Many partition relations below density

MANY P AR TITION RELA TIONS BELOW DENSITY SH918 SAHARON SHELAH Abstract. W e force 2 λ to b e l arge and for m an y pairs in the inte rv al ( λ, 2 λ ) a strong version of the p olarized partition relations hold. W e apply this to prob- lems in general topology . E.g. consisten tly , ev ery 2 λ is successor of singular and for ev ery H ausdorﬀ regular space X , hd( X ) ≤ s ( X ) +3 , hL( X ) ≤ s ( X ) +3 and better when s ( X ) is r egular, via a half-gr aph partition r elations. F or the case s ( X ) = ℵ 0 we get hd( X ), hL( X ) ≤ ℵ 2 . Minor changes i n July , 2020. V ersion 2023-08-16. See https://shelah.l ogic.at/papers/918/ for p ossible up dates. Date : August 16, 2023. 2010 Mathematics Subje ct Classiﬁc ation. Primary 03E35, 03E02; Secondary: 54A25, 54A35. Key wor ds and phr ases. set theory , indep ende nce, forcing, partition r elations, top ological car- dinal inv ariants, hereditary density , hereditary Li ndelof. The author tha nks Al ice Le onhardt f or the b eautiful t yping for the journal version up to 2019. Research supported by the Uni ted States-Israel Binational Science F oundation (Grant No. 2002323) . F or the changes after 2019, the author would like to thank the typist for his work and is also grateful for the generous funding of t yping services donat ed b y a person who wishes to remain anonymous. First Typed - 06/Dec/21. 1 2 SAHARON S HELAH Annot a ted C ontent § 0 Int ro duction, pg. 3. § 1 A Criterion for Strong Polarized Partition Relations , pg. 5. [W e g iv e suﬃcient conditions for having strong versions of p olarized parti- tion relatio ns after for cing.] § 2 The forcing, pg. 12. [Assume GCH for simplicity and p a parameter with λ < µ regular and Θ ⊆ Reg ∩ [ λ, µ + ) a nd we deﬁne Q p which adds µ Cohen subsets to λ but hav e many kinds of supp orts, one for each θ ∈ Θ, inﬂuencing the or der .] § 3 Applying the criterion, pg. 25. [The main result is that (cardinal arithmetic is c hang e d just by making 2 λ = µ and) using § 1 we prov e the stro ng version of p olarized partition relations hold in many instances.] MANY P AR TITION RELA TIONS BELO W DENSITY SH918 3 0. Introduction Out motiv a tion is a problem in ge ne r al topolo gy and for this we g et a consistency result in the partitio n ca lc ulus. In Juhasz-Shelah [JS08] was prov e d: if ( ∀ µ < λ )( µ ℵ 0 < λ ) then there is a c.c.c. forcing notion that adds a reg ular top ological space, hereditarily Lindel¨ of of density λ . A natural question as ked ther e ([JS0 8]) is: Problem 0.1. Assume ℵ 1 < λ ≤ 2 ℵ 0 . Do es there exist (i.e., pro v ably in ZFC) a hereditary Lindel¨ of regular space of density λ ? On cardinal inv ar ian ts in gener al top ology see [Juh80]. W e prove the consistency of a negative a nsw er, in fact o f stro nger results by pr o ving the consistency of strong v ariants of p olarized partition r elations (the half-gr aphs, see b elow). They are strong enough to reso lv e the questio n ab out hereditary density (and her editary Lindel¨ of ). Mor eo ver, if λ = λ <λ < µ = µ <µ (and G.C.H. holds in [ λ, µ )), then there is a forcing extension making 2 λ ≥ µ neither adding new ( < λ )- sequences no r collapsing cardinals such that for many pairs λ ∗ < µ ∗ in the interv al we hav e the appropria te partition rela tions. An earlier r e sult is in the pap er [She88, Theo rem 1.1, pg.357] and it states the following: if λ > κ > µ are r egular cardinals, λ > κ ++ , then there is a ca rdinal and coﬁnality preserv ing for cing that makes 2 µ = λ and κ ++ → ( κ ++ , ( κ ; κ ) κ ) 2 in addition to the main result there 2 λ → [ λ ] 2 3 , see mor e in [She89], [She92], [She96], [She00]. The a pplied notion of for c ing ( Q, ≤ ) is the following: p ∈ Q if p is a function from a subset dom( p ) ∈ [ λ ] ≤ κ int o Add( µ, 1 ) − {∅} where Add( µ, 1) denotes the forcing adding a Cohen subset of µ . p ≤ q if dom( p ) ⊇ dom( q ) , p ( α ) ≤ q ( α ) for α ∈ do m( q ) and |{ α ∈ dom( q ) : p ( α ) 6 = q ( α ) }| < µ . F or simu ltaneo usly many n - place p olarized partition relation Shela h-Stanley [SS01] deals with it but there ar e pr o blems ther e, so we do not rely on it. Our main result in gener al topo logy is Theorem 3.1 0, by it: c o nsisten tly , G.C.H. fails badly (2 µ is a successor o f a limit cardinal > µ except when µ is strong limit singular and then 2 µ = µ + ) and hd( X ), hL( X ) are ≤ s ( X ) +3 for ev ery Hausdorﬀ regular X and | X | ≤ 2 (hd( X )) + , w ( X ) ≤ 2 (hL( X )) + for any Hausdorﬀ X . (Usually s ( X ) +2 suﬃce so in pa r ticular “ X is her editary Lindel¨ of ⇒ X has density ≤ ℵ 2 ”. Concerning partition relations we give a gene r alization of the ear lier result expla ined ab o ve, namely , the consistency of 2 ℵ 0 = λ a nd µ ++ → ( µ, ( µ ; µ ) µ ) 2 simult aneo usly holding for each regular cardinal µ such that µ ++ ≤ λ . This gives a model in which though GCH fails badly , we hav e strong eno ugh partition re la tions implying that the hereditary densit y and the hereditar y Lindel¨ of num b ers of a T 3 space X are bo unded b y s ( X ) +3 where s ( X ) stands for sprea d. The notio n of forcing ( P , ≤ ) used for the argument is de ﬁned as follows. F or each regular ca rdinal µ < λ deﬁne the following equiv alence relation E µ on λ . xE µ y iﬀ x + µ = y + µ . Let [ x ] µ denote the equiv a lence class of x . p ∈ P if p is a function from some set dom( p ) ⊆ λ into { 0 , 1 } suc h that | [ x ] µ ∩ dom( p ) | < µ holds for every successor µ < λ, x < λ . p ≤ q if p ⊇ q and fo r every successor µ < λ we have: |{ [ x ] µ : ∅ 6 = dom( q ) ∩ [ x ] µ 6 = Dom( p ) ∩ [ x ] µ }| < µ. 4 SAHARON S HELAH This notion of forcing ( P , ≤ ), in a most r emark able w ay , imitates co ncurren tly several diﬀerent posets ( Q, ≤ ) a s deﬁned ab ov e. Not s urprisingly , in order to show that ( P, ≤ ) is cardinal and coﬁnality preserving , the author uses ideas s imilar to those in [She88]. In o rder to prove the main claim, that is, the partition relatio n, w e use the following trick: we ﬁnd a condition ¯ p suc h that the dense sets w e are in terested in are all dense below ¯ p . It suﬃces, therefo r e, to sho w that forcing with the part below ¯ p gives the req uir ed result, and this reduces the problem to showing that a certain notion of forcing ( R, ≤ ) forces the sought-for-partition r elation where | R | is sma ll (compared to µ ). As ( R, < ) is close to the p oset ( Q, < ) of [She88], an elemen tar y sub-mo del a rgument similar to the one there applies . The exp osition of the method is axio matic; the author fo r m ulates the most general situation where this metho d works, and then sp eciﬁes it to the situation sketc hed ab o ve. This is not nece s sarily the optimal descr iption for those who are only in- terested in the applica tion given. There is, ho wev er, reason fo r the p eculiar way of presenting this pro of: we w o uld like to include this metho d in to the to ol kit set, and simply quote it at p ossible later a pplications. Recall (ﬁrst app eared in Erd¨ os-Ha jnal [EH78], but pro ba bly raised b y Galvin in letters in the mid seventies): Deﬁnition 0.2. 1) λ → ( µ ; µ ) 2 κ means that: for every c : [ λ ] 2 → κ there are ε and α i , β i for i < µ such that: ( a ) ε < κ, ( b ) if i < j < µ then α i < β i < α j < λ, ( c ) if i ≤ j < µ then c { α i , β j } = ε . 2) W e ca n re place µ b y an ordina l and if κ = 2 w e may omit it. Deﬁnition 0.3. 1) Le t λ → ( µ, ( µ ; µ ) κ ) 2 means that: for every c : [ λ ] 2 → 1 + κ there are ε and α i , β i for i < µ such that: ( a ) ε < κ, ( b ) α i < β i < α j < λ for i < j < µ, ( c ) 0 if ε = 0 then i < j ⇒ c { α i , α j } = ε , so we ca n for get the β i ’s, ( c ) 1 if ε ≥ 1 then i ≤ j ⇒ c { α i , β j } = ε . 2) In part (1) if κ = 1 w e ma y omit it. Abov e replacing µ b y “ < µ ” m ea ns “for every ξ < µ we hav e ....”. W e thank Shimoni Garti for many cor rections and Istv an Juhasz for questions and historical remarks ; we may cont inue this research in [S + ]. MANY P AR TITION RELA TIONS BELO W DENSITY SH918 5 1. Strong pol arized p a r tition rela tions W e deal with suﬃcient co nditions on a forcing notion for preserving suc h partition relations. F or this, w e use a n expansion of a forcing notion. Instead of the usual pair ( Q, ≤ Q ), namely , the underlying set and the par tia l order , w e use a quadruple of the form Q = ( Q, ≤ Q , ≤ pr Q , ap Q ). The “pr” sta nds for pure, a nd the “ap” stands for a pure. Both are included (as partial orders) in Q . Discussion 1.1. W e deﬁne (b elow) the notion of “( λ, θ , ξ )-forcing” to give a suﬃ- cient condition for appropria te case s o f the partition rela tions deﬁned a bov e to hold. W e s ta rt with the quadruple Q = ( Q, ≤ Q , ≤ pr Q , ap Q ) such that q ∈ Q ⇒ ap Q ( q ) ⊆ Q and ≤ Q , ≤ pr Q are quasi or ders o n Q . The idea is that if r ∈ ap Q ( q ) then r and q are compatible in Q , close to “ r is a n a- pure ex tension o f q ”. Deﬁnition 1.2. 1) W e say that Q is a ( χ + , θ , ξ )-forcing no tio n when χ + , θ are regular uncountable ca rdinals, ξ an or dinal and ⊛ below holds; in writing ( χ + , θ , < ζ ) w e mean that ⊛ holds for every ξ < ζ ; also we can replac e χ + by λ : ⊛ ( a ) Q = ( Q, ≤ Q , ≤ pr Q , ap Q ) , ( b ) Q = ( Q, ≤ Q ) is a forcing notion (i.e. a qua si o rder, so  Q means  Q , and p ∈ Q means p ∈ Q and V Q means V Q and G ˜ is the Q - na me of the g eneric set), ( c ) ≤ pr Q is a quasi or der on Q and p ≤ pr Q q implies p ≤ Q q , ( d )( α ) ap Q is a function with domain Q , ( β ) for q ∈ Q w e hav e 1 q ∈ ap Q ( q ) ⊆ Q, ( γ ) r ∈ ap Q ( q ) ⇒ r , q a r e co mpatible in Q ; moreov er, ( γ ) + if r ∈ ap Q ( q ) ∧ q ≤ pr Q q + then q + , r ar e compatible in Q moreov er there is r + ∈ a p Q ( q + ) such that q +  Q “ r + ∈ G ˜ Q ⇒ r ∈ G ˜ Q ” 2 , ( e ) ( Q, ≤ pr Q ) is ( < θ )-complete, i.e. any ≤ pr Q -increasing sequence of length < θ has a ≤ pr Q -upp e r b ound is Q , ( f ) ( Q, ≤ pr Q ) satisﬁes the χ + -c.c., ( g ) if ¯ q = h q ε : ε < θ i is ≤ pr Q -increasing then 3 for stationary many limit ordinals ζ < θ , the sequence ¯ q ↾ ζ has an exact ≤ pr Q -upp e r b ound, see part (2) b elow, ( h ) if h q ε : ε < θ i is ≤ pr Q -increasing and p ε ∈ a p Q ( q ε ) for ε < θ a nd ξ < θ then for some ζ < θ we hav e q ζ  Q “if p ζ ∈ G ˜ Q then ξ ≤ otp { ε < ζ : p ε ∈ G ˜ Q } ”, ( i ) if q ∈ Q then ap Q ( q ) has cardina lit y < θ , 1 it is natural to demand q ∈ ap Q ( q ), but not really necessary (if we do not demand it, this just complicates a little ⊛ ( C )( d )) . 2 no harm in asking that r ≤ pr Q s and s ∈ ap Q ( q + ) and q + ≤ s for some s . Wh y this do es not foll o w from our assumption? By the present demand r + , q + ha ve a common ≤ -upp er b ound which i s s , so s  “ q + , r + ∈ G ˜ Q hence r ∈ G ˜ Q ” so without loss of generalit y r ≤ s , but this does not say q ≤ pr Q s . 3 Note that: we can restrict ourselve s to the case q 0 ∈ I , where I is a dense s ubset of Q . Also we can restri ct ourselv es to the set of ¯ q sequences which is the set of plays of a s uitable game wi th one play er using a ﬁxed strategy , etc. 6 SAHARON S HELAH ( j ) if q ∗ ≤ r then there is a ( q ∗ , r )-witness ( q , p ) which mea ns: • 1 q ∗ ≤ pr Q q , • 2 p ∈ ap Q ( q ∗ ) , • 3 q  Q “ p ∈ G ˜ ⇒ r ∈ G ˜ ”. 2) Assume Q satisﬁes claus e s (a )-(e) of pa rt (1 ). Let ¯ q = h q ε : ε < δ i b e a ≤ pr Q -increasing s equence of conditions , δ < θ a limit ordina l. W e say that q is an exact ≤ pr Q -upp e r bo und of ¯ q when ε < δ = ℓg ( ¯ q ) ⇒ q ε ≤ pr Q q and: ( ∗ ) ¯ q, q if p ∈ ap Q ( q ) then for so me ε < δ and p ′ ∈ ap Q ( q ε ), w e have  Q “if q , p ′ ∈ G ˜ Q then p ∈ G ˜ Q ”. R emark 1.3 . Can we w eaken claus e (i) o f ⊛ of 1.2(1) to “car dinalit y ≤ θ ” ? 1) Her e it mostly does not matter, but in one p oin t of the pro of o f 1.4 it do es: in proving ⊛ 4 there, choo s ing ζ ( ∗ ) such tha t it will b e pos sible to c ho ose ε ( ∗ ). 2) There is a price for dema nding a strict ine q ualit y . The pr ice is (in 2.13(1)) that, r e calling κ = κ y , instea d of using ap y ( q ) = { r : q ≤ ap κ r ∈ Q y } w e use ap y ( q ) = { r : q ≤ ap κ r ∈ Q y and supp κ ( q , r ) ⊆ supp θ ( p y α y ( q ) , q ) } . Claim 1.4. If Q is a ( χ + , θ , ξ ∗ ) -for cing notion, κ < θ = cf( θ ) and χ = χ <θ then χ + → ( ξ ∗ , ( ξ ∗ ; ξ ∗ ) κ ) 2 holds in V Q . R emark 1.5 . W e c a n replace χ + by “reg ular χ ′ such tha t α < χ ′ ⇒ | α | <θ < χ ′ ”. Pr o of. Let λ ∗ be large enough (so in particular Q , θ, . . . , ∈ H ( λ + ∗ )). Cho ose a well ordering < ∗ λ + ∗ of the set H ( λ + ∗ ). Recalling Deﬁnition 1.2 clea rly θ > ℵ 0 , hence without loss of genera lit y κ is inﬁnite, so 1 + κ = κ . T ow ar d co n tradiction a ssume p ∗  Q “ c ˜ is a function f ro m [ χ + ] 2 to κ ” is a coun- terexample. W e now c ho ose ¯ M s uc h that: ⊛ 1 ( a ) ¯ M = h M α : α ≤ θ i , ( b ) M α ≺ ( H ( λ + ∗ ) , ∈ ) , ( c ) M α has cardinality χ , ( d ) [ M α ] <θ ⊆ M α if α is non-limit, ( e ) M α is ≺ -incr easing contin uous, ( f ) Q , p ∗ , c ˜ belo ng to M α and χ + 1 ⊆ M α , ( g ) ¯ M ↾ ( α + 1) ∈ M α +1 . Note that χ = χ <θ implies θ < χ + , so let ⊛ 2 δ ∗ := min ( χ + \ M θ ). W e sha ll now prov e: ⊛ 3 if q ∈ Q and ϕ ( x, y ) ∈ L θ ,θ is a formula with parameters from M θ such that ( H ( λ + ∗ ) , ∈ , < ∗ λ + ∗ ) | = ϕ [ δ ∗ , q ] then for some pair ( δ, q ′ ) ∈ M θ we have: ( a ) δ < δ ∗ , ( b ) ( H ( λ + ∗ ) , ∈ , < ∗ λ + ∗ ) | = ϕ [ δ, q ′ ], ( c ) q ′ , q has a co mmon ≤ pr Q -upp e r bo und. MANY P AR TITION RELA TIONS BELO W DENSITY SH918 7 Wh y ⊛ 3 holds? Let ¯ r = h r ζ : ζ < ζ ∗ i list Q , each member app earing χ + times, now without loss of g eneralit y ¯ r ∈ M 0 so necessar ily we can ﬁnd ζ 1 ∈ ζ ∗ \ M θ such that q = r ζ 1 and let ζ 2 = min( M θ ∩ ( ζ ∗ + 1) \ ζ 1 ), of course ζ ∗ ∈ M θ and ζ 2 ∈ M θ and ζ 1 < ζ 2 ∧ cf( ζ 2 ) > χ . Let Y = { q ′ ∈ Q : ( H ( λ + ∗ ) , ∈ , < ∗ λ + ∗ ) | = ( ∃ x )( ϕ ( x, q ′ ) ∧ x ∈ χ + ) } . Recall that χ <θ = χ , so ⊙ 3 . 1 Y ∈ M θ , Y ⊆ Q a nd q ∈ Y . Now we a sk: ⊙ 3 . 2 is there Z ⊆ Y of cardina lit y ≤ χ s uc h that for ev ery q ′′ ∈ Y for a t least one q ′ ∈ Z the pair ( q ′ , q ′′ ) is ≤ pr Q -compatible? Assume tow ard co n tradiction that the answer is ne g ativ e, then in particular | Y | > χ and w e can choose r ε ∈ Y b y induction on ε < χ + such that ζ < ε ⇒ the pair ( r ζ , r ε ) is ≤ pr Q -incompatible. Wh y? In stage ε try to use Z := { r ζ : ζ < ε } , so Z ⊆ Y has ca rdinalit y ≤ | ε | ≤ χ , so s ome r ε ∈ Y can s erv e a s q ′′ in ⊙ 3 . 2 , by our as sumption tow ard c o n tradiction. Hence h r ε : ε < χ + i cont ra dict clause (f ) of Deﬁnition 1.2(1). So the answer to ⊙ 3 . 2 is yes , hence there is such Z ∈ M θ , but χ + 1 ⊆ M θ hence Z ⊆ M θ . So apply t he property of Z , with q standing for q ′′ , so there is q ′ ∈ Z ⊆ Q ∩ M θ such tha t the pair ( q ′ , q ) is ≤ pr Q -compatible; but Z ⊆ Y he nc e by the deﬁnition of Y there is δ ∈ χ + such that ( H ( λ + ∗ ) , ∈ , < ∗ λ + ∗ ) | = ϕ [ δ, q ′ ], and as q ′ ∈ Z ⊆ M θ without loss o f generality δ ∈ M θ , hence δ ∈ χ + ∩ M θ so b y the deﬁnition of δ ∗ we hav e δ < δ ∗ ; so ⊛ 3 holds indeed. Next (but its pro of will take a while) ⊛ 4 if q 0 ∈ Q is ab ov e p ∗ then for some triple ( q 1 , p, ι ) we ha ve: ( a ) q 0 ≤ pr Q q 1 , ( b ) ι < κ , ( c ) p ∈ ap Q ( r ) for some r satisfying q 0 ≤ pr Q r ≤ pr Q q 1 , ( d ) if ι = 0 then p ≤ q 1 , ( e ) if q satisﬁes q 1 ≤ pr Q q and ϕ ( x, y ) ∈ L θ ,θ is a formula with para meters from M θ satisﬁed b y the pair ( δ ∗ , q ) in the mo del ( H ( λ + ∗ ) , ∈ , < ∗ λ + ∗ ), then we ca n ﬁnd q ′ , q ′′ , δ such that the septuple q = ( q , p, ι, ϕ ( x, y ) , q ′ , q ′′ , δ ) satisﬁes the following: ⊠ q • 1 δ < δ ∗ (hence δ ∈ M θ ), • 2 ( H ( λ + ∗ ) , ∈ , < ∗ λ + ∗ ) | = ϕ [ δ, q ′ ], • 3 if ι = 0 then: ( α ) q ≤ pr Q q ′′ , ( β ) q ′ ≤ pr Q q ′′ , ( γ ) q ′′  “ c ˜ { δ, δ ∗ } = 0”. • 4 if ι ∈ (0 , κ ), then q ≤ pr Q q ′′ and q ′′  “ p ∈ G ˜ Q ⇒ c ˜ { δ, δ ∗ } = ι ∧ q ′ ∈ G ˜ Q ”. Wh y? Assume toward con tradictio n that ⊛ 4 fails. W e le t h S ε : ε ≤ θ i b e a ⊆ - increasing contin uous sequence o f subsets of θ with S θ = θ , | S ε +1 \ S ε | = θ , | S 0 | = θ 8 SAHARON S HELAH and min( S ε +1 \ S ε ) ≥ ε . Now we try to choose ( q ∗ ε , x ε , ϕ ε ) b y induction on ε < θ (but ϕ ε is ch os en in the ( ε + 1)-th stag e ) such tha t: ⊙ 4 . 1 ( α ) q ∗ ε ∈ Q and h q ∗ ζ : ζ ≤ ε i is ≤ pr Q -increasing , ( β ) q ∗ 0 = q 0 , ( γ ) if ε is a limit ordina l ( < θ ) and h q ∗ ζ : ζ < ε i has an ex act ≤ pr Q -upp e r bo und (see part (2) of Deﬁnition 1.2) then q ∗ ε is an exact ≤ pr Q -upp e r b ound of it, ( δ ) x ε = h ( p ∗ ξ , ι ξ ) : ξ ∈ S ε i lists { ( p, ι ) : ι < κ a nd p ∈ ap Q ( q ∗ ζ ) for some ζ such tha t ζ = 0 ∨ ζ < ε } , here we use clause (i) of 1.2(1) reca lling q ∗ ζ ∈ ap Q ( q ∗ ζ ), by clause (d)( β ) of 1.2(1) so 1 ≤ | ap Q ( q ∗ ζ ) | < θ , ( ε ) for successo r ordina l ε = ζ + 1, let ( q ∗ ζ +1 , ϕ ζ ( x, y )) exemplify that the triple ( q ∗ ζ , p ∗ ζ , ι ζ ) do es not satisfy demand (e) on ( q 1 , p, ι ) in ⊛ 4 , i.e.: ( ∗ ) q ∗ ζ ≤ pr Q q ∗ ζ +1 and ϕ ζ ( x, y ) ∈ L θ ,θ is a formula with parameter s fr om M θ which the pair ( δ ∗ , q ∗ ζ +1 ) satis ﬁe s in ( H ( λ + ∗ ) , ∈ , < ∗ λ + ∗ ) but we cannot ﬁnd q ′ , q ′′ , δ such that the se ptuple q ζ +1 := ( q ∗ ζ +1 , p ∗ ζ , ι ζ , ϕ ε ( x, y ) , q ′ , q ′′ , δ ) satisﬁes ⊠ q ζ +1 . W e sho w that the induction can be carried out. Assume w e are stuc k at ε . Now if ε = 0 we can satisfy claus es ( α ) + ( β ) and recalling 1 ≤ | ap Q ( q 0 ) | < θ w e can choose x 0 to satisfy clause ( δ ) and since ( γ ) , ( ε ) are v acuous we are done. Suppo se ε > 0. F or limit ε we can c ho ose q ∗ ε as re q uired in clause ( α ) by clause (e) of Deﬁnition 1.2( 1); also clause ( γ ) is relev ant but cause s no problem; and lastly , we ca n c ho ose x ε and since clause ( ε ) is v acuous for limit ordinals, we are done again. So ε is a successor, let ε = ζ + 1, so q ∗ ζ was deﬁned. Now if we cannot choose ( q ∗ ζ +1 , ϕ ζ ( x, y )) = ( q ∗ ε , ϕ ζ ( x, y )) then the triple ( q ∗ ζ , p ∗ ζ , ι ζ ) is as r equired from the triple ( q 1 , p, ι ) in ⊛ 4 . But this is impossible (by our a ssumption tow a rd contradiction), so we can ﬁnd ( q ∗ ζ +1 , ϕ ζ ( x, y )) as r equired; and ag ain we ca n c ho ose x ε as for ε = 0. So it is enough to ge t a contradiction from the assumption that we can car ry o ut the induction. But by clause (g) of Deﬁnition 1.2(1) the set S := { ζ < θ : ζ is a limit ordinal and the sequence h q ∗ ε : ε < ζ i has an exact ≤ pr Q -upp e r b ound } is stationar y . As S is stationary noting ⊙ 4 . 1 ( δ ) a nd r ecalling clause (i) of Deﬁnition 1.2(1) whic h gives | a p Q ( q ∗ ε ) | < θ = cf( θ ) for ε < θ , clea r ly fo r some limit o r dinal ζ ( ∗ ) ∈ S we hav e: if ι < κ ( < θ ) and p ∈ ∪{ ap Q ( q ∗ ε ) : ε < ζ ( ∗ ) } then for unboundedly many ε < ζ ( ∗ ) w e hav e ( p ∗ ε , ι ε ) = ( p, ι ). Let ϕ ( x , y ) ∈ L θ ,θ express all t he prop erties that the pair ( δ ∗ , q ∗ ζ ( ∗ ) ) satisﬁes and are use d b elo w, i.e., ( ∃ y 0 , . . . , y ζ ( ∗ ) )[ x ∈ χ + ∧ y = y ζ ( ∗ ) ∧ V ε<ζ ≤ ζ ( ∗ ) y ε ≤ pr Q y ζ ∧ V ε<ζ ( ∗ ) ϕ ε ( x, y ε +1 ) ∧ ( y ζ ( ∗ ) is an exact ≤ pr Q -upp e r b ound of h y i : i < ζ ( ∗ ) i )]. So ( ∗ ) ( H ( λ + ∗ ) , ∈ , < ∗ λ + ∗ ) | = ϕ [ δ ∗ , q ∗ ζ ( ∗ ) ]. By ⊛ 3 we ca n ﬁnd a pa ir ( δ, q ′ ) such that: MANY P AR TITION RELA TIONS BELO W DENSITY SH918 9 ⊙ 4 . 2 ( a ) δ < δ ∗ hence δ ∈ M θ and q ′ ∈ M θ , ( b ) ( H ( λ + ∗ ) , ∈ , < ∗ λ + ∗ ) | = ϕ [ δ, q ′ ], ( c ) q ′ , q ∗ ζ ( ∗ ) are ≤ pr Q -compatible. Let q ′′ be such that ( d ) q ′ ≤ pr Q q ′′ and q ∗ ζ ( ∗ ) ≤ pr Q q ′′ . Let h q ′ ζ : ζ ≤ ζ ( ∗ ) i exemplify ϕ [ δ, q ′ ] a nd without loss of generality { q ′ ζ : ζ ≤ ζ ( ∗ ) } ⊆ M θ , in particular, ε ≤ ζ ( ∗ ) ⇒ q ′ ε ≤ pr Q q ′ ζ ( ∗ ) = q ′ ≤ pr Q q ′′ and, of co urse, ε ≤ ζ ( ∗ ) ⇒ q ∗ ε ≤ pr Q q ∗ ζ ( ∗ ) ≤ pr Q q ′′ . Case 1 : q ′′  Q “ c ˜ { δ, δ ∗ } = 0”. There is ε < ζ ( ∗ ) such tha t ι ε = 0. W e get contradiction to the c ho ice of the ( q ∗ ε +1 , ϕ ε ). Wh y? Let us chec k that the septuple q = ( q ∗ ε +1 , q ∗ ε +1 , 0 , ϕ ε ( x, y ) , q ′ ε +1 , q ′′ , δ ) is s uc h that ⊠ q holds. F or • 1 : Recall ⊙ 4 . 2 ( a ) F or • 2 : By ⊙ 4 . 1 ( ε )( ∗ ) we hav e ( H ( λ + ∗ ) , ∈ , < ∗ λ + ∗ ) | = ϕ ε ( δ ∗ , q ∗ ε +1 ) by the c hoice of ϕ ( x, y ) and of h q ′ ζ : ζ ≤ ζ ( ∗ ) i we have ( H ( λ + ∗ ) , ∈ , < ∗ λ + ∗ ) | = ϕ ε [ δ, q ′ ε +1 ] as requir ed. F or • 3 ( α ) : it means q ∗ ε +1 ≤ pr Q q ′′ which holds a s q ∗ ε +1 ≤ pr Q q ∗ ζ ( ∗ ) by ⊙ 4 . 1 ( α ) and q ∗ ζ ( ∗ ) ≤ pr Q q ′′ by ⊙ 4 . 2 ( d ). F or • 3 ( β ) : it means q ′ ε +1 ≤ pr Q q ′′ which has b een prov ed just befor e “Case 1”. F or • 3 ( γ ) : it mea ns q ′′  “ c ˜ { δ, δ ∗ } = 0” which holds b y the case as sumption F or • 4 : it is v a ccuous. So indeed ⊠ q holds contradicting the choice of ( q ∗ ε +1 , ϕ ε ), see ⊙ 4 . 1 ( ε ). Case 2 : Not Case 1. Cho ose ( q + , ι ) such that q + ∈ Q , q ∗ ζ ( ∗ ) ≤ Q q ′′ ≤ Q q + and q +  Q “ c ˜ { δ, δ ∗ } = ι ” where ι ∈ (0 , κ ), we use “ not Case 1” . By clause (j) of ⊛ of Deﬁnition 1.2 applied with ( q ∗ ζ ( ∗ ) , q + ) here standing for ( q ∗ , r ) there, we can ﬁnd a pair ( s, p ) such that ⊙ 4 . 3 ( a ) p ∈ ap Q ( q ∗ ζ ( ∗ ) ), ( b ) q ∗ ζ ( ∗ ) ≤ pr Q s , ( c ) s  Q “ p ∈ G ˜ Q ⇒ q + ∈ G ˜ Q ”. As q ∗ ζ ( ∗ ) is a n exact ≤ pr Q -upp e r b ound of h q ∗ ε : ε < ζ ( ∗ ) i becaus e ζ ( ∗ ) ∈ S and p ∈ ap Q ( q ∗ ζ ( ∗ ) ), see par t (2) of Deﬁnition 1.2, there is a pair ( p ′ , ε ( ∗ )) such that: ⊙ 4 . 4 ( a ) ε ( ∗ ) < ζ ( ∗ ), ( b ) p ′ ∈ ap Q ( q ∗ ε ( ∗ ) ), ( c )  Q “if q ∗ ζ ( ∗ ) , p ′ ∈ G ˜ Q then p ∈ G ˜ Q ”. So by the choice o f ζ ( ∗ ) for some ζ < ζ ( ∗ ) which is > ε ( ∗ ) w e hav e ( p ∗ ζ , ι ζ ) = ( p ′ , ι ). Let q = ( q ∗ ζ +1 , p ∗ ζ , ι ζ , ϕ ζ ( x, y ) , q ′ ζ , s, δ ). This s e ptuple s atisﬁes ⊠ q bec ause: F or • 1 : Recall ⊙ 4 . 2 ( a ). F or • 2 : as in case 1. F or • 3 : it is v a ccuous. 10 SAHARON S HELAH F or • 4 : it means ﬁrst q ∗ ζ +1 ≤ pr Q s which holds as q ∗ ζ +1 ≤ pr Q q ∗ ζ ( ∗ ) by ⊙ 4 . 1 ( α ) a nd q ∗ ζ ( ∗ ) ≤ pr Q s by ⊙ 4 . 3 ( b ). Second, s  “ p ∗ ζ ∈ G ˜ Q ⇒ c ˜ { δ, δ ∗ } = ι ” which holds as p ∗ ζ = p ′ and a ssuming G ⊆ Q is generic over V if s, p ′ ∈ G then b y ⊙ 4 . 3 ( b ) also q ∗ ζ ( ∗ ) ∈ G hence by ⊙ 4 . 4 ( c ) also p ∈ G hence by ⊙ 4 . 3 ( c ) also q + ∈ G hence by the choice of q + in the b eginning of the case we ha ve V [ G ] satisﬁes c ˜ [ G ] { δ, δ ∗ } = ι . Third, s  “ p ∗ ζ ∈ G ˜ Q ⇒ q ′ ζ ∈ G ˜ Q ” which holds as p ∗ ζ = p ′ and assuming G ⊆ Q is generic ov er V , if s, p ′ ∈ G then as ab ov e q + ∈ G hence by the choice o f q + in the beg inning of the cas e also q ′′ ∈ G hence by ⊙ 4 . 2 ( d ) also q ′ ∈ G hence b y the c hoice of ϕ and of h q ′ ζ : ζ ≤ ζ ( ∗ ) i we have q ′ ζ ∈ G as r equired. Hence we get a con tradiction to t he c hoice of ( q ∗ ζ +1 , ϕ 3 ). So we a re done proving ⊛ 4 . Let the triple ( q ∗ , p ∗ , ι ∗ ) satisfy the de ma nds on ( q 1 , p, ι ) in ⊛ 4 for q 0 = p ∗ and let r ∗ be as guaranteed by clause (c) of ⊛ 4 so ⊙ p ∗ ≤ pr Q r ∗ ≤ pr Q q ∗ and p ∈ ap Q ( r ∗ ). Now we cho ose q ζ , q ′ ζ , q ′′ ζ , q ′′′ ζ , r ζ , p ζ , α ζ , β ζ by induction on ζ < θ such that: ⊛ 5 ( a ) q ζ ∈ Q , ( b ) h q ξ : ξ ≤ ζ i is ≤ pr Q -increasing , ( c ) q 0 = q ∗ , ( d ) α ζ < β ζ < δ ∗ and ε < ζ ⇒ β ε < α ζ , ( e ) ( q ′ ζ , q ′′ ζ , α ζ ) is as ( q ′ , q ′′ , δ ) is guar an teed to b e in clause (e) of ⊛ 4 with q ζ here standing for q there (and of c ourse p ∗ , ι ∗ here stands for p, ι there) a nd a suitable ϕ , hence, ( α ) α ξ , β ξ < α ζ < δ ∗ for ξ < ζ , ( β ) q ζ ≤ pr Q q ′′ ζ , ( γ ) the pair ( α ζ , q ′ ζ ) ∈ M θ is similar eno ug h to ( δ ∗ , q ζ ), ( δ ) if ι ∗ > 0 then q ′′ ζ  “ if p ∗ ∈ G ˜ Q then c ˜ { α ζ , δ ∗ } = ι ∗ and q ′ ζ ∈ G ˜ Q ”, ( ε ) if ι ∗ = 0 then q ′ ζ ≤ pr Q q ′′ and q ′′ ζ | = “ c ˜ { α ζ , δ ∗ } = ι ∗ and q ′′ ζ | = “ c ˜ { α ε , α ζ } = ι ∗ ” for ε < ζ . ( f ) the quadruple ( β ζ , r ζ , p ζ , q ′′′ ζ ) ∈ M θ is similar enough to the quadruple ( δ ∗ , r, p ∗ , q ′′ ζ ), i.e.: ( α ) β ζ ∈ ( α ζ , δ ∗ ), ( β ) the pair ( q ′′′ ζ , q ′′ ζ ) is ≤ pr Q -compatible, ( γ ) p ζ ∈ ap Q ( r ζ ) and r ζ ≤ pr Q q ′′′ ζ , ( δ ) q ′′′ ζ  Q “if p ζ ∈ G ˜ Q then c ˜ { α ε , β ζ } = ι ∗ for ε ≤ ζ ”. ( g ) q ′′ ζ ≤ pr Q q ζ +1 and q ′′′ ζ ≤ pr Q q ζ +1 . [Wh y can we carry out the induction? Note that q ′ ζ , . . . , β ζ are chosen in the ( ζ + 1)- th step. F or ζ = 0 just let q 0 = q ∗ so the only relev ant cla us es (a ),(c) are s atisﬁed. F or ζ limit only clause (b) is relev ant and we can ch o ose q ζ by clause (e) o f Deﬁnition 1.2. W e a re left with ζ successor, let ζ = ξ + 1. MANY P AR TITION RELA TIONS BELO W DENSITY SH918 11 W e ﬁrst cho ose ( q ′ ξ , q ′′ ξ , α ξ ) as requir e d in clause (e) of ⊛ 5 using appropriate ϕ and ⊛ 4 ( e ) for o ur ( q ∗ , p ∗ , ι ∗ ). Clearly in ⊛ 5 clause (e) holds as well as the seco nd statement in clause (d). In particular , ( e )( δ ) comes from ⊛ 4 ( e ), and ( e )( ε ) comes from ϕ , i.e. as ε < ζ ⇒ q ε ≤ pr Q q ζ . Second, w e c ho ose ( β ξ , r ξ , p ξ , q ′′′ ξ ) as required in claus e (f ) of ⊛ 5 . [Wh y? W e can ﬁnd ( β ξ , r ξ , p ξ , q ′′′ ξ ) ∈ M θ similar enough to ( δ ∗ , r, p ∗ , q ′′ ξ ), using ( ∗ ) 3 with ( δ ∗ , q ′′ ζ ) here standing for ( δ ∗ , q ) there a nd q ′′′ ζ here standing for q ′ in the conclusio n of ⊛ 3 (and r ξ , p ξ are gotten by exis ten tial quantiﬁers in cho o sing ϕ which holds as r ∗ , p ∗ witness). First, n ote that α ζ < δ ∗ holds as α ζ ∈ M θ hence α and β ξ < δ ∗ but β ξ ∈ M θ so β ξ < δ ∗ so clause ( f )( α ) holds. Seco nd, q ′′′ ζ , q ′′ ζ are ≤ pr Q -compatible by ⊛ 3 ( c ) hence clause ( f )( β ) holdsa. Third, the para llel o f ( f )( γ ) holds for ( p ∗ , r ∗ ) by the choice of r ∗ and as q ∗ = q 0 ≤ pr Q q ζ ≤ pr Q q ′′ ζ . F ourth, the pa r allel of ( f )( δ ) holds for ( q ′′ ζ , p ∗ ) by ( e )( δ ). Third, as q ′′ ξ , q ′′′ ξ are ≤ pr Q -compatible there is q ζ = q ξ +1 as required in clause (g). So we can satisfy ⊛ 5 . Now we apply clause (h) of Deﬁnition 1.2(1) to the sequence h ( q ε , p ε ) : ε < θ i hence there is ζ < θ as there, so a s p ε ∈ ap Q ( q ε ) the conditions p ε , q ε are compatible in Q hence they hav e a common upp er b ound r ∈ Q hence by the choice of h ( p ε , q ε ) : ε < θ i above, r  Q “ ξ ∗ ≤ o tp { ε < ζ : q ε , p ε ∈ G ˜ Q } ”. So r  Q “the sequence h ( α ε , β ε ) : ε < ζ and q ε , p ε ∈ G ˜ Q i is as required” noting that: • if ι ∗ ≥ 0 , then q ζ +1 | = “ c ˜ { α ε , β ζ } ” = ι ∗ for ε ≤ ζ , • if ι ∗ = 0 , then q ζ +1 | = “ c ˜ { α ε , α ζ } = ι ∗ for ε ≤ ζ . So we are done.  1 . 4 12 SAHARON S HELAH 2. Many strong polarized p ar tition rela tions W e ca n b elow say mo re o n strongly inaccessible θ ∈ Θ. Hyp othesis 2.1. Let p = ( λ, µ, Θ , ¯ ∂ ) s atisfy: ( a ) λ = λ <λ < µ = µ <µ , ( b ) Θ ⊆ [ λ, µ ] is a set of regular cardinals with λ, µ ∈ Θ , ( c ) ¯ ∂ = h ∂ θ : θ ∈ Θ i is an increa sing seque nce o f car dinals suc h that: ( α ) ∂ θ = cf( ∂ θ ) , ( β ) ∂ θ = ( ∂ θ ) <∂ θ , ( γ ) ∂ θ ≤ θ and if θ < κ ar e fro m Θ then ∂ θ < ∂ κ , ( δ ) ∂ θ ≥ κ if κ ∈ (Θ ∩ θ ) , ( ε ) if θ = λ then ∂ θ = λ . ( d ) notation: let θ , Υ , κ v ary on Θ . The reader may concentrate on (see 3.4): Example 2. 2. A ss ume ( a ) V satisﬁes G.C.H. from λ to µ , i.e., ∂ ∈ [ λ, µ ) ⇒ 2 ∂ = ∂ + , ( b ) λ = λ <λ < µ = µ <µ , ( c ) Θ := { θ + : λ ≤ θ < µ } ∪ { λ, µ } and, ( d ) ∂ θ = θ for every θ ∈ Θ, so in 2.4(5) b elow in this example we hav e ∂ θ = min { θ + , µ } . F or the rest of this section p , i.e. λ, µ, Θ , ¯ ∂ a r e ﬁxe d. Example 2. 3. A s above by Θ : = { κ : κ = λ or κ = µ or κ = λ ω (1+ α )+1 < µ for some n < } Deﬁnition 2.4. 1) F or κ ∈ Θ, let E κ be the equiv alenc e relation on µ deﬁned by ( ∗ ) iE κ j iﬀ i + κ = j + κ . 2) F o r any car dinal κ ∈ [ λ, µ ] deﬁne E <κ as Eq λ ∪ S { E θ : θ ∈ Θ ∩ κ } . F o r suc h κ , if κ / ∈ Θ , let E κ = E <κ . 3) F o r i < µ a nd κ ∈ Θ let [ i ] κ = i/E κ = the E κ -equiv a lence class of i , and for A ⊆ µ , let A/E κ = { i/E κ : i ∈ A } . F or i < µ, A ⊆ µ we say that i /E κ is represented in A iﬀ A ∩ ( i/E κ ) 6 = ∅ . If A ⊆ B ⊆ µ , we say that i/E κ grows from A to B iﬀ ∅ 6 = A ∩ ( i/E κ ) 6 = B ∩ ( i/E κ ). If w e write functions p, q instead of A, B we mean dom( p ), dom( q ) resp ectively . 4) Note that for all i, j < µ we have iE µ j . Thus, the follo wing deﬁnition mak es sense: if i, j are < µ we let κ ( i, j ) b e the minimal κ ∈ Θ such that iE κ j . 5) Suppos e κ ∈ Θ , let ∂ κ = min { ∂ θ : κ < θ ∈ Θ } if κ < µ and ∂ κ = µ if κ = µ. (Notice that κ is just a n index in ∂ κ , and this is no t car dinal ex ponentiation.) Thu s, in pa rticular, Observ ation 2.5 . 1) F or i, j < µ we have: κ ( i, j ) is wel l deﬁne d and for i, j < µ, θ ∈ [ λ, µ ) we have iE θ j ⇔ θ ≥ κ ( i, j ) as: MANY P AR TITION RELA TIONS BELO W DENSITY SH918 13 ( ∗ ) if θ < κ ar e b oth fr om Θ , then E θ r eﬁn es E κ and, in fact, e ach E κ - e quivalenc e class is the un ion of κ many E θ -e qu iva lenc e classes. 2a) If κ < θ ar e fr om Θ then ∂ κ ≤ ∂ θ ; use d in 2.9(1). 2b) ∂ θ < ∂ θ exc ept p ossibly for θ = µ (stil l ∂ µ ≤ µ = ∂ µ ); r e c al l 2.1(c) ( γ ) . 2c) s up(Θ ∩ κ ) ≤ ∂ κ for κ ∈ Θ ; r e c al l 2 .1 ( c )( δ ) 2d) ∂ θ = ( ∂ θ ) <∂ θ for θ ∈ Θ; re c al l 2.1(c) ( β ) , 2e) If κ ∈ Θ t hen e ach E <κ -e qu iva lenc e class h as c ar dinality ≤ ∂ κ (by (2c)); use d in the pr o of of 2.9(3)). 3a) ∂ λ = λ . 3b) I f θ < κ ar e suc c essive elements of Θ then ∂ θ = ∂ κ . 3c) I f κ ∈ Θ and S (Θ ∩ κ ) is a singular c ar dinal, then ∂ κ ≥ ( S (Θ ∩ κ )) + . Deﬁnition 2.6. 1) The forcing notion Q : = Q p = ( Q , ≤ Q ) = ( Q p , ≤ Q p ), but we may omit p when c lear fro m the co n text, is deﬁned by: ( A ) q ∈ Q iﬀ ( a ) q is a (partial) function fro m µ to { 0 , 1 } , ( b ) if i < µ and κ ∈ Θ, then the cardinalit y o f ( i/E κ ) ∩ dom( q ) is < ∂ κ (note: taking κ = µ , the cardina lity o f dom( q ) is < ∂ µ ≤ µ ), ( B ) p ≤ q o r p ≤ Q q iﬀ ( a ) p ⊆ q , i.e. dom( p ) ⊆ dom( q ) and α ∈ dom( p ) ⇒ p ( α ) = q ( α ) ( b ) for every θ ∈ Θ the s et { A ∈ µ/E θ : A g rows from p to q } has ca rdi- nality < ∂ θ . 2) F or κ ∈ Θ \ { µ } and p, q ∈ Q , let: ( A ) p ≤ pr p ,κ q or p ≤ pr κ q iﬀ : ( a ) p ≤ q and, ( b ) no E κ -equiv a lence cla ss grows fr om p to q . ( B ) p ≤ ap p ,κ q or p ≤ ap κ q iﬀ : ( a ) p ≤ q , ( b ) dom( q ) /E κ = dom( p ) /E κ . 3) F or κ = µ and p, q ∈ Q , let: ( A ) p ≤ pr µ q iﬀ p = q . ( B ) p ≤ ap µ q iﬀ p ≤ q . 4) Let Q κ = Q p ,κ = ( Q, ≤ Q , ≤ pr κ , ap κ ) where ap κ = ap p ,κ is the function with domain Q such that a p κ ( q ) = { q ′ : q ≤ ap κ q ′ } ; so Q κ as a forcing notion is Q . 5) Let ≤ us p ,κ = ≤ us κ = ≤ p be ≤ Q p for κ ∈ Θ. R emark 2.7 . Clearly Q κ is r elated to § 1, and if κ is the las t member of Θ ∩ µ we can use it (enough if Θ = { λ, µ } , but not in gener al, so we shall use a v ar ian t). Claim 2.8. Conc erning Deﬁnition 2.6 ( a ) ( α ) if κ ∈ Θ , then ≤ , ≤ pr κ , ≤ ap κ ar e p artial or derings of Q , ( β ) p ≤ pr κ q ⇒ p ≤ q and p ≤ ap κ q ⇒ p ≤ q , ( γ ) if κ = µ then ≤ ap κ = ≤ , ( δ ) if κ = µ then ≤ pr κ is the e quality. 14 SAHARON S HELAH ( b ) ( α ) if p 1 , p 2 ∈ Q and they ar e c omp atible as fu n ctions, then p 1 ∪ p 2 ∈ Q ; ( β ) mor e over, letting q = p 1 ∪ p 2 , if clause (b) of 2.6(1)(B) holds b etwe en p k and q , for k = 1 , 2 , then q is the lub, in Q , of p 1 and p 2 . ( c ) if p ≤ q a nd κ ∈ Θ , then ther e ar e r, s ∈ Q su ch that: ( α ) p ≤ pr κ r ≤ ap κ q , ( β ) p ≤ ap κ s ≤ pr κ q , ( γ ) q = r ∪ s , ( δ ) q i s the ≤ -lub of r , s . ( d ) if q ∈ Q then : ( α ) ∅ ≤ q (and ∅ , the empty fun ction, ∈ Q p ), ( β ) ( ∀ r )( q ≤ r ≡ q ≤ ap µ r ) , ( γ ) κ ∈ Θ \ { µ } ⇒ ∅ ≤ pr κ q , ( δ ) ∅ 6 = q ⇒ ∅  ap κ q fo r any κ ∈ Θ \{ µ } . ( e ) if κ 1 ≤ κ 2 ar e b oth fr om Θ , then : ≤ pr κ 2 ⊆≤ pr κ 1 and ≤ ap κ 1 ⊆≤ ap κ 2 , ( f ) if κ ∈ Θ and p ≤ ap κ q and p ≤ pr κ r , t hen : ( α ) q ∪ r is a wel l deﬁne d function ∈ Q, ( β ) p ≤ ( q ∪ r ) , ( γ ) q ≤ pr κ ( q ∪ r ) , ( δ ) r ≤ ap κ ( q ∪ r ) , ( ε ) q ∪ r is a ≤ -lu b of q , r in Q p . ( g ) if κ ∈ Θ , p ≤ pr κ q i ( i = 1 , 2) and q 1 , q 2 ar e c omp atible in Q (even just as functions), then p ≤ pr κ ( q 1 ∪ q 2 ) , ( h ) if p ≤ ap κ q k for k = 1 , 2 , and q 1 , q 2 ar e c omp atible in Q (even just as functions), then q k ≤ ap κ ( q 1 ∪ q 2 ) for k = 1 , 2 , ( i ) ( α ) if { p ε : ε < ζ } has an ≤ -u pp er b ound then ∪{ p ε : ε < ζ } is an upp er b ound, ( β ) similarly for ≤ pr κ , ≤ ap κ , ( γ ) assume p ε ∈ Q for every ε < ζ , and p ε , p ξ has a c ommon ≤ x κ -upp er b ound for any ε, ξ < ζ ; then the union of { p ε : ε < ζ } is a ≤ x κ -lub when x = us,a p and ζ < λ ( δ ) if { p ε : ε < ζ } ⊆ Q has a c ommon ≤ pr κ -upp er b ound and ζ < ∂ κ , then { p ε : ε < ζ } has a ≤ pr κ -lub - t he union ( j ) if p ≤ ap κ q then Dom( q ) \ Dom( p ) has c ar dinality < ∂ κ , ( k ) if p 1 ≤ ap κ p 3 and p 1 ≤ p 2 ≤ p 3 then p 1 ≤ ap κ p 2 and p 2 ≤ ap κ p 3 , ( l ) if p 1 ≤ pr κ p 2 , p ℓ ≤ ap κ q ℓ for ℓ = 1 , 2 and q 1 ∪ q 2 is a function, t hen q := q 1 ∪ q 2 is a ≤ -lub of q 1 , q 2 and q 2 ≤ ap κ q , q 1 ≤ q , ( m ) assume p 1 , p 2 ar e c omp atible in Q then ther e is a p air ( q , t ) such that: • 1 p 1 ≤ pr κ q , • 2 p 2 ≤ ap κ t , • 3 q  “ t ∈ G ˜ ⇒ p 1 ∈ G ˜ ” , • 4 q , t ar e c omp atible and we say ( q , t ) is a witness for ( p 1 , p 2 ) . ( n ) if h p ℓ α : α < δ i is ≤ pr κ -incr e asing for ℓ = 1 , 2 , δ a limit or dinal of c oﬁnality < ∂ κ and α < δ ⇒ p 1 α ≤ ap κ p 2 α then S α<δ p 1 α ≤ ap κ ( S α<δ p 2 α ) . Pr o of. Straig h tforward. E.g. MANY P AR TITION RELA TIONS BELO W DENSITY SH918 15 Clause (i) : So a ssume x ∈ { us, pr, ap } and κ ∈ Θ and { p ε : ε < ζ } ⊆ Q and q ∈ Q is an ≤ x κ -upp e r bo und of { p ε : ε < ζ } . Let p := ∪{ p ε : ε < ζ } t hen we shall pr o ve that p ∈ Q and p is a ≤ x κ -upp e r bo und of { p ε : ε < ζ } ; this clea rly suﬃces for proving sub-clauses ( α ) , ( β ) o f clause (i), and the ≤ x κ -lub pa r t, i.e. s ub-clauses ( γ ) , ( δ ) are left to the reader ; for ( γ ) , ( δ ), see 2.9(1B),(1A). Now ( ∗ ) 1 p is a well deﬁned function with domain ⊆ µ and p ⊆ q . [Wh y? As ε < ζ ⇒ p ε ⊆ q , i.e. a s functions (by 2.6(1)(B)(a)) clear ly p ⊆ q , as functions, so p is a w ell deﬁned function with domain ⊆ dom( q ) but dom( q ) ⊆ µ by 2.6(A)(a).] ( ∗ ) 2 if i < µ and θ ∈ Θ then the cardinality of ( i/E θ ) ∩ dom( p ) is < ∂ θ . [Wh y? Recall p ⊆ q ∈ Q , see abov e so as q ∈ Q b y 2.6(1)(a) w e ha ve | ( i/E θ ) ∩ dom( p ) | ≤ | ( i/E θ ) ∩ dom( q ) | < ∂ θ .] ( ∗ ) 3 p ∈ Q . [Wh y? By ( ∗ ) 1 + ( ∗ ) 2 recalling 2.6(1)(A).] ( ∗ ) 4 p ε ⊆ p for ε < ζ . [Wh y? By the choice of p .] ( ∗ ) 5 if ε < ζ and θ ∈ Θ then { A ∈ µ/E θ : A grows from p ε to p } has car dinalit y < ∂ θ . [Wh y? Be c ause, reca lling p ⊆ q , this set is included in { A ∈ µ/E θ : A grows from p ε to q } whic h has ca rdinalit y < ∂ θ bec ause p ε ≤ q which holds as p ε ≤ x κ q .] ( ∗ ) 6 p ε ≤ p for ε < ζ . [Wh y? By ( ∗ ) 4 + ( ∗ ) 5 recalling 2.6(1)(B).] ( ∗ ) 7 if x = us then p is a ≤ -upp er bo und of { p ε : ε < ζ } . [Wh y? By ( ∗ ) 3 + ( ∗ ) 6 .] ( ∗ ) 8 if x = pr and ε < ζ then p ε ≤ pr κ p . [Wh y? If κ = µ then ≤ pr κ is eq ualit y and p ε ≤ pr κ q hence p ε = q but p ε ⊆ p ⊆ q hence p ε = p s o this is trivial, hence assume κ < µ . W e h ave to check 2.6(2)(A), now clause (a) there holds by ( ∗ ) 6 and c la use (b) there holds as no E κ -equiv a lence class grows from p ε to q (as p ε ≤ pr κ q ) and p ⊆ q .] ( ∗ ) 9 if x = pr then p is a ≤ x κ -upp e r bo und of { p ε : ε < ζ } . [Wh y? By ( ∗ ) 8 .] ( ∗ ) 10 if x = ap and ε < ζ then p ε ≤ ap κ p . [Wh y? If κ = µ then ≤ ap κ = ≤ us κ and we are done b y ( ∗ ) 7 . Assume κ < µ . W e have to chec k 2.6(2)(B). First, claus e (a ) there holds by ( ∗ ) 6 . Second, clause (b) there holds b ecause if A ∈ dom( p ) /E κ then A ∩ dom( p ) 6 = ∅ b y the deﬁnition, hence A ∩ dom( q ) 6 = ∅ a s p ⊆ q b y ( ∗ ) 1 , but this implies A ∩ Dom( p ε ) 6 = ∅ b ecause p ε ≤ ap κ q , as required.] 16 SAHARON S HELAH ( ∗ ) 11 if x = ap then p is a ≤ x κ -upp e r b ound of { p ε : ε < ζ } . [Wh y? By ( ∗ ) 10 .] The ≤ x κ -lub parts are easy to o, for a limit ordinal δ see 2.9(1A). Clause (j): Let U = { A : A ∈ µ/E κ and A grows from p to q } . Recalling Deﬁnition 2 .6(1)(B)(b) clearly , as p ≤ q , we hav e |U | < ∂ κ . But as p ≤ ap κ q necessarily dom( q ) \ dom( p ) is included in ∪{ A : A ∈ U } . Also as q ∈ Q b y Deﬁnition 2 .6(1)(A )(b) w e have A ∈ U ⇒ | A ∩ dom( q ) | < ∂ κ . So dom( q ) \ dom( p ) is included in ∪{ A ∩ dom( q ) : A ∈ U } , a unio n o f < ∂ κ sets each of cardinality < ∂ κ . But ∂ κ is regular by 2.1 (C)( β ), so we ar e done.] Clause (m) : As p 1 , p 2 are compatible in Q , there is r ∈ Q such that p 1 ≤ r, p 2 ≤ r . Cho ose t = ∪{ r ↾ ( i/E κ ) : i/E κ grow from p 2 to r } ∪ p 2 , so t ∈ Q and p 2 ≤ ap κ t ≤ pr κ r . Cho ose q = ∪{ r ↾ ( i/E κ ) : i/E κ do es no t grow from p 1 to r } ∪ r , so q ∈ Q and p 1 ≤ pr κ q ≤ apr κ r . Now chec k.  2 . 8 Claim 2.9. L et κ ∈ Θ . 1) ( Q, ≤ pr κ ) is ( < ∂ κ ) -c omplete and in fact i f ¯ p = h p α : α < δ i is < pr κ -incr e asing, δ a limit o r dinal < ∂ κ then p δ := ∪{ p α : α < δ } is a ≤ pr κ -lub and a ≤ - lub of ¯ p ; we use κ < θ ∈ Θ ⇒ ∂ κ ≤ ∂ θ , se e 2.5(2a). 1A) If γ ( ∗ ) < ∂ κ and p α ∈ Q f or α < γ ( ∗ ) and p α , p β has a c ommon ≤ pr κ -lub f or any α, β < γ ( ∗ ) t hen p ∗ = ∪{ p α : α < γ ( ∗ ) } is a ≤ pr κ -lub of { p α : α < γ ( ∗ ) } . 1B) If γ ( ∗ ) < λ then (1A) holds for ≤ ap κ . 2) If k ∈ Θ and p ∈ Q then Q p,k := Q p ,p,k = ( { q : p ≤ ap κ q } , < ap κ ) satisﬁes 4 the ( ∂ κ ) + -c.c. 3) Mor e over if h p α : α < ∂ + κ i i s ≤ pr κ -incr e asing c ontinuous and p α ≤ ap κ q α for α < ∂ + κ , then for some α < β the c onditions q α , q β ar e c omp atible in Q , mor e over ther e is r such that q α ≤ r and q β ≤ ap κ r and p α = p β ⇒ q α ≤ ap κ r ∧ q β ≤ ap κ r . 4) Assu me p ∈ Q p , χ = | A | < ∂ κ , κ ∈ Θ and p  “ f ˜ is a function f r om A ∈ V to V ”. Then we c an ﬁnd q su ch that: ( α ) p ≤ pr κ q , ( β ) if a ∈ A then I q,f ˜ ,a := { r : q ≤ ap κ r and r for c es a value to f ˜ ( a ) } is pr e dense over q i n Q q , ( γ ) mor e over some subset I ′ q,f ˜ ,a of I q,f ˜ ,a of c ar dinality ≤ ∂ κ is pr e dense over q in Q q , (r e al ly fol lows). Pr o of. 1) By (1 A). 1A) Let q α,β be a common ≤ pr κ -upp e r b ound of p α , p β for α, β < γ ( ∗ ). Why is p ∗ ∈ Q ? Let us chec k Deﬁnition 2.6(1)(A). Clearly p ∗ is a partia l function from µ to { 0 , 1 } so clause (a) there holds. F or chec king clause (b) there, assume θ ∈ Θ and A ∈ µ/E θ . First, assume θ ≤ κ and A ∩ dom( p ∗ ) 6 = ∅ then for so me α < γ ( ∗ ) w e hav e A ∩ dom( p α ) 6 = ∅ , hence A ∩ dom( p ∗ ) = ∪{ A ∩ Dom( p β ) : β < γ ( ∗ ) } ⊆ ∪{ A ∩ 4 compare with [SS01, 1.8] MANY P AR TITION RELA TIONS BELO W DENSITY SH918 17 dom( q α,β ) : β < γ ( ∗ ) } , but p α ≤ pr κ q α,β and A ∩ dom( p α ) 6 = ∅ hence A ∩ dom( q α,β ) = A ∩ Dom( p α ). T og ether A ∩ dom( p ∗ ) is equa l to A ∩ dom( p α ) which, b ecause p α ∈ Q , has cardinality < ∂ θ as required in clause (b) of Deﬁnition 2.6(1)(A). Second, of cours e, if A ∩ dom( p ∗ ) = ∅ this holds, to o. Third, assume θ > κ , then α < γ ( ∗ ) ⇒ p α ∈ Q ⇒ | A ∩ dom( p α ) | < ∂ θ , hence | A ∩ dom( p ∗ ) | = | A ∩ S α<γ ( ∗ ) dom( p α ) | ≤ P α<γ ( ∗ ) | A ∩ do m( p α ) | whic h is < ∂ θ as γ ( ∗ ) < ∂ κ ≤ ∂ θ = cf( ∂ θ ), s o ag ain the desir ed conclusion of clause (b) of Deﬁnition 2.6(1)(A) holds. T og ether indeed p ∗ ∈ Q . Wh y α < γ ( ∗ ) ⇒ p α ≤ p ∗ ? W e hav e to chec k 2.6(1)(B), obviously clause (a) there holds. Clause (b) there is prov ed as ab o ve. Wh y α < γ ( ∗ ) ⇒ p α ≤ pr κ p ∗ ? W e hav e to chec k Deﬁnition 2.6(2)(A), now clause (a) there was just prov ed a nd cla us e (b) there holds as in the pro of of “ p ∗ ∈ Q ”. Next we show that p ∗ is a ≤ pr κ -lub of ¯ p , so a ssume q ∈ Q and α < δ ⇒ p α ≤ pr κ q . T o show p ∗ ≤ pr κ q w e have to c heck clauses (B)(a),(b) of 2.6(1) and ( A)(b) of 2.6(2). As p ∗ = ∪{ p α : α < γ ( ∗ ) } , clear ly p ∗ ⊆ q as a function so 2.6(1)(B)(a) ab ov e holds. Also if A ∈ µ/E κ and A is repr esen ted in p ∗ then it is r e presen ted in p α for some α < γ ( ∗ ), but p α ≤ pr κ q so q ↾ A = p α ↾ A but ( p α ↾ A ) ⊆ ( p ∗ ↾ A ) ⊆ ( q ↾ A ) hence q ↾ A = p ∗ ↾ A a s r equired in 2 .6(2)(A) (b). Lastly , when θ ∈ Θ, 2.6(1)(B)(b) holds: if θ ≤ κ b ecause mo re was just prov ed and if θ > κ it is proved as in the pro of of p ∗ ∈ Q . 2) This is a sp ecial case of (3) when h p α : α < ∂ + κ i is constant (re c a lling 2.8(h)). 3) So in particular p i ≤ ap κ q i for i < ∂ + κ . Hence b y clause (j) of Claim 2.8 the set u i := d om( q i ) \ dom( p i ) has ca rdinalit y < ∂ κ . Hence by the ∆-system lemma (recalling that ( ∂ κ ) <∂ κ = ∂ κ by 2.1(c)( β )) for so me un b ounded U ⊆ ∂ + κ the sequence h u i : i ∈ U i is a ∆- system, with hear t u ∗ . Moreov er, since 2 | u ∗ | ≤ ∂ <∂ κ κ = ∂ κ < ∂ + κ , we can assume that q i ↾ u ∗ = q ∗ for every i ∈ U . As ea c h E <κ -class has cardinality ≤ ∂ κ (see 2.5(2)(c),(e)), without loss of g enerality for ev ery i 6 = j from U , if α ∈ u i \ u ∗ then α/E <κ is disjoint to u j . Now by 2.8(h) for ev ery i, j ∈ U , the function q = q i ∪ q j is a ≤ ap κ -lub of q i , q j for part (2), i.e. when p i = p j . Also it is easy to chec k that for i < j, q is a ≤ - lub of q i , q j which is ≤ ap κ -ab o ve q j for part (3). 4) If κ = µ then ≤ ap κ = ≤ by clause 2.8(a)( γ ), reca ll Q p = ( { q ∈ Q : p ≤ q } , ≤ Q p ) so q = p can serve, as Q p satisﬁes the ∂ + κ -c.c. b y par t (2); s o we sha ll assume κ < µ . Recall that ∂ κ < ∂ κ by 2.5(2)(b). As | A | < ∂ κ = cf( ∂ κ ), b y part (1) o f the claim and clause (f ) o f Claim 2.8 it is enough to consider the ca se A = { a } . Now w e try to choo s e p i , r i , b i by induction on i < ∂ + κ , but r i , b i are chosen in stage i + 1 together with p i +1 , such that: ⊛ ( a ) p 0 = p , ( b ) h p j : j ≤ i i is ≤ pr κ -increasing , ( c ) p i +1 ≤ ap κ r i , ( d ) p i +1  “if r i ∈ G ˜ Q then f ˜ ( a ) = b i ”, ( e ) p i +1  “if r i ∈ G ˜ Q then for no j < i do we have r j ∈ G ˜ Q ”, ( f ) if i is a limit, then p i is the union so a ≤ pr κ -lub of h p j : j < i i . F or i = 0 just use clause (a) of ⊛ . 18 SAHARON S HELAH F or i limit use clause (f ) of ⊛ recalling part (1) of the claim and the fact tha t ∂ + κ ≤ ∂ k . F or i = j + 1 , try to c ho ose q i such that: p j ≤ q i and q i  “ r i 1 / ∈ G ˜ Q for i 1 < j ” . If w e cannot, we hav e s uc c eeded, i.e. p i is as required from q with I p i ,f ˜ ,a = { p i ∪ r j : j < i } . If we can, let ( b j , r j ) b e such t hat q i ≤ r j and r j forces f ˜ ( a ) = b j ; clearly poss ible . By clause (c) of Claim 2.8 a pplied to the pa ir ( p j , r j ) w e ch o ose 5 p i such that p j ≤ pr κ p i ≤ ap κ r j and clearly we hav e ca rried o ut the induction. But if we carry the induction then we ge t a contradiction by part (3). So we hav e to b e stuck for some i < ∂ + κ , and as s aid ab o ve we then get the desir ed co nclusion.  2 . 9 Conclusion 2 .10 . F orcing with Q p : ( a ) do es not collapse cardinals except p ossibly cardinals from the set Ω p = { θ : λ < θ ≤ µ and for no κ ∈ Θ do w e hav e ∂ κ < θ ≤ ∂ κ } , so µ / ∈ Ω p , ( b ) doe s not change coﬁnalities / ∈ Ω p , moreov e r if it ch ang es the coﬁnality o f θ ∈ Reg to χ < θ then there is θ 1 ∈ Ω p such that χ ≤ θ < θ 1 , moreov er, [ χ, θ 1 ] ∩ Reg ⊆ Ω p , ( c ) do es not add new sequences of le ngth < λ , ( d ) do es not c hang e 2 θ for θ / ∈ [ λ, µ ), ( e ) makes 2 λ = µ , ( f ) also the s e t Ω ′ p := ∪{ ( κ 1 , 2 sup(Θ ∩ κ ) ]: for s ome κ ∈ Θ , Θ ∩ κ has no last mem b er, so sup(Θ ∩ κ ) is stro ng limit and κ 1 = min(Reg \ sup(Θ ∩ κ )) } , is O.K. in clauses (a ),(b), ( g ) Q p has car dinalit y µ and satisﬁes the ∂ + µ -c.c., recalling ∂ µ ≤ µ . Pr o of. First, Q p is ( < λ )-complete hence it adds no new sequences to λ> V , i.e. clause (c) holds so cardinals ≤ λ are preser v ed a s well as coﬁnalities ≤ λ as well as 2 θ for θ < λ . Second, | Q p | = µ as p ∈ Q p ⇒ p is a function from dom( p ) ⊆ µ to { 0 , 1 } , see 2.6(1)(A)(a) and | dom( p ) | < ∂ µ = µ b y 2.6(1)(A)(b) and µ <µ = µ b y ?? (a). Third, by 2.9(2) the forcing no tion Q p satisﬁes the ∂ + µ -c.c. but Q = Q p when p = ∅ so Q satisﬁe s the ∂ + µ -c.c. and of course ∂ µ ≤ µ . This gives clauses (g) and (d) (recalling (c)). F ourth, for clause (e), for any α < µ let η ˜ α ∈ λ 2 b e deﬁned by p  “ η α ( i ) = ℓ ” iﬀ i < λ ∧ ˆ α + i ∈ do m( p ) ∧ ˆ ℓ = p ( α + i ). By densit y indeed  Q “ η ˜ α ∈ λ 2” and  Q “ η ˜ α 6 = η ˜ β ” for α 6 = β < µ , so clearly clause (e) holds. Fifth, use 2.9(2),(4) to prov e clauses (a) and (b), to ward co n tradiction assume θ is regular in V and θ 1 is no t in Ω p but p  Q “ χ = cf ( θ ) < θ 1 ≤ θ ”. If θ ≤ λ or just χ < λ use clause (c), if θ > µ use clause (g) so necessar ily λ ≤ χ < θ 1 ≤ θ ≤ µ . By the choice o f Ω p there is κ ∈ Θ such that ∂ κ < θ 1 ≤ ∂ κ and χ + ∂ κ < θ 1 ≤ θ ; now without loss of g enerality p  “ f ˜ : χ → θ has range un b ounded in θ ”. Apply 5 we can use r ′ j suc h that p j ≤ ap κ r ′ j ≤ pr κ r j suc h that r j is the ≤ -lub of r ′ j , p i +1 , ma y be helpful but not needed no w. MANY P AR TITION RELA TIONS BELO W DENSITY SH918 19 2.9(4) with ( p, χ, f ˜ , κ ) here standing for ( p, A, f ˜ , κ ) there a nd get q , hI q,f ˜ ,α : α < χ i as there. By 2.9(3) w e hav e |I q,f ˜ ,α | ≤ ∂ κ and ∪{I q,f ˜ ,α : α < χ } ha s cardinality ≤ χ + ∂ κ < θ 1 . In a n y ca se, in V the set { β : for s ome α < χ and q 1 “ f ˜ ( α ) 6 = β ” } has cardinality < θ 1 ≤ θ , contradiction. So clauses (a),(b) holds. W e are left with clause (f ), it is not r eally needed, still nice to hav e. Now if θ ∈ Reg ∩ ( λ, µ ] is in Ω ′ p and κ witness it then neces sarily Θ ∩ κ , w hich is not empty has no last element so if θ 1 < θ 2 are fro m Θ ∩ θ then θ 1 ≤ ∂ θ 2 = ( ∂ θ 2 ) <∂ θ 2 ≤ θ 2 hence sup(Θ ∩ θ ) is strong limit. If θ = κ use claus e (b). If θ ≥ 2 <κ we repeat the pr o ofs a bov e for ≤ pr <κ where ≤ pr <κ = ∩{≤ pr θ : θ ∈ Θ ∩ θ } , ≤ ap <κ = { ( p, q ) : p ≤ q and α ∈ dom( p ) \ do m( p ) ⇒ ( ∃ θ ∈ Θ ∩ θ )(( α/E θ ∩ do m ( p )) 6 = ∅} .  2 . 10 Deﬁnition 2.11. 1 ) If p ≤ q a nd κ ∈ Θ let supp κ ( p, q ) := ∪{ i/E κ : i ∈ dom( q ) \ dom( p ) } so of cardinality < ∂ κ . 2) W e say y = h κ, ¯ p, ¯ u i = h κ y , ¯ p y , ¯ u y i is a rea sonable p -par ameter when : ⊛ 1 ( a ) κ ∈ Θ but κ < µ, ( b ) θ = θ y = min(Θ \ κ + y ) , notice that θ is well d eﬁned, as κ y < µ and µ ∈ Θ, ( c ) ¯ p = h p α : α < γ i is a non-empty ≤ pr θ -increasing contin uous sequence, so w e wr ite γ = γ y , ¯ p = ¯ p y and p α = p y α , ( c ) ¯ u = h u α : α < γ i is ⊆ -increas ing contin uous, so u α = u y α , ¯ u = ¯ u y , ( d ) u α ⊆ ∪{ i /E κ : i ∈ Dom( p α ) } for α < γ , ( e ) | u α | ≤ ∂ κ for α < γ . 3) F o r y as a bov e we deﬁne Q y as ( Q y , ≤ y , ≤ pr y , ap y ) (so Q y = ( Q y , ≤ y ) is Q y as a forcing notion), where: ⊛ 2 ( a ) Q y := { q : fo r some α < γ y we have p α ≤ ap θ q and supp θ ( p α , q ) ⊆ u α } , ( b ) ≤ y = ≤ p ↾ Q y , ( c ) for q ∈ Q y , let α y ( q ) = min { α < γ y : p α ≤ ap θ q and supp θ ( p α , q ) ⊆ u α } , ( d ) the tw o-place relatio n ≤ pr y is deﬁned by p ≤ pr y q iﬀ: ( α ) p, q ∈ Q y , ( β ) p ≤ pr p ,κ q . ( e ) for q ∈ Q y let ap y ( q ) = ap Q y ( q ) = { r ∈ Q y : q ≤ ap κ r and supp κ ( q , r ) ⊆ s upp θ ( p α y ( q ) , q ) } . Observ ation 2.12. L et y b e a r e asonable p -p ar ameter. 0) If p 1 ≤ p 2 ≤ q 2 ≤ q 1 and κ 1 ≥ κ 2 ar e fr om Θ then supp κ 2 ( p 2 , q 2 ) ⊆ supp κ 1 ( p 1 , q 1 ) . 0A) If p 1 ≤ p 2 ≤ p 3 then supp κ 1 ( p 1 , p 3 ) = supp κ 1 ( p 1 , p 2 ) ∪ supp κ 1 ( p 2 , p 3 ) . 1) F or q ∈ Q y the or dinal α y ( q ) is wel l de ﬁne d < γ y . 2) If q 1 ≤ y q 2 ar e fr om Q y then α y ( q 1 ) ≤ α y ( q 2 ) . 2A) If q 1 ∈ Q y and q 1 ≤ ap p ,κ q 2 then q 2 ∈ Q y , q 1 ≤ y q 2 and α y ( q 1 ) = α y ( q 2 ) . 3) If p ≤ pr y r and q ∈ ap y ( p ) then s := q ∪ r b elongs t o Q y , s ∈ ap y ( r ) and q ≤ pr y s . Pr o of. 0), 0A) Should be ea sy . 20 SAHARON S HELAH 1) By the deﬁnitions of q ∈ Q y and of α y ( q ). 2) F or ℓ = 1 , 2 letting α ℓ = α y ( q ℓ ) we ha ve p α ℓ ≤ ap θ q ℓ ∧ supp θ ( p α ℓ , q ℓ ) ⊆ u α ℓ . If α 2 < α 1 then p α 2 ≤ p α 1 ≤ ap θ q 1 ≤ q 2 ∧ p α 2 ≤ ap θ q 2 hence p α 2 ≤ ap θ q 1 (b y 2.8(k)) and supp θ ( p α 2 , q 1 ) ⊆ supp θ ( p α 2 , q 2 ) ⊆ u α 2 by the deﬁnition of 2.11(1) of supp, contradicting the choice of α 1 . 2A) W e kno w p α y ( q 1 ) ≤ ap p ,κ q 1 by the deﬁnition o f α y ( q 1 ) but we assume q 1 ≤ ap p ,κ q 2 and ≤ pr p ,κ is a quasi order hence p α y ( q 1 ) ≤ ap p ,κ q . So b y the deﬁnition q 2 ∈ Q y ∧ α y ( q 1 ) ≥ α y ( q 2 ). Also clear ly q 1 ≤ p q 2 hence q 1 ≤ y q 2 hence b y part (2), α y ( q 1 ) ≤ α y ( q ∗ ), together we a re done. 3) Let κ = κ y and θ = θ y , p α = p y α . By Deﬁnition 2.1 1(3)(e),(f ) we kno w that p ≤ pr p ,κ r and p ≤ ap p ,κ q . By Claim 2.8(f ) w e k no w that s ∈ Q p and p ≤ ap p ,κ q ≤ pr p ,κ s and p ≤ pr p ,κ r ≤ ap p ,κ s recalling s = q ∪ r , no te ( ∗ ) 1 the ordinal β := α y ( r ) < γ y is w ell deﬁned. [Wh y? As r ∈ Q y .] ( ∗ ) 2 α y ( s ) = α y ( r ) = β . [Wh y? As p ∈ Q y the or dinal α := α y ( p ) < γ y is well deﬁned and by part (2) we hav e α ≤ β . So c learly p β ≤ ap p ,θ r b y the choice o f β and r ≤ ap p ,κ s as said ab ov e, hence by ?? (e) reca lling κ < θ , we have ≤ ap p ,κ ⊆≤ ap θ hence r ≤ ap p ,θ s , so together p β ≤ ap p ,θ s . Also s = q ∪ r hence supp θ ( r , s ) ⊆ supp θ ( p, q ) and as q ∈ ap y ( p ) necessarily p ≤ ap p ,κ q hence p ≤ ap p ,θ q hence by part (2A) supp θ ( p, q ) ⊆ supp θ ( p α , q ) ⊆ u y α y ( q ) = u y α y ( p ) = u y α but u y α ⊆ u y β as α ≤ β . T o gether supp θ ( r , s ) ⊆ u β , and by the choice of β clear ly supp θ ( p β , r ) ⊆ u β hence supp θ ( p β , s ) ⊆ supp θ ( p β , r ) ∪ supp θ ( r , s ) ⊆ u β ∪ u β = u β . As w e hav e sho wn earlier that p β ≤ ap p ,θ s it follows that s ∈ Q y and α y ( s ) ≤ β . But r ≤ p s hence by part (2) we know that β = α y ( r ) ≤ α y ( s ) so neces sarily α y ( s ) = α y ( r ) = β , i.e. ( ∗ ) holds .] So p α y ( s ) ≤ ap p ,θ s and supp θ ( p α y ( s ) , s ) = supp θ ( p β , s ) ⊆ u β = u α y ( s ) so together s ∈ Q y , the ﬁrst s tatemen t in the conclusio n. Also q ≤ pr y s , for this check ( e )( α ) + ( β ) o f Deﬁnition 2.11(3); for clause ( α ): q ∈ Q y is a ssumed, s ∈ Q y was just pr o ved; for clause ( β ) “ q ≤ pr p ,κ s ” was prov ed in the beg inning o f the pro of; s o the third statemen t in the conclusion holds. Lastly , we c he ck that s ∈ ap y ( r ), for this w e hav e to check the tw o demands in 2.11( 3)(f ), no w “ s ∈ Q y ” w as pro ved ab ov e, “ r ≤ ap p ,κ s ” w as pro ved in the beg inning o f the pro of and “supp κ ( r , s ) ⊆ supp θ ( p α y ( r ) , s )” holds as supp κ ( r , s ) ⊆ supp θ ( r , s ) ⊆ supp θ ( p α y ( r ) , s ) = supp θ ( p β , s ) = supp θ ( p α y ( s ) , s ) is as require d.  2 . 12 Claim 2 .13. 1) Assume κ < θ ar e suc c essive memb ers of Θ p and ( ∀ α < ∂ θ )( | α | <∂ κ < ∂ θ ) and y is a r e asonable p -p ar ameter, κ = κ y henc e θ y = θ and ¯ p y is ≤ pr θ - incr e asing (henc e also ≤ pr κ -incr e asing) and γ y is a suc c essor or a li mit or dinal of c oﬁnality ≥ ∂ θ . Then Q y is a ( ∂ + θ , ∂ θ , < ∂ θ ) -for cing. 2) If in addition γ y = α ∗ + 1 then p α ∗  “ G ˜ Q ∩ Q y is a su bset of Q y generic over V ” . MANY P AR TITION RELA TIONS BELO W DENSITY SH918 21 Pr o of. 1) W e sho uld chec k fo r Q = Q y (deﬁned in 2 .11) each of the cla us es of Deﬁnition 1.2. Let p α = p y α , u α = u p α . Clause (a) : T rivia l, just Q y has the right fo rm, a qua druple. Clause (b) : ( Q y , ≤ y ) is a forcing no tion. Wh y? By ⊛ 2 ( b ) + ( c ) from 2 .11 (3), i.e. Q y is a non-empty subset of Q p bec ause γ y > 0 so p y 0 = p ∈ Q y and ≤ y being ≤ Q p ↾ Q y is a quasi or der. Clause (c) : ≤ pr y is a quasi or der on Q y and p ≤ pr y q ⇒ p ≤ y q ⇒ p ≤ p q . Wh y? The ﬁrst half holds b e c ause if p 1 ≤ pr y p 2 ≤ pr y p 3 then : we should chec k that p 1 ≤ pr y p 3 , i.e. clauses ( α ) , ( β ) o f ⊛ 2 ( e ) of 2.11(3) hold. N ow clause ( α ) is obvious, for clause ( β ) note p 1 ≤ pr p ,κ p 2 ≤ pr p ,κ p 3 and ≤ pr p ,κ is a pa rtial or der of Q p , so p 1 ≤ pr p ,κ p 3 , and hence ( β ) ho lds. The second part of clause (c) whic h sa ys p ≤ pr y q ⇒ p ≤ y q (rec a lling Claim 2.8(a)( β )) holds by the deﬁnition of ≤ y , ≤ pr y in ⊛ 2 ( c ) , ( e ) of 2 .11( 3). Clause (d)( α ): ap y is a function with doma in Q y . Wh y? By ⊛ 2 ( f ) of 2.11(3). Clause (d)( β ) : if q ∈ Q y then q ∈ ap y ( q ) ⊆ Q y . Wh y? B y ⊛ 2 ( f ) of 2.11(3) trivia lly ap y ( q ) ⊆ Q y . Also we can c heck that q ∈ ap y ( q ) : q ∈ Q y by an a ssumption and q ≤ ap κ q as ≤ ap κ is a quasi order o n Q p and “supp κ ( q , q ) ⊆ supp θ ( p α y ( q ) , q )” trivially b ecause supp κ ( q , q ) = ∅ . Clause (d)( γ ) : if r ∈ ap y ( q ) and q ∈ Q y then r , q are compatible in Q y . Wh y? As r ∈ ap y ( q ) ⇒ ( q ≤ ap κ r ∧ { r, q } ⊆ Q y ) ⇒ q ≤ y r . Clause (d)( γ ) + : if r ∈ ap y ( q ) and q ≤ pr y q + then q + , r ar e co mpa tible in ( Q y , ≤ y ), moreov er there is r + ∈ a p Q y ( q + ) such that q +  Q y “ r + ∈ G ˜ Q y ⇒ r ∈ G ˜ Q y ”. This follows fro m 2.1 2(3), by deﬁning s = r + = r ∪ q + , whic h g iv es more. Clause (e) : ( Q y , ≤ pr y ) is ( < ∂ θ )-complete, reca lling ∂ θ = ∂ κ . So assume h q ε : ε < δ i is ≤ pr y -increasing and δ is a limit ordinal < ∂ θ ; now ( Q p , ≤ pr κ ) is ( < ∂ κ )-complete by Claim 2.9(1 ) and h q ε : ε < δ i is also ≤ pr p ,κ -increasing by clause ⊛ 2 ( e )( β ) o f Deﬁnition 2.11(3) hence q δ := ∪{ q ε : ε < δ } is a ≤ pr p ,κ -lub of the seque nc e b y 2.9(1). No w h α ε := α y ( q ε ) : ε < δ i is an ≤ -increasing sequence of ordinals < γ y by Obser v ation 2.12(2). Also by an a ssumption o f 2.1 3(1), the ordinal γ y is a s uccessor or dinal or limit of coﬁnality ≥ ∂ θ but then δ < cf( γ y ). So in bo th c a ses α ∗ = sup { α ε : ε < δ } is an ordinal < γ y . B ut ¯ p y is ≤ pr p ,κ -increasing contin uo us hence p α ∗ = ∪{ p α ε : ε < δ } and s imilarly u α ∗ = ∪{ u α ε : ε < δ } . No w easily q δ is a ≤ ap θ -extension of p y α ∗ , and supp θ ( p y α ∗ , q δ ) ⊆ ∪{ supp θ ( p α y ( q ε ) , q ε ) : ε < δ } ⊆ ∪{ u α ε : ε < δ } = u α ∗ which has cardinality < ∂ θ set each hence q δ ∈ Q y . Easily q δ is as required. Clause (f ) : ( Q y , ≤ pr y ) satisﬁes the ∂ + θ -c.c. Wh y? Let q ε ∈ Q y for ε < ∂ + θ , so α ε := α y ( q ε ) is well de ﬁned and without loss of generality h α ε : ε < ∂ + θ i is constant or increa sing; also p α ε ≤ ap θ q ε so by Deﬁnition 2.6 the set s upp θ ( p α y ( q ε ) , q ε ) has ca rdinalit y < ∂ θ , so by the ∆-sys tem lemma, as in the pro of of 2.9(3) there are ε (1) < ε (2) < ∂ + θ such tha t: ( ∗ ) if i 1 ∈ supp θ ( p α ε (1) , q ε (1) ) and i 2 ∈ supp θ ( p α ε (2) , q ε (2) ) then 22 SAHARON S HELAH ( α ) if i 1 = i 2 then q ε (1) ( i ) = q ε (2) ( i ) ( β ) if i 1 E κ i 2 then i 1 , i 2 ∈ supp θ ( p α ε (1) , q ε (1) ) ∩ supp θ ( p α ε (2) , q ε (2) ). So ε (1) < ε (2) , α ε (1) ≤ α ε (2) , p α ε (1) ≤ ap θ q ε (1) , p α ε (2) ≤ ap θ q ε (2) . Hence q := q ε (1) ∪ q ε (2) belo ngs to Q p is a ≤ ap θ -lub of { q ε (1) , q ε (2) } and q α ε (2) ≤ ap θ q hence q ∈ Q y . Also if i ∈ dom( q ) \ Dom( p ε ( ℓ ) ) then i/E κ is disjoint to dom( p ε ( ℓ ) ) by ( ∗ )( β ); this implies p ε ( ℓ ) ≤ pr κ q which means p ε ( ℓ ) ≤ pr y q by 2.11(3)(e), for ℓ = 1 , 2 so q ε (1) , q ε (2) are indeed commpatible in ( Q y , ≤ pr y ). Clause (g) : if ¯ q = h q ε : ε < ∂ θ i is ≤ pr y -increasing , then for stationa rily many limit ζ < ∂ θ the sequence ¯ q ↾ ζ has an exact ≤ pr y -upp e r b ound (recalling that ∂ θ here stands for θ in Deﬁnition 1.2). Wh y? W e pr o ve more, that if cf( ζ ) = ∂ κ and h q ε : ε < ζ i is ≤ pr y -increasing then the union q = ∪{ q ε : ε < ζ } is an ex a ct ≤ pr y -upp e r b ound. This suﬃces as ∂ κ < ∂ θ and b oth are regula r. N ow by 2.12(2) the sequence h α y ( q ε ) : ε < ζ i is ≤ - increasing hence h u α y ( q ε ) : ε < ζ i is ⊆ -increa sing and letting α ∗ = ∪{ α y ( q ε ) : ε < ζ } we hav e α ∗ < γ y as γ y is a successo r ordinal o r limit of co ﬁnalit y ≥ ∂ θ ; hence u α ∗ = ∪{ u α y ( q ε ) : ε < ζ } , see 2.11(2)(c). By the pro of o f c lause (e) which we have proved ab ov e, cle arly q ∈ Q y and is a ≤ pr y -upp e r bo und of h q ε : ε < ζ i . But what a bout “exact”? w e should check Deﬁnition 1.2(2). So ass ume p ∈ ap y ( q ) a nd we should prove that for some ε < ζ and p ′ ∈ a p y ( q ε ) we hav e  Q y “if q , p ′ ∈ G ˜ Q y then p ∈ G ˜ Q y ”. Note that q ≤ ap p ,κ p and supp θ ( q , p ) ⊆ u α ∗ by the deﬁnit ion of a p y ( q ), hence u := supp κ ( q , p ) is a subset of supp θ ( q , p ) ⊆ u y α ∗ of ca rdinalit y < ∂ κ . As h u y α ε : ε < ζ i is ⊆ -increasing with union u y α ∗ necessarily for some ε < ζ we have u ⊆ u α ε . Let p ′ = p ↾ dom( p ε ), and chec k (as in earlier cases ). Clause (h) : if h q ε : ε < ∂ θ i is ≤ pr y -increasing and r ε ∈ ap y ( q ε ) for ε < ∂ θ and ξ < ∂ θ then for some ζ < ∂ θ we hav e q ζ  Q y “if r ζ ∈ G ˜ Q y then ξ ≤ otp { ε < ζ : p ε ∈ G ˜ Q y } ”. This follows fro m 2.9(3). Clause (i) : ap y ( q ) has cardinality < ∂ θ . Should be clear as α < ∂ θ ⇒ | α | <∂ κ < ∂ θ by an assumption of the claim and α < ∂ θ ⇒ | u α | < ∂ θ (see 2.11(3)(f )) a nd the deﬁnition o f ap y ( q ) in ⊛ 2 ( e ) of 2.11(3). Let α = α y ( q ) so α < γ y and | a p y ( q ) | = |{ s : q ≤ ap κ s and supp κ ( q , s ) ⊆ sup θ ( p α y ( q ) , q ) }| ≤ | s upp θ ( p α y ( q ) , q ) | <κ but | supp θ ( p α y ( q ) , q ) | < ∂ θ and s o by a n assumption of the claim | supp θ ( p α y ( q ) , q ) | <κ < ∂ θ so we are do ne. Clause (j) : Let q ∗ ≤ y r , so α ≤ β where α := α y ( q ∗ ) , β := α y ( r ). By 2.8(c) we can ﬁnd a pair ( q , p ) such tha t q ∗ ≤ pr p ,κ q ≤ ap p ,κ r , q ∗ ≤ ap p ,κ p ≤ pr p ,κ r , r = p ∪ q . Now chec k. 2) Let Q ′′ = { p : p α ∗ ≤ ap θ p } . So clearly Q ′′ ⊆ Q y and then ( ∀ p ∈ Q y )( ∃ q ∈ Q ′′ )[ p ≤ y q ], b y claus e (f ) of Claim 2.8, i.e. Q ′′ is a dens e subse t of Q y (b y ≤ Q y = ≤ Q p ↾ Q y ). Really q 1 ∈ Q ′′ ∧ q 1 ≤ q 2 ∈ Q y ⇒ q 2 ∈ Q ′′ by 2.12(2). Suppo se I is a dens e op en subset o f Q y so I 1 := I ∩ Q ′′ is dense in Q y . MANY P AR TITION RELA TIONS BELO W DENSITY SH918 23 Let G b e a subse t o f Q g eneric over V such that p α ∗ belo ngs to it. If I ∩ G 6 = ∅ w e are done, otherwise some q 1 ∈ G is incompatible (in Q ) with every q ∈ I . As G is directed there is q 2 ∈ G s uch that p α ∗ ≤ q 2 ∧ q 1 ≤ q 2 . As p α ∗ ≤ q 2 by cla use (c) of Claim 2.8 there is a r 2 ∈ Q suc h that p α ∗ ≤ ap θ r 2 ≤ pr θ q 2 . So r 2 ∈ Q ′′ hence by the as sumption on I there is r 3 ∈ I such that r 2 ≤ r 3 . Now as r 3 ∈ I necessarily p α ∗ ≤ ap θ r 3 and of course p α ∗ ≤ r 2 ≤ r 3 hence b y claus e (k) of Claim 2.8 we ha ve r 2 ≤ ap θ r 3 . Recalling r 2 ≤ pr κ q 2 it follows by clause (f ) o f 2.8 that there is q 3 ∈ Q such that q 2 ≤ q 3 ∧ r 3 ≤ q 3 hence q 3  “ G ˜ ∩ I 6 = ∅ ” a nd q 1 ≤ q 3 , contradicting the choice of q 1 .  2 . 13 Claim 2.14. If κ ∈ Θ \{ µ } , θ = min(Θ \ κ + ) and θ = µ ⇒ ∂ θ < µ and ( ∀ α < ∂ θ )[ | α | <∂ κ < ∂ θ ] and ξ < ∂ θ , σ < ∂ θ then  Q p “ ∂ + θ → ( ξ , ( ξ ; ξ ) σ ) 2 ” . Pr o of. Let σ < ∂ θ and ξ < ∂ θ and we shall prov e  Q p “ ∂ + θ → ( ξ , ( ξ ; ξ ) σ ) 2 ”. T ow ard this as sume c ˜ is a Q p -name and q ∗ ∈ Q p forces that c ˜ is a function from [ ∂ + θ ] 2 to 1 + σ . Now we sha ll apply Claim 2.9(4) with θ here standing for κ ther e. W e choose ( p i , u i ) by induction on i < ∂ + θ such that: ⊛ 1 ( a ) p i ∈ Q p is ≤ pr θ -increasing contin uous with i and p 0 = q ∗ (so if θ = µ, then V i p i = q ∗ bec ause ≤ pr θ is equality), ( b ) for every i < j < ∂ + θ the set I i,j is predense ab o ve p j +1 where I i,j = { r : p j +1 ≤ ap θ r and r forces a v alue to c ˜ { i, j }} , ( c ) more over I i,j has a subset I ′ i,j of cardina lit y ≤ ∂ θ which is predense ov er p j +1 , ( d ) u i is ⊆ -increas ing contin uous and u i ⊆ ∪{ α/E κ : α ∈ Dom( p i ) } and | u i | ≤ ∂ θ for i < ∂ + θ , ( e ) α ∈ u i ⇒ ( α/E κ ) ⊆ u i , ( f ) q ∈ I ′ i,j ⇒ supp κ ( p j +1 , q ) ⊆ u j +1 . [Wh y is this p ossible? F or i = 0 let p 0 = q ∗ , for i limit let u i = ∪{ u j : j < i } a nd i < ∂ + θ , a nd we like to applly 2.9(1) with κ there standing for θ her e, so if ∂ + θ ≤ ∂ θ this is ﬁne, o therwise b y 2.5(2)(h) necessa rily θ = µ ∧ ∂ θ = µ = 2 θ contradicting an assumption. Lastly , if i = ι + 1 then w e ha ve to deal with c ˜ { ζ , ι } for ζ < ι , i.e. with ≤ ∂ θ names of o rdinals < σ . So we apply 2.9(4) with ( p ι , ι, h c ˜ ( j, ι ) : j < ι i , θ ) here standing for ( p, A, f ˜ , κ ) there and get p i , hI j,ι , I ′ j,ι : j < ι i here standing for q , hI q,f ˜ ,a , I ′ q,f ˜ ,a : a ∈ A i there . So the relev a n t parts of clauses (a),(b),(c) hold. Deﬁne u i as in clauses (d),(e),(f ) p ossible as |I ′ j,ι | ≤ ∂ θ , r ∈ I ′ j,ι ⇒ | supp κ ( p i , q ) | ≤ ∂ κ < ∂ θ . So w e a re done car rying the induction.] Let ¯ p = h p i : i < ∂ + θ i and ¯ u = h u i : i < ∂ + θ i . So this will help to tra nslate the problem from the forcing Q to the for c ing Q y . W e deﬁne y = ( κ, h p α : α < ∂ + θ i , h u α : α < ∂ + θ i ), so: ⊛ 2 y is a reasonable p -parameter . [Wh y? Check, see Deﬁnition 2.11(2).] ⊛ 3 Q y is a ( ∂ + θ , ∂ θ , < ∂ θ )-forcing. [Wh y? By Claim 2.13(1).] Now for i < j < ∂ + θ , 24 SAHARON S HELAH ( ∗ ) ( a ) I i,j is predense in Q y , ( b ) if q 1 , q 2 ∈ I i,j or just ∈ Q y , then q 1 , q 2 are compatible in Q p iﬀ they are compatible in Q y . [Wh y? The ﬁrst clause (a) holds b y our deﬁnitions. F or the sec o nd clause (b), assume q 1 , q 2 ∈ Q y . If they ar e compatible in Q y , then clear ly they are co mpatible in Q p . T o show the other direction, let q be q 1 ∪ q 2 . If q ∈ Q y we are done, since q 1 , q 2 ≤ y q . So le t us pro ve that q ∈ Q y . Denote α 1 = α y ( q 1 ) , α 2 = α y ( q 2 ) and without loss of gener a lit y α 1 ≤ α 2 . So p α 1 ≤ ap θ q 1 , p α 2 ≤ ap θ q 2 and also p α 1 ≤ pr θ p α 2 , a nd it follows from 2.8(f )( δ ) that p α 2 ≤ ap θ q . Moreov er , supp θ ( p α 2 , q ) ⊆ supp θ ( p α 1 , q 1 ) ∪ supp θ ( p α 2 , q 2 ) ⊆ u α 1 ∪ u α 2 = u α 2 . T o gether, q ∈ Q y and we are done.] So we can deﬁne a Q y -name c ˜ ′ as follows; fo r q ∈ Q y q  Q y “ c ˜ ′ { i, j } = t ” iﬀ q  Q p “ c ˜ { i, j } = t ” . So b y ( ∗ )  Q y “ c ˜ ′ : [ ∂ + θ ] 2 → σ ” . Now by cla im 1.4 fo r some Q y -name and a sequence h α ˜ ε , β ˜ ε : ε < ξ i w e hav e  Q y “ the sequence h α ˜ ε , β ˜ ε : ε < ξ i is as required in Deﬁnition 0.3 (for ∂ + θ → ( ξ , ( ξ ; ξ ) σ ) 2 )” . So for each ε < ξ there is a maximal antic hain J ε of Q y of elements forcing a v alue to ( α ˜ ε , β ˜ ε ) by Q y . But Q y satisﬁes the ∂ + θ -c.c. so |J ε | ≤ ∂ θ hence for some α ∗ < ∂ + θ we hav e: ( ∗ ) J ε ⊆ { q : ( ∃ α ≤ α ∗ )( p α ≤ ap Q q ) } for any ε < ξ . Recall that (by 2.1 3) ( ∗ ) p α ∗  “ G ˜ Q ∩ Q y ↾ ( α ∗ +1) is a subset of Q y ↾ ( α ∗ +1) generic ov er V ”. so we ar e done.  2 . 14 R emark 2.15 . 1 ) W e can replace th e expone nt 2 by n ≥ 2, so g e tting suitable po larized pa rtition r elations; w e intend to contin ue elsewhere. 2) F or exa ct such results prov able in ZFC see [EHMR84] a nd [She81 ]. MANY P AR TITION RELA TIONS BELO W DENSITY SH918 25 3. Simul t aneous P ar tition Rela tions and General topology Recall (to s implify r esults we deﬁne hL + ( X ) > λ > cf ( λ ) using a n elab orate deﬁ- nition for reg ula rs). Deﬁnition 3.1. Let X b e a top ological space. W e deﬁne: ( a ) the density o f X is d ( X ) : = min {| S | : S ⊆ X and S is dense in X } , ( b ) the her e ditary density o f X is: hd( X ) = sup { λ : X has a subspace of densit y ≥ λ } , ( c ) hd + ( X ) = c hd( X ) = sup { λ + : X has a subspace o f densit y ≥ λ } , ( d ) X is not λ -Lindel¨ o f if there is a family {U α : α < λ } o f op en subsets of X whose union is X but w ⊆ λ ∧ | w | < λ ⇒ ∪{U α : α ∈ w } 6 = X , ( e ) the hereditarily Lindel¨ of num b er o f X is: hL( X ) = c hL( X ) = sup { λ : there ar e x α ∈ X and U α ∈ op en( X ) for α < λ , such that x α ∈ U α and α < β ⇒ x β / ∈ U α } , ( f ) hL + ( X ) = sup { λ + : there are x α ∈ X , U α for α < λ as ab ov e } , ( g ) the spread of X is s ( X ) = sup { λ : X has a disc r ete s ubs e t with λ po ints } , (h) s + ( X ) = ˆ s ( X ) = sup { λ + : X has a discre te subspa c e with λ p oints } . Our starting p oin t was the following problem (0.1) of Juhasz-Shelah [JS08]. Problem 3.2. Assume ℵ 1 < λ < 2 ℵ 0 . Do es there exist a hereditarily Lindel¨ of Hausdorﬀ regula r space of density λ ? W e a nsw er negatively b y a consistency res ult but then lo ok a gain at related prob- lems on hereditary densit y , Lindel¨ ofness and spread; our main theorem is 3.10 getting consistency for all car dinals. W e also try to clar ify the relatio nships of this and related partition r elations to χ → [ θ ] 2 2 κ, 2 , recalling that b y [She88], consisten tly , e.g. 2 ℵ 0 → [ ℵ 1 ] 2 n, 2 for n < ω . Now, s e e 3.1 3 b elow, 2 ℵ 0 → [ ℵ 1 ] 2 n, 2 implies 2 ℵ 0 → ( ℵ 1 , ( ℵ 1 ; ℵ 1 ) n ) 2 and b y 3.14 it implies γ < ℵ 1 ⇒ 2 ℵ 0 → ( γ ) 2 n , see o n the consistency of this B aumgartner-Ha jnal in [BH73], and Galvin in [Gal7 5]. On cardinal inv aria n ts in genera l top ology , in pa rticular, s ( X ),hd( X ),hL( X ), see Juhasz [Juh80]; in particula r reca ll the o b vious. Observ ation 3.3. F or a Haus dorﬀ top olo gic al sp ac e X : ( a ) hL( X ) ≥ s ( X ) , ( b ) hd( X ) ≥ s ( X ) , ( c ) for λ re gular, X is her e ditarily λ -Lindel¨ of (i.e. every subsp ac e is λ -Lindelof ) iﬀ ther e ar e x α ∈ X and U α for α < λ as in (e) of Deﬁnition 3.1, ( d ) we cho ose the se c ond statement in (c) as the deﬁnition of “ X is her e ditarily λ -Lindel¨ of” t hen 3.7, 3.9 b elow hold also for λ singular. Conclusion 3 .4 . Assume λ = λ <λ < µ = µ <µ and GCH holds in [ λ, µ ], s o λ ≤ θ = cf( θ ) ≤ µ ⇒ θ = θ <θ and { λ, µ } ⊆ Θ ⊆ Reg ∩ [ λ, µ ] and for θ ∈ Θ we let ∂ θ = θ and let p = ( λ, µ, Θ , h ∂ θ : θ ∈ Θ i ). Then ( a ) p is a s requir e d in Hyp othesis 2 .1, ( b ) the forcing notion Q p satisﬁes: ( α ) Q p is of cardinality µ ( β ) Q p is ( < λ )-complete (hence no new sequence of length < λ is added), 26 SAHARON S HELAH ( γ ) no ca rdinal is collapsed, no coﬁnality is changed, ( δ ) in V Q p we hav e λ = λ <λ , 2 λ = µ and χ / ∈ [ λ, µ ) ⇒ 2 χ = (2 χ ) V , ( ε ) if κ < θ ar e succe ssiv e mem b ers o f Θ a nd θ is not a s uccessor of singular (or just θ = χ + ⇒ χ <κ = χ ) then λ → ( ξ , ( ξ ; ξ ) σ ) 2 for any ξ , σ < θ . Pr o of. By 2.1 0 and 2 .14.  3 . 4 The topo logical consequences from 3.4 in 3.5 hold by 3.7 and 3.9 b elo w, that is Conclusion 3 .5 . W e can add in 3.4 that ( b )( ζ ) if θ ∈ [ λ, µ ) ∩ Θ is the successor of the regular κ then for any Hausdor ﬀ regular top ological space X , we ha ve hd( X ) ≥ θ + ⇒ s + ( X ) ≥ θ and also hL( X ) ≥ θ + ⇒ s + ( X ) ≥ θ so recalling θ = κ + we hav e hd( X ) ≥ θ + ⇒ hL( X ) ≥ s ( X ) ≥ κ, hL( X ) ≥ θ + ⇒ hd( X ) ≥ s ( X ) ≥ κ ( η ) if θ ∈ ( λ, µ ] is a limit cardinal then hd( X ) ≥ θ ∨ hL( X ) ≥ θ ⇒ s ( X ) ≥ θ . Observ ation 3.6. 1) If λ 1 → ( ξ 1 ; ξ 1 ) 2 κ 1 and λ 2 ≥ λ 1 , ξ 2 ≤ ξ 1 , κ 2 ≤ κ 1 then λ 2 → ( ξ 2 ; ξ 2 ) 2 κ 2 . 1A) Similarly for λ → ( ξ , ( ξ ; ξ ) κ ) 2 . 2) If λ → ( ξ , ( ξ ; ξ ) κ ) 2 then λ → ( ξ ; ξ ) 2 1+ κ . 3) λ → ( ξ + ξ ; ξ + ξ ) 2 κ implies ( λ, λ ) → ( ξ , ξ ) 1 , 1 κ ; the p olarize d p artition. Claim 3.7. X has a discr ete subsp ac e o f size µ , i.e. s + ( X ) > µ (henc e is not her e ditarily µ -Lindel¨ of ) when : ( a ) λ → ( µ, ( µ ; µ )) 2 ( b ) X is a Hausdorﬀ, mor e over a r e gular (= T 3 ) top olo gic al sp ac e ( c ) X has a subsp ac e of density ≥ λ . R emark 3.8 . The pro ofs of 3.7, 3.9 ar e similar to older pro ofs. Pr o of. X has a subspace Y with density ≥ λ , b y clause (c) of the assumption. W e choose x α , C α by induction on α < λ such that ⊛ ( α ) x α ∈ Y , ( β ) C α = the closure of { x β : β < α } , ( γ ) x α / ∈ C α . This is p ossible as Y has densit y ≥ λ . Let u 1 α be an ope n neigh b orho o d of x α disjoint to C α . Let u 2 α be an ope n neighborho o d of x α whose clo s ure, cℓ ( u 2 α ) is ⊆ u 1 α . Why do es it exist? As X is a regular (= T 3 ) space. W e deﬁne c : [ λ ] 2 → { 0 , 1 } a s fo llo ws: ( ∗ ) if α < β then c { α, β } = 1 iﬀ x β ∈ u 2 α . By the assumption λ → ( µ, ( µ ; µ )) 2 at least one of the following cases o ccurs. Case 1 : There is a n increasing sequence h α ε : ε < µ i of ordinals < λ such that ε < ζ < µ ⇒ c { α ε , α ζ } = 0. This means t hat ε < ζ < µ ⇒ x α ζ / ∈ u 2 α ε . But if ε < ζ < µ then u 2 α ζ is an o pen neighborho o d o f x α ζ included in u 1 α ζ which is disjoint to C α ζ and x α ε ∈ C α ζ so x α ε / ∈ u 2 α ζ . MANY P AR TITION RELA TIONS BELO W DENSITY SH918 27 Lastly , x α ε ∈ u 2 α ε by the choice of u 2 α ε . T ogether we ar e done, i.e. h ( x α ε , u 2 α ε ) : ε < µ i is as required. Case 2 : There is a sequence h ( α ε , β ε ) : ε < µ i such that: ( ∗ ) 1 ε < ζ < µ ⇒ α ε < β ε < α ζ < λ. ( ∗ ) 2 ε < ζ ⇒ c { α ε , β ζ } = 1, really ε ≤ ζ suﬃce. So, ( ∗ ) 3 ε < ζ ⇒ x β ζ ∈ u 2 α ε . but now for every ε < µ let ( ∗ ) 4 y ε := x β 2 ε and u 3 ε := u 2 β 2 ε \ cℓ ( u 2 α 2 ε +1 ). So, ( a ) u 3 ε = u 2 β 2 ε \ cℓ ( u 2 α 2 ε +1 ) is op en (as op en min us clo sed). ( b ) y ε ∈ u 3 ε . [Wh y? Reca ll y ε = x β 2 ε belo ngs to u 2 β 2 ε (b y the choice of u 2 β 2 ε ) and not to u 1 α 2 ε +1 (as u 1 α 2 ε +1 is disjo in t to C α 2 ε +1 while x β 2 ε ∈ C α 2 ε +1 ) hence not to cℓ ( u 2 α 2 ε +1 ) b eing a subset of u 1 α 2 ε +1 . T ogether y ε belo ngs to u 2 β 2 ε \ cℓ ( u 2 α 2 ε +1 ) = u 3 ε .] ( c ) if ε < ζ < µ then y ζ / ∈ u 3 ε . [Wh y? Now y ζ = x β 2 ζ belo ngs to u 2 α 2 ε +1 by ( ∗ ) 3 as 2 ε + 1 < 2 ζ which follows fr om ε < ζ hence y ζ belo ngs to cℓ ( u 2 α 2 ε +1 ) hence y ζ / ∈ u 3 ε by the deﬁnition o f u 3 ε .] ( d ) if ζ < ε < µ then y ζ / ∈ u 3 ε . [Wh y? As u 3 ε ⊆ u 2 β 2 ε and the latter is disjoint to C β 2 ε to which x β 2 ζ = y ζ belo ngs.] T og ether h ( y ε , u 3 ε ) : ε < µ i e x empliﬁes that we are do ne .  3 . 7 Claim 3.9. X has a discr ete subsp ac e of size µ when : ( a ) λ → ( µ, ( µ ; µ )) 2 , ( b ) X is a Hausdorﬀ mor e over a r e gular (= T 3 ) top olo gic al sp ac e, ( c ) hL + ( X ) > λ , i.e. if λ is a r e gular c ar dinal this me ans that X is not her e ditarily λ -Lindel¨ of Pr o of. Similar to 3.7. W e choose h ( x α , u 1 α ) : α < λ i such that u 1 α is an o pen subset of X, x α ∈ u 1 α and u 1 α ∩ { x β : β ∈ ( α, λ ) } = ∅ . W e can choo se them as hL + ( X ) > λ . W e then choo se an op en neigh b orho o d u 2 α of x α such that cℓ ( u 2 α ) ⊆ u 1 α . W e then deﬁne c : [ λ ] 2 → { 0 , 1 } a s fo llo ws ( ∗ ) if α < β then c { α, β } = 1 iﬀ x α ∈ u 2 β . W e co n tinue as in the pro of of 3.7, but now, in Case 2 ( ∗ ) ′ 3 ε < ζ ⇒ x α ε ∈ u 2 β ζ and let ( ∗ ) ′ 4 y ε := x α 2 ε , u 3 ε := u 2 α 2 ε \ cℓ ( u 2 β 2 ε +1 ).  3 . 9 Now we co me to our ma in result. 28 SAHARON S HELAH Theorem 3.1 0. T he Main The or em . It is c onsistent (using no lar ge c ar dinals) t ha t: ( ∗ ) ( α ) 2 µ is µ + if µ is str ong limit singular and always 2 µ is the su c c essor of a singular c ar dinal, ( β ) for every µ we have µ ≤ χ < 2 µ ⇒ 2 χ = 2 µ , ( γ ) hd( X ) ≥ θ ⇔ hL( X ) ≥ θ ⇔ s( X ) ≥ θ f or any limit c ar dinal θ and H ausdorﬀ r e gular (= T 3 ) top olo gic al sp ac e X, ( δ ) hd( X ) ≤ s( X ) +3 and hL( X ) ≤ s ( X ) +3 for any Hausdorﬀ r e gular (= T 3 ) top olo gic al sp ac e, ( ε ) in ( δ ) we c an r eplac e s( X ) +3 by s( X ) +2 exc ept when s ( X ) is r e gular, ( ζ ) in p articular, if X is a (Hausdorﬀ r e gu lar t op olo gic al sp ac e which is) Lindel¨ of or of c ountable density or ju s t s ( X ) = ℵ 0 then hd( X ) + hL( X ) ≤ ℵ 2 , ( η ) if X is a Ha us dorﬀ sp ac e 6 then | X | < 2 (hd( X ) + ) ( θ ) if X is a Hausdorﬀ sp ac e then w ( X ) ≤ 2 (hL( X ) + ) ( ι ) if 2 µ > µ + then µ ++ → ( ξ , ( ξ ; ξ ) µ ) 2 for ξ < µ + . R emark 3.11 . In the Theo rem 3.10 a bov e: 1) If we use less s harp results in § 1, § 2, § 3 we s ho uld a bov e just use (hd( X )) + n ( ∗ ) for large enough n ( ∗ ). 2) W e may like to improv e claus e ( η ) to ≤ 2 hd( X ) . If b elow we choo s e µ ε +1 strongly inaccessible (so w e need to assume V | = “there are un b oundedly many s tr ong inaccessible car dinals and cla use ( α ) is changed”), no thing is lost, we ha ve λ ε +1 = µ ε +1 then we can add: ( η ) + for any Hausdorﬀ space X , | X | < 2 hd( X ) except (p ossibly) when hd ( X ) is strong limit singula r. 3) Similarly for clause ( θ ) ab out w ( X ) ≤ 2 hL( X ) . 4) Pr obably using large cardinal we ca n eliminate also the ex ceptional case in ( η ) + ; it seemed that a similar situation is the one in Cummings-Shelah [CS95], but w e hav e no t loo k ed in to this. 5) W e ma y wonder whether in clause ( ζ ) we can replace ℵ 2 by ℵ 1 and s imilarly for other cardinals , hop efully see [S + ]. Pr o of. W e can assume V satisﬁes G.C.H. W e c ho ose h ( λ ε , µ ε ) : ε a n ordinal i s uc h that: ⊛ ( a ) λ 0 = µ 0 = ℵ 0 , ( b ) λ ε < cf( µ ε +1 ) < µ ε +1 , ( c ) λ ε +1 is the ﬁrst regular ≥ µ ε +1 , ( d ) for limit ε w e hav e λ ε is the ﬁrst reg ular cardina l ≥ µ ε := ∪{ λ ζ : ζ < ε } . Now let p ε = ( λ ε , λ ε +1 , Θ ε , ¯ ∂ ε ) where Θ ε , ¯ ∂ ε are deﬁned by Θ ε = Reg ∩ [ λ ε , λ ε +1 ] , ¯ ∂ ε = h ∂ ε θ : θ ∈ Θ ε i , ∂ ε θ = θ , so are chosen as in 3.4. So h p ε : ε an ordinal i is a class. W e deﬁne an Easton supp ort iteration h P ε , Q ˜ ε : ε ∈ Or d i so ∪{ P ε : ε ∈ Ord } is a class forcing, choos ing the P ε -name Q ˜ ε such tha t 6 is i nteresting b ecause usually 2 χ = 2 ( χ + ) , see clause ( α ) MANY P AR TITION RELA TIONS BELO W DENSITY SH918 29  P ε “ Q ˜ ε = Q p ε , i.e. Q ˜ ε is deﬁned as in Deﬁnition 2.6 fo r the parameter p ε (in the universe V P ε of cours e ” . As in V P ε section tw o is applicable for p ε so in V P ε +1 , the conclusions of 3.4, 3.5 hold a nd 2 λ ε = λ ε +1 so cardinal ar ithmetic should be clear, in particular, clause ( α ) holds. Of co urse, forcing with P ∞ / P ε +1 do es no t c hange those conclusio ns as it is λ ε +1 -complete. In V P ∞ we hav e eno ugh cases of θ + → ( ξ , ( ξ ; ξ )) 2 , i.e. clause ( γ ) by 2.14. So, ﬁr s t, if χ ≥ s( X ) b elongs to [ λ ε , µ ε +1 ) and is regular we hav e χ +2 → ( χ ; ( χ ; χ )) 2 and hd( X ), hL( X ) ≤ χ +2 . But if s ( X ) ∈ [ λ ε , µ ε +1 ) then s ( X ) + < µ ε +1 recalling µ ε is singular hence hd( X ) , hL( X ) ≤ s ( X ) +3 < µ ε +1 . Second, if χ = s ( X ) belongs to no such interv al then χ + = λ ε , χ = µ ε > cf( µ ε ) for some ε hence reca lling λ ε = λ <λ ε ε = 2 χ (in V P ∞ ) we hav e the conclus ion. So clause ( δ ) follows hence also clauses ( γ ) , ( ε ). Let us deal with clause ( η ), let χ = hd( X ). First, if χ ∈ [ λ ε , µ ε +1 ) we get hL( X ) ≤ χ +3 < µ ε +1 hence | X | ≤ 2 χ +3 = 2 χ by the classical inequality of de-Gro ot, ( | X | ≤ 2 hL( X ) ; see [Juh80]). Second, if χ b elongs to no such in terv al, then χ = µ ε ∧ χ + = λ ε , 2 µ ε = 2 χ for some ε . So | X | ≤ 2 2 hL( X ) ≤ 2 2 χ = 2 χ + as required. Clause ( θ ) is prov ed similarly .  3 . 10 Theorem 3.1 2 . If i n V ther e is a class of (str ongly) inac c essible c ar dinals, then in some for cing extension ( ∗ ) ( α ) 2 µ is µ + when µ is a st r ong limit singular c ar dinal and is a we akly inac c essible c ar dinal otherwise. ( ∗ ) ( β ) − ( ι ) as in The or em 3.10. Pr o of. As in the pro of of Theo r em 3 .10 .  Claim 3.13. Assu m e χ → [ θ ] 2 2 κ, 2 wher e κ ≥ 2 , χ ≤ 2 λ and λ = λ <λ < θ = cf( θ ) . Then χ → ( θ, ( θ ; θ ) κ ) 2 . Pr o of. Let c : [ χ ] 2 → κ be g iv en. Let η α ∈ λ 2 for α < χ be pairwise distinct. W e deﬁne d : [ χ ] 2 → 2 κ by: for α < β < χ let d { α, β } b e 2 ε + ℓ when c { α, β } = ε and ℓ = 1 iﬀ ℓ 6 = 0 iﬀ η α < lex η β (i.e. η α ( ℓg ( η α ∩ η β )) < η β ( ℓg ( η α ∩ η β )). As we are assuming χ → [ θ ] 2 2 κ, 2 there is U ∈ [ χ ] θ such that Rang( d ↾ [ U ] 2 ) has ≤ 2 members, without loss of genera lit y otp( U ) = θ . If the n um b er of members of Ra ng( d ↾ [ U ] 2 ) is one w e are done, so assume it is { 2 ε 0 + ℓ 0 , 2 ε 1 + ℓ 1 } where ε 0 , ε 1 < κ and ℓ 0 , ℓ 1 < 2. But w e cannot hav e ℓ 1 = ℓ 2 by the Sierpinsk i colour ing prop erties as θ > λ hence without loss o f generality ℓ 0 = 0 , ℓ 1 = 1 . If ε 0 = ε 1 = 0 we are done, as then Case ( c ) 0 of Deﬁnition 0.2(2) holds , so assume ℓ ∈ { 0 , 1 } ⇒ ε ℓ 6 = 0. Let Λ = { η ∈ λ> 2 : for θ o rdinals α ∈ U w e ha ve η ⊳ η α } . Now Λ has tw o ⊳ -incomparable members (o therwise we get a contradiction by cf( θ ) > λ ) say ν 0 , ν 1 ∈ Λ are ⊳ -incompa rable and witho ut lo s s of generality ν 0 < lex ν 1 . So, ( ∗ ) if ν 0 E η α and ν 1 ⊳ η β and α < β then c { α, β } = ε 0 ( ∗ ) if ν 1 ⊳ η α , ν 0 ⊳ η α and α < β then c { α, β } = ε 1 . 30 SAHARON S HELAH As θ is regular a nd otp( U ) = θ w e can cho ose α ε , β ε by induction on ε < θ such that: ⊙ ( a ) α ε ∈ U and α ε > sup { β ζ : ζ < ε } , ( b ) ν 0 < η α ε , ( c ) β ε ∈ U is > α ε , ( d ) ν 1 ⊳ η β α . So Case ( c ) 1 of Deﬁnition 0.2(2) ho lds . So we a re done.  3 . 13 W e ca n re ma rk als o Claim 3. 14. Assume λ = λ <λ < cf( θ ) and χ ≤ 2 λ and χ → [ θ ] 2 2 κ, 2 . Then for every or dinal γ < λ + we have χ → ( γ ) 2 κ . Pr o of. Without los s o f gener alit y κ ≥ 2. So let c : [ χ ] 2 → κ . Cho ose h η α : α < χ i and d a s in the pro of o f 3.13 and let U ⊆ χ of order t yp e θ and { 2 ε 0 , 2 ε 1 + 1 } b e as there so ε 0 , ε 1 < κ . As { η α : α ∈ U } is a subset of λ> 2 o f car dinalit y θ > λ = λ <λ clearly (e.g. prove by induction on γ < λ + that) for every such U there is U ′ ⊆ U of order type γ such that h η α : α ∈ U ′ i is < lex -increasing . So U ′ is as r equired, i.e. c ↾ [ { η α : α ∈ U ′ } ] 2 is constantly ε 0 (of c ourse also ε 1 is O.K . if we use < lex -decreasing seq uence).  3 . 14 R emark 3 .15 . If w e us e versions of χ → [ θ ] 2 κ, 2 with privilege p ositions for the v alue 0, we can get co rresp onding b etter results in 3.13, 3 .14 . MANY P AR TITION RELA TIONS BELO W DENSITY SH918 31 References [BH73] J. Baumgartn er and A. Ha j nal, A pr o of (involving martin ’s axiom) of a p artition r elation , Polsk a Ak ademia Nauk. F undamenta M athe maticae 78(3 ) (1973), 193–203. [CS95] James Cummings and Saharon Shelah, Car dinal invariants ab ove the c ontinuum , A nn. Pure Appl. Logic 75 (1995), no. 3, 251–268, arXi v: math/9509228. MR 1355135 [EH78] Paul Erd˝ os and A ndras Ha jnal, Emb e dding the or ems for gr aphs establishing ne gative p artition r e lations , P erio d.Math.Hungar. 9 (1978), 205–230. [EHMR84] Paul Erd˝ os, Andras Ha jnal, A . M at ´ e, and Richard Rado, Combinatorial set t he ory: Partition r elations for c ar dinals , Studies in Logic and the F oundation of Math., vol. 106, Nor th– Holl and Publ. Co, Am s terdam, 1984. [Gal75] F. Galvin, On a p artition the or e m of b aumgartner and hajnal , Inﬁnite and Fini te Sets (Colloq. , Keszthely , 1973; dedicated to P . Erd˝ os on his 60th Bi rthda y) (Amsterdam), Collo q. Math. Soc. J´ anos Bolya i, vol. 10, North-Holland, 1975, pp. 711–729. [JS08] Istv´ an Juh´ asz and Saharon Shelah, Her ed itarily Lindel¨ of sp ac es of singular den- sity , Studia Sci. Math. Hungar. 45 (2008), no. 4, 557–562, arXiv: math/0702295. MR 2641451 [Juh80] Istv an Juh´ asz, Car dinal functions in top olo gy —ten ye ars later. se c ond e dition , Math- ematical Centre T r act s, vol. 123, Mathematisc h Centrum, Amsterdam, 1980. [S + ] S. Shelah et al., Tb a , In preparation. Preliminary num b er: Sh:F884. [She81] Saharon Shelah, Canonization t he or e ms and applic ations , J. Symbolic Logic 46 (1981), no. 2, 345–353. M R 613287 [She88] , Was Sierpi´ nski right? I , Israel J. Math. 62 (1988 ), no. 3, 355–380. MR 955139 [She89] , Consistency of p ositiv e p artition t he or e ms for gr aphs and mo dels , Set theory and its appli cat ions (T oronto, ON, 1987), Lecture Notes in Math., vol. 1401, Springer, Berlin, 1989, pp. 167–193. MR 1031773 [She92] , St r ong p artition r elations b elow the p ower set: c onsiste nc y; was Sierpi ´ nski right? II , Sets, graphs and n umbers (Budapest, 1991) , Colloq. Math. Soc. J´ anos Bolya i, vol. 60, North-Holland, Amsterdam, 1992, arXi v: math/9201 244 , pp. 637–668. MR 1218224 [She96] , Was Sierpi´ nski right? III. Can co ntinuum-c.c. t i mes c.c.c. be c ontinuum- c.c.? , Ann. Pure Appl. Logic 7 8 (1996), no . 1-3, 25 9–269, arXiv: math/9509226. MR 1395402 [She00] , Was Sierpi´ nski right? IV , J. Sym b olic Logic 65 (2000), no. 3, 1031–105 4, arXiv: math/9712282. M R 1791363 [SS01] Saharon Shelah and Lee J. Stanley , F or ci ng many p ositiv e p olarize d p artition r elations b e twe en a c ar dinal and its p owerset , J. Sym b olic Logic 66 (2001) , no. 3, 1359–1370, arXiv: math/9710216. M R 1856747 Einstein Institute of Ma thema tics, Edmond J. Safra Campus, Giv a t Ra m, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel, and, Dep ar tment of Ma thema tics, Hill Center - Busch Cam pus, R utgers, The St a te University of New Jersey, 110 Frel- inghuysen Ro ad, Pisca t a w ay, NJ 08854-801 9 US A Email addr ess : shel ah@math.huji.a c.il URL : http://shela h.logic.at

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