) whenever s1 , s 2 are
similar over n which means:
(i) F H OP s 2 ,s 1 (ζ ) n [¯ pn ] = F (ζ ) n [¯ pn ] for ζ ∈s1 ,
(ii) H OP s 2 ,s 1 preserves the relations sp F (ζ 1), F (ζ 2 ) < sp F (ζ 3 ),
F (ζ 4) and F (ζ 3 ) sp (F (ζ 1), F (ζ 2 )) = i (in the interesting
case ζ 3 = ζ 1 , ζ 2 implies i = 0).
G Stage/Claim: Player I has a winning strategy in this game.Proof. As the game is closed, it is determined, so we assume player II hasa winning strategy , and eventually we shall get a contradiction. We deneby induction on n, r n and Φ n such that(a) r n
∈Rn , r n ≤ r n +1 ,(b) Φ n is a nite set of initial segments of plays of the game,(c) in each member of Φ n player II uses his winning strategy,(d) if y belongs to Φ n then it has the form py, , h y, , qy, : ≤ m(y) ; let
hy = hy,n y and T y = T qy ,m (y ) ; also T y ⊆n ≥ 2, qy,s ≤ r n
s for s ∈fs T y .(e) Φn ⊆ Φn +1 , Φn is closed under taking the initial segments and the
empty sequence (which too is an initial segment of a play) belongs toΦ0 .
(f) For any y ∈ Φn and T, h either for some z ∈ Φn +1 , n z = n y + 1,y = z (n y + 1), T z = T and hz = h or player I has no legal ( n y + 1) th
move pn
, hn
(after y was played) such that T pn = T , hn
= h, and pns = r n
s for s ∈fs T (or always ≤ or always ≥ ).
8/3/2019 Saharon Shelah- Strong Partition Realations Below the Power Set: Consistency Was Sierpinski Right? II
19There is no problem to carry the denition. Now r n
s : n < ω dene afunction d∗: if η1 , . . . , ηk ∈
m 2 are distinct then d∗( η1 , . . . , ηk ) = c iff forevery (equivalently some) ζ 1 < · · · < ζ k from B , η F (ζ ) and r k
{ζ 1 ,...,ζ k }“d˜ k ({ζ 1 , . . . , ζ k }) = c”.
Now apply 2.7(2) to this coloring, get T ∗⊆ω> 2 as there. Now playerI could have chosen initial segments of this T ∗ (in the n th move in Φn ) andwe get easily a contradiction.
H Stage: We x a winning strategy for player I (whose existence is guar-anteed by stage G).
We dene a forcing notion Q∗. We have ( r,y,f ) ∈Q∗ iff
(i) r ∈P cna δ
(ii) y = p , h , q : ≤ m(y) is an initial segment of a play of Gm in whichplayer I uses his winning strategy
(iii) f is a nite function from B to {0, 1} such that f − 1({1}) ∈fs T y (whereT y = T qm ( y ) ).
(iv) r = qy,m (y )f − 1 ({1}) .
The Order is the natural one.
I Stage: If J ⊆ P cna δ
is dense open then {(r,y,f ) ∈Q∗ : r ∈J } is dense inQ∗.
Proof. By 3.8(1) (by the appropriate renaming).
J Stage: We dene Q δ in V P δ as {(r,y,f ) ∈Q∗ : r ∈G˜
P δ }, the order isas in Q∗.
The main point left is to prove the Knaster condition for the partialordered set Q∗= Qˆ P δ , Q
˜ δ, a δ , eδ demanded in the denition of K 1 . This
will follow by 3.8(3) (after you choose meaning and renamings) as done instages K,L below.
K Stage: So let i < δ , cf(i) = ℵ1 , and we shall prove that P +δ+1 /P i satisesthe Knaster condition. Let pα ∈ P ∗δ+1 for α < ω 1 , and we should nd p ∈P i , p P i “there is an unbounded A ⊆ {α : pα i ∈G
˜ P i } such that for any α, β ∈A, pα , pβ are compatible in P ∗δ+1 / G
˜ P i ”.
Without loss of generality:
(a) pα ∈P cnδ+1 .
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<k ∗ j α = j , f ( j α ) = f ( j β ), F ( j α )) m(yα ) = F ( j β ) m(yβ ).The main problem is the compatibility of the qy α ,m (y α ) . Now by the
denition Θ α (in stage E) and 3.8(3) this holds.
L Stage: If c ⊂ δ + 1 is closed for Q∗, then P ∗δ+1 /P cnc satises the Knaster
condition.
If c is bounded in δ, choose a successor i ∈(sup c, δ) for Q i ∈K 1 . Weknow that P i /P cnc satises the Knaster condition and by stage K, P ∗δ+1 /P i
also satises the Knaster condition; as it is preserved by composition wehave nished the stage.
So assume c is unbounded in δ and it is easy too. So as seen in stage J,we have nished the proof of 3.1.
Theorem 3.11. If λ ≥ ω , P is the forcing notion of adding λ Cohen reals then
(∗)1 in V P , if n < ω d : [λ]≤ n → σ, σ < ℵ0 , then for some c.c.c. forcing notion Q we have Q “ there are an uncountable A ⊆ λ and an one-to-one F : A →ω 2 such that d is F -canonical on A” (see notation in§2).
(∗)2 if in V , λ ≥ µ →wsp (κ)ℵ0 (see [Sh289]) and in V P , d : [µ]≤ n → σ,σ < ℵ0 then in V P for some c.c.c. forcing notion Q we have Q “ thereare A ∈[µ]κ and one-to-one F : A →ω 2 such that d is F -canonicalon A” (see §2, ).
(∗)3 if in V , λ ≥ µ →wsp (ℵ1)nℵ2
and in V P d : [µ]≤ n → σ, σ < ℵ0 then inV P for every α < ω 1 and F : α →ω 2 for some A ⊆ µ of order type α and F : A →ω 2, F (β )
def = F (otp( A ∩ β )) , d is F -canonical on A.
(∗)4 in V P
, 2ℵ0
→ (α, n )3
for every α < ω 1 , n < ω . Really, assuming V |=GCH, we have ℵn 13
→ (α, n ) see [Sh289].
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21Proof. Similar to the proof of 3.1. Supercially we need more indiscerni-bility then we get, but getting M u : u ∈[B ]≤ n we ignore d({α, β }) whenthere is no u with {α, β } ∈M u .
Theorem 3.12. If λ is strongly inaccessible ω-Mahlo, µ < λ , then for some
c.c.c. forcing notion P of cardinality λ, V P
satises (a) MA µ
(b) 2ℵ0 = λ = 2 κ for κ < λ(c) λ → [ℵ1 ]nσ,h (n ) for n < ω , σ < ℵ0, h(n) is as in 3.1.
Proof. Again, like 3.1.
4. Partition theorem for trees on large cardinals
Lemma 4.1 Suppose µ > σ + ℵ0 and(∗)µ for every µ-complete forcing notion P , in V P , µ is measurable.Then(1) for n < ω , P r f
eht (µ,n,σ ).
(2) P r f eht (µ, < ℵ0 , σ), if there is λ > µ , λ → µ+
<ω
2.
(3) In both cases we can have the P r f ehtn version, and even choose the
< ∗α : α < µ in any of the following ways.(a) We are given < 0
α : α < µ , and we let for η, ν ∈α 2 ∩T , α ∈SP (T )( T is the subtree we consider):η <∗α ν if and only if clpT (η) < 0
(b) We are given < 0α : α < µ , we let that for ν, η ∈α 2∩T , α ∈SP (T ):
η <∗α ν if and only if n (β + 1) < 0β +1 ν (β + 1) where β = sup( α ∩ SP (T )) .
Remark. 1) (∗)µ holds for a supercompact after Laver treatment. Onhypermeasurable see Gitik Shelah [GiSh344].2) We can in (∗)µ restrict ourselves to the forcing notion P actually used.For it by Gitik [Gi] much smaller large cardinals suffice.
3) The proof of 4.1 is a generalization of a proof of Harrington to HalpernLauchli theorem from 1978.
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Conclusion 4.2. In 4.1 we can get P r f ht (µ,n,σ ) (even with (3)).
Proof of 4.2. We do the parallel to 4.1(1). By ( ∗)µ , µ is weakly compacthence by 2.6(2) it is enough to prove P r f
aht (µ,n,σ ). This follows from 4.1(1)by 2.6(1).
Proof of Lemma 4.1. 1), 2). Let κ ≤ ω, σ(n) < µ , dn ∈Colnσ (n ) (µ> 2) for
n < κ .
Choose λ such that λ → (µ+ )< 2κ2µ (there is such a λ by assumption
for (2) and by κ < ω for (1)). Let Q be the forcing notion ( µ> 2, ), andP = P λ be {f : dom(f ) is a subset of λ of cardinality < µ , f (i) ∈ Q}ordered naturally. For i ∈dom( f ), take f (i) = <> ; Let η
˜ ibe the P-name
for {f (i) : f ∈G˜ P }. Let D
˜be a P-name of a normal ultralter over µ (in
V P ). For each n < ω , d ∈Colnσ (n ) (µ> 2), j < σ (n) and u = {α 0 , . . . , α n − 1},
where α 0 < · · · < α n − 1 < λ , let A˜
jd (u) be the P λ -name of the set
Ajd (u) = i < µ : η
˜ αi : < n are pairwise distinct and
j = d(ηα 0 i , . . . , η α n − 1 i) .
So A˜
jd (u) is a P λ -name of a subset of µ, and for j (1) < j (2) < σ (n) we have
P λ “A˜
j (1)d (u) ∩ A
˜j (2)d (u) = ∅, and j<σ (n ) A
˜jd (u) is a co-bounded subset of
µ”. As P “D is µ-complete uniform ultralter on µ”, in V P there is exactlyone j < σ (n) with Aj
d (u) ∈D . Let j˜ d
(u) be the P -name of this j .
Let I d (u) ⊆ P be a maximal antichain of P , each member of I d (u)forces a value to j
˜ d(u). Let W d (u) = {dom( p) : p ∈I d (u)} and W (u) =
{W d n (u) : n < κ }. So W d (u) is a subset of λ of cardinaltiy ≤ µ as well asW (u) (as P satises the µ+ -c.c. and p ∈P ⇒ | dom( p)| < µ ).
As λ → (µ++ )< 2κ2µ , dn ∈ Coln
σ n (µ> 2) there is a subset Z of λ of cardinality µ++ and set W + (u) for each u ∈[Z ]<κ such that:
(i) W + (u1) ∩ W + (u2) = W + (u1 ∩ u2),
(ii) W (u) ⊆W + (u) if u ∈[Z ]<κ ,
(iii) if |u1 | = |u2 | < κ and u1 , u2 ⊆ Z then W + (u1) and W + (u2) have thesame order type and note that H [u1 , u2 ]
def = H OP W + (u 1 ) ,W + (u 2 ) , induces
naturally a map from P u1
def = { p ∈ P : dom( p) ⊆ W + (u1)} to
P u2def = { p∈P : dom( p) ⊆W + (u2)}.
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For any generic G ⊆ P λ to which p belongs, β < α and ordinalsi0 < · · · < i n − 1 from Z such that h− 1(i ) β : < n are pairwise distinctwe have that
B{i : <n } ,β = ξ < µ : dn (ηi 0 ξ , . . . , η i n − 1 ξ) = c(u, h∗) ,
belongs to D [G], where u = {h− 1 (i ) β : < n} and h∗ : u → s(|u |) isdened by h∗(h− 1 (i ) β ) = H OP
{i : <n } ,s (n ) (i ). Really every large enoughβ < µ can serve so we omit it. As D [G] is µ-complete uniform ultralter onµ, we can nd ξ ∈(ζ, κ) such that ξ ∈B u for every u ∈[α 2]n , n < κ . Welet for ν ∈α 2, F α (ν ) = η
˜ h ( i )[G] ξ, and we let pu = p0
u except when u = {ν },
then: pu (i) = p0
u (i) i = γ (0)F α +1 (ν ) i = γ (0)
.
For α + 1, α is a successor: First for η ∈α − 1 2 dene F (ηˆ ) = F α (η)ˆ .Next we let {(u i , h i ) : i < i ∗}, list all pairs ( u, h ), u ∈[α 2]≤ n , h : u → s(|u |),one-to-one onto. Now, we dene by induction on i ≤ i∗, pi
u (u ∈ [α 2]<κ )such that :(a) pi
u ∈R | u | ,(b) pi
u increases with i,(c) for i + 1, (vii) holds for ( u i , h i ),(d) if ν m ∈α 2 for m < n , n < κ , ν m (α − 1) : m < n are pairwise distinct,
then p{ν m (α − 1) : m<n } ≤ p0{ν m :m<n } ,
(e) if ν ∈α 2, ν (α − 1) = then p0{ν } (0) = F α (ν (α − 1))ˆ .
There is no problem to carry the induction.Now F α +1
α 2 is to be dened as in the second case, starting withη → pi∗
{η} (η).
For α = 0, 1: Left to the reader.So we have nished the induction hence the proof of 4 .1(1), (2).
3) Left to the reader ( the only inuence is the choice of h in stage of theinduction).
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The negative results here suffice to show that the value we have for 2 ℵ0 in
§3 is reasonable. In particular the Galvin conjecture is wrong and that forevery n < ω for some m < ω, ℵn → [ℵ1]mℵ0
.
See Erdos Hajnal Mate Rado [EHMR] for
Fact 5.1. If 2<µ < λ ≤ 2µ , µ → [µ]nσ then λ → [(2<µ )+ ]n +1
σ .
This shows that if e.g. in 1.4 we want to increase the exponents, to 3(and still µ = µ<µ ) e.g. µ cannot be successor (when σ ≤ ℵ0) (by [Sh276],3.5(2)).
Denition 5.2. P r np (λ,µ, σ), where σ = σn : n < ω , means that
there are functions F n : [λ]n
→ σn such that for every W ∈ [λ]µ
for some n, F n ([W ]n ) = σ(n). The negation of this property is denoted by NP r np (λ,µ, σ).
If σn = σ we write σ instead of σn : n < ω .
Remark 5.2A. 1) Note that λ → [µ]<ωσ means: if F : [λ]<ω → σ then for
some A ∈ [λ]µ , F ([A]<ω ) = σ. So for λ ≥ µ ≥ σ = ℵ0 , λ → [µ]<ωσ , (use
F : F (α ) = |α |) and P r np (λ,µ,σ ) is stronger than λ → [µ]<ωσ .
2) We do not write down the monotonicity properties of P r np — they are obvious.
Claim 5.3 1) We can (in 5.2) w.l.o.g. use F n,m : [λ]n → σn for n, m < ωand obvious monotonicity properties holds, and λ ≥ µ ≥ n.2) Suppose NP r np (λ,µ,κ ) and κ → [κ]nσ or even κ → [κ]<ω
σ . Then the following case of Chang conjecture holds:
(*) for every model M with universe λ and countable vocabulary, there is an elementary submodel N of M of cardinality µ,
|N ∩ κ | < κ
3) If NP r np (λ, ℵ1 ,ℵ0) then (λ, ℵ1) → (ℵ1 ,ℵ0).
Proof. Easy.
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26Theorem 5.4. Suppose P r np (λ0 , µ, ℵ0 ), µ regular > ℵ0 and λ1 ≥ λ0 , andno µ ∈(λ0 , λ 1) is µ -Mahlo. Then P r np (λ1 , µ, ℵ0).
Proof. Let χ = 8(λ1 )+ , let {F 0n,m : m < ω } list the denable n-place functions in the model ( H (χ ),∈, < ∗
χ ), with λ0 ,µ ,λ 1 as parameters,let F 1
n,m(α
0, . . . , α
n − 1) (for α
0, . . . , α
n − 1< λ
1) be F 0
n,m(α
0, . . . , α
n − 1) if
it is an ordinal < λ 1 and zero otherwise. Let F n,m (α 0 , . . . , α n − 1 ) (forα 0 , . . . , α n − 1 < λ 1 ) be F 0n,m (α 0 , . . . , α n − 1 ) if it is an ordinal < ω and zerootherwise. We shall show that F n,m (n,m < ω ) exemplify P r np (λ1 , µ, ℵ0)(see 5.3(1)).
So suppose W ∈ [λ1 ]µ is a counterexample to P r (λ 1 , µ, ℵ0) i.e. for non,m,F n,m ([W ]n ) = ω. Let W ∗ be the closure of W under F 1n,m (n,m < ω ).Let N be the Skolem Hull of W in (H (χ ),∈, < ∗
χ ), so clearly N ∩ λ 1 = W ∗.Note W ∗⊆ λ1 , |W ∗| = µ. Also as cf(µ) > ℵ0 if A ⊆W ∗, |A| = µ then forsome n,m < ω and u i ∈[W ]n (for i < µ ), F 1n,m (u i ) ∈A and [i < j < µ ⇒
F 1n,m (u i ) = F 1n,m (u i )]. It is easy to check that also W 1 = {F 1n,m (u i ) : i < µ }is a counterexample to P r (λ 1 ,µ ,σ ). In particular, for n, m < ω , W n,m ={F 1n,m (u) : u ∈[W ]n } is a counterexample if it has power µ. W.l.o.g. W is a
counterexample with minimal δ def = sup( W ) = ∪{α +1 : α ∈W }. The abovediscussion shows that |W ∗∩ α | < µ for α < δ . Obviously cf δ = µ+ . Letα i : i < µ be a strictly increasing sequence of members of W ∗, converging
to δ, such that for limit i we have α i = min( W ∗− j<i (α j + 1). LetN = i<µ N i , N i N , |N i | < µ , N i increasing continuous and w.l.o.g.N i ∩ δ = N ∩ α i .
α Fact: δ is > λ 0 .Proof. Otherwise we then get an easy contradiction to P r (λ 0 ,µ ,σ )) aschoosing the F 0n,m we allowed λ0 as a parameter.β Fact: If F is a unary function denable in N , F (α) is a club of α for everylimit ordinal α(< λ 1) then for some club C of µ we have
(∀ j ∈C \ { min C })(∃i1 < j )(∀i ∈(i1 , j ))[i ∈C ⇒ α i ∈F (α j )].
Proof. For some club C 0 of µ we have j ∈C 0 ⇒ (N j , {α i : i < j }, W )(N, {α i : i < µ }, W ).
We let C = C 0 = acc( C ) (= set of accumulation points of C 0).We check C is as required; suppose j is a counterexample. So j =
sup( j ∩ C ) (otherwise choose i1 = max( j ∩ C )). So we can dene, byinduction on n, i n , such that:
8/3/2019 Saharon Shelah- Strong Partition Realations Below the Power Set: Consistency Was Sierpinski Right? II
27(a) in < i n +1 < j(b) α i n ∈F (α j )(c) (α i n , α i n +1 ) ∩ F (α j ) = ∅.
Why ( C 0)? |= “ F (α j ) is unbounded below α j ” hence N |= “ F (α j ) is
unbounded below α j ”, but in N , {α i : i ∈C 0 , i < j } is unbounded belowα j .Clearly for some n,m,α j ∈W n,m (see above). Now we can repeat the
proof of [Sh276,3.3(2)] (see mainly the end) using only members of W n,m .Note: here we use the number of colors being ℵ0 .
β + Fact: Wolog the C in Fact β is µ.Proof: Renaming.
γ Fact: δ is a limit cardinal.Proof: Suppose not. Now δ cannot be a successor cardinal (as cf δ = µ ≤λ0 < δ ) hence for every large enough i, |α i | = |δ|, so |δ| ∈W ∗⊆ N and|δ|+ ∈W ∗.
So W ∗∩ |δ| has cardinality < µ hence order-type some γ ∗< µ . Choosei∗< µ limit such that [ j < i ∗⇒ j + γ ∗< i ∗]. There is a denable functionF of (H (χ ),∈, < ∗χ ) such that for every limit ordinal α, F (α) is a club of α ,0 ∈F (α), if |α | < α , F (α) ∩ |α | = ∅, otp( F (α)) = cf α .
So in N there is a closed unbounded subset C α j = F (α j ) of α j of ordertype ≤ cf α j ≤ | δ|, hence C α j ∩ N has order type ≤ γ ∗, hence for i∗ chosenabove unboundedly many i < i ∗, α i ∈C α i ∗ . We can nish by fact β + .
δ Fact: For each i < µ , α i is a cardinal.Proof: If |α i | < i then |α i | ∈N i , but then |α i |+ ∈N i contradicting to Factγ , by which |α i |+ < δ , as we have assumed N i ∩ δ = N ∩ α i .
ε Fact: For a club of i < µ , α i is a regular cardinal.(Proof: if S = {i : α i singular } is stationary, then the function α i → cf(α i )is regressive on S . By Fodor lemma, for some α∗< δ , {i < µ : cf α i < α ∗} isstationary. As |N ∩ α∗| < µ for some β ∗, {i < µ : cf α i = β ∗} is stationary.Let F 1,m (α) be a club of α of order type cf( α), and by fact β we get acontradiction as in fact γ .
ζ Fact: For a club of i < µ , α i is Mahlo.Proof: Use F 1,m (α ) = a club of α which, if α is a successor cardinal or
inaccessible not Mahlo, then it contains no inaccessible, and continue as infact γ .
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28ξ Fact: For a club of i < µ , α i is α i -Mahlo.Proof: Let F 1,m (0) (α ) = sup {ζ : α is ζ -Mahlo}. If the set {i < µ : α i is notα i -Mahlo} is stationary then as before for some γ ∈N , {i : F 1,m (0) (α i ) = γ }is stationary and let F 1,m (1) (α ) — a club of α such that if α is not ( γ + 1)-Mahlo then the club has no γ -Mahlo member. Finish as in the proof of fact
δ.
Remark 5.4.A. We can continue and say more.
Lemma 5.5 1) Suppose λ > µ > θ are regular cardinals, n ≥ 2 and(i) for every regular cardinal κ, if λ > κ ≥ θ then κ → [θ]<ω
σ (1) .(ii) for some α(∗) < µ for every regular κ ∈(α(∗), λ ), κ → [α (∗)]n
σ (2) .Then(a) λ → [µ]n +1
σ where σ = min {σ(1) , σ(2)},(b) there are functions d2 : [λ]n +1 → σ(2), d1 : [λ]3 → σ(1) such that for
every W ∈[λ]µ , d1 ([W ]3) = σ(1) or d2 ([W ]n +1 ) = σ(2) .2) Suppose λ > µ > θ are regular cardinals, and(i) for every regular κ ∈[θ, λ ), κ → [θ]<ω
σ (1) ,(ii) sup{κ < λ : κ regular } → [µ]n
σ (2) .Then(a) λ → [µ]2n
σ where σ = min {σ(1) , σ(2)}(b) there are functions d1 : [λ]3 → σ(1), d2 : [λ]2n → σ(2) such that for every W ∈[λ]µ , d1 ([W ]3) = σ(1) or d2 ([W ]2n = σ(2).
Remark. The proof is similar to that of [Sh276] 3.3,3.2.Proof. 1) We choose for each i, 0 < i < λ i , C i such that: if i is a successorordinal, C i = {i − 1, 0}; if i is a limit ordinal, C i is a club of i of order typecf i, 0∈C i , [cf i < i ⇒ cf i < min( C i − { 0})] and C i \ acc(C i ) contains onlysuccessor ordinals.
Now for α < β , α > 0 we dene by induction on , γ + (β, α ), γ − (β, α ),and then κ(β, α ), ε(β, α ).(A) γ +0 (β, α ) = β , γ −0 (β, α ) = 0.(B) if γ + (β, α ) is dened and > α and α is not an accumulation point of
C γ +
(β,α
)then we let γ −
+1(β, α ) be the maximal member of C γ +
(β,α
)which is < α and γ ++1 (β, α ) is the minimal member of C γ + (β,α ) which
8/3/2019 Saharon Shelah- Strong Partition Realations Below the Power Set: Consistency Was Sierpinski Right? II
(explanation for (c): if εm (β, α ) < α this is easy (check the denition)and if εm (β, α ) = α , necessarily ξ = α and it is trivial).(d) if ≤ m then ε (β, α ) ≤ εm (β, α )For a regular κ ∈(α (∗), λ ) let g1
κ : [κ]<ω → σ(2) exemplify κ → [θ]<ωσ (1)
and for every regular cardinal κ ∈ [θ, λ ) let g2κ : [κ]n → σ(2) exemplify
κ → [α(∗)]nσ (2) . Let us dene the colourings:
Let α 0 > α 1 > .. . > α n . Remember n ≥ 2.Let n = n(α 0 , α 1 , α 2) be the maximal natural number such that:
(i) εn (α 0 , α 1) < α 0 is well dened,(ii) for ≤ n, γ − (α 0 , α 1) = γ − (α 0 , α 2).
We dene d2(α 0 , α 1 , . . . , α n ) as g2κ (β 1 , . . . , β n ) where
κ = cf ( γ +n (α 0 ,α 1 ,α 2 ) (α 0 , α 1)) ,
β =otp α ∩ C γ +n ( α 0 ,α 1 ,α 2 ) (α 0 ,α 1 ) .
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If the set w = γ ∈ C γ +n (α 0 ,α 1 ) : γ > i (∗), γ = min γ/E is nite,
we let d1(α 0 , α 1 , α 2) be g1κ ({β γ : γ ∈w}) where κ = C γ +
n (α 0 ,α 1 ) , β γ =
otp γ ∩ C γ +n (α 0 ,α 1 ) .
We have dened d1 , d2 required in condition ( b) ( though have not yetproved that they work) We still have to dene d (exemplifying λ → [µ]n +1 ).Let n ≥ 3, for α 0 > α 1 > .. . > α n , we let d(α 0 , . . . , α n ) be d1(α 0 , α 1 , α 2) if wdened during the denition has odd number of members and d2(α 0 , . . . , α n )otherwise.
Now suppose Y is a subset of λ of order type µ, and let δ = sup Y . LetM be a model with universe λ and with relations Y and {(i, j ) : i ∈C j }. LetN i : i < µ be an increasing continuous sequence of elementary submodels
of M of cardinality < µ such that α(i) = α i = min( Y \ N i ) belongs to N i +1 ,sup( N ∩ α i ) = sup( N ∩ δ). Let N =
i<µN i . Let δ(i) = δi
def = sup( N i ∩ α i ),
so 0 < δ i ≤ α i , and let n = n i be the rst natural number such that δi anaccumulation point of C i
def = C γ +n (α i ,δ ( i )) , let εi = εn ( i ) (α i , δi ). Note that
γ +n (α i , δi ) = γ +n (α i , ε i ) hence it belongs to N .
Case I: For some (limit) i < µ , cf(i) ≥ θ and (∀γ < i )[γ + α(∗) < i ] suchthat for arbitrarily large j < i , C i ∩ N j is bounded in N j ∩ δ = N j ∩ δj .This is just like the last part in the proof of [Sh276],3.3 using g1
κ and d1 forκ = cf( γ +n i (α i , δi ).
Case II: Not case I .Let S 0 = {i < µ : (∀α < i )[γ + α(∗) < i ], cf(i) = θ}. So for every i ∈S 0
for some j (i) < i , (∀ j ) j ∈( j (i), i) ⇒ C i ∩ N j is unbounded in δj . But as
C i ∩ δi is a club of δi , clearly (∀ j ) j ∈( j (i), i) ⇒ δj ∈C i .
We can also demand j (i) > ε n (α ( i ) ,δ ( i )) (α (i), δ(i)).As S 0 is stationary, (by not case I) for some stationary S 1 ⊆ S 0 and
n(∗), j (∗) we have (∀i ∈S 1) j (i) = j (∗) ∧n(α(i), δi ) = n(∗) .
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31Choose i(∗) ∈ S 1 , i(∗) = sup( i(∗) ∩ S 1), such that the order type of
S 1 ∩ i(∗) is i(∗) > α (∗). Now if i2 < i 1 ∈S 1 ∩ i(∗) then n(α i (∗) , α i 1 , α i 2 ) =
n(∗). Now L i (∗)def = otp( α i ∩ C i (∗) ) : i ∈S 1 ∩ i(∗) are pairwise distinct
and are ordinals < κdef = |C i (∗) |, and the set has order type α(∗). Now apply
the denitions of d2 and g2κ on L i (∗) .
2) The proof is like the proof of part (1) but for α 0 > α 1 > · · · we letd2(α 0 , . . . , α 2n − 1 ) = g2
κ (β 0 , . . . , β n ) where
β def = otp( C γ +
n (β 2 ,β 2 +1 ) (β 2 , β 2 +1 ) ∩ β 2 +1 )
and in case II note that the analysis gives µ possible β ’s so that we canapply the denition of g2
κ .
Denition 5.7. Let λ →stg [µ]nθ mean: if d : [λ]n → θ, and α i : i < µ is
strictly increasingly continuous and for i < j < µ , γ i,j ∈[α i , α i +1 ) then
θ = d(w) : for some j < µ, w ∈[{γ i,j : i < j }]n .
Lemma 5.8. 1) ℵt → [ℵ1]n +1ℵ0
for n ≥ 1.2) ℵn →stg [ℵ1]n +1
ℵ0for n ≥ 1.
Proof. 1) For n = 2 this is a theorem of Torodcevic, and if it holds forn ≥ 2 by 5.5(1) we get that it holds for n+1 (with n, λ, µ, θ, α (∗), σ(1),σ(2) there corresponding to n + 1, ℵn +1 , ℵ1 , ℵ0 ,ℵ0 , ℵ0 ,ℵ0 here).2) Similar.
References
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Saharon ShelahInstitute of MathematicsThe Hebrew University Jerusalem, Israel