Small universal families of graphs on ℵ ω+1
SSMALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 JAMES CUMMINGS, MIRNA DˇZAMONJA, AND CHARLES MORGAN
Abstract.
We prove that it is consistent that ℵ ω is strong limit,2 ℵ ω is large and the universality number for graphs on ℵ ω +1 issmall. The proof uses Prikry forcing with interleaved collapsing. Introduction If µ is an infinite cardinal, a universal graph on µ is a graph withvertex set µ which contains an isomorphic induced copy of every suchgraph. More generally, a family F of graphs on µ is jointly universal ifevery graph on µ is isomorphic to an induced subgraph of some graphin F . We denote by u µ the least size of a jointly universal family ofgraphs on µ , and record the easy remarks that u µ ≤ µ and that if u µ ≤ µ then u µ = 1. If µ = µ <µ , then by standard results in modeltheory there exists a saturated (and hence universal) graph on µ . Itfollows that under GCH and the hypothesis that µ is regular, u µ = 1.A standard idea in model theory (the construction of special models )shows that under GCH we have u µ = 1 for singular µ as well: wefix (cid:104) µ i : i < cf( µ ) (cid:105) a sequence of regular cardinals which is cofinal in µ , build a graph G which is the union of an increasing sequence ofinduced subgraphs G i where G i is a saturated graph on µ i , and argueby repeated applications of saturation that G is universal.Questions about the value of u µ when µ < µ <µ have been investigatedby several authors. We refer the reader to papers by Dˇzamonja andShelah [4, 3], Kojman and Shelah [6], Mekler [10] and Shelah [13].We will consider the case when µ is a successor cardinal κ + and2 κ > κ + . When κ is regular it is known that: James Cummings was partially supported by NSF grant DMS-1101156. MirnaDˇzamonja thanks EPSRC for their support through their grant EP/I00498 andLeverhulme Trust for a Research Fellowship for the period May 2014 to May 2015.Charles Morgan thanks EPSRC for their support through grant EP/I00498. Cum-mings, Dˇzamonja and Morgan thank the Institut Henri Poincar´e for their supportthrough the “Research in Paris” program during the period 24-29 June 2013. Theauthors thank Jacob Davis for his useful comments on draft versions of this paper. a r X i v : . [ m a t h . L O ] A ug JAMES CUMMINGS, MIRNA DˇZAMONJA, AND CHARLES MORGAN (1) It is possible to produce models where u κ + is arbitrarily large[6], for example by adding many Cohen subsets of κ over amodel of GCH.(2) It is possible to produce models where κ <κ = κ , 2 κ is arbitrarilylarge and u κ + = κ ++ [4] by iterated forcing over a model ofGCH.The question whether we can have u κ + = 1 when 2 κ > κ + remainsmysterious for general values of κ , though it is known [10, 13] to havea positive solution for κ = ω .When κ is singular then questions about u κ + become harder, sincewe have fewer forcing constructions available. Dˇzamonja and Shelah[4] found a line of attack on this kind of question, where the key ideais that we will prepare a large cardinal κ by means of iterated forcingwhich preserves its large cardinal character, and only at the end of theconstruction will we force to make κ become a singular cardinal. Bythis method Dˇzamonja and Shelah produced models where κ is singularstrong limit of cofinality ω , 2 κ is arbitrarily large and u κ + ≤ κ ++ .In [4] the final step in the construction is Prikry forcing, so that inthe final model κ is still rather large by some measures, for exampleit is still a cardinal fixed point. In this paper we will use a forcingposet defined by Foreman and Woodin [5] which will make κ become ℵ ω . In some joint work with Magidor and Shelah [2], we obtain similarresults where the final step is a form of Radin forcing which changesthe cofinality of κ to uncountable values such as ω .Our main result is this: it is consistent relative to a supercompactcardinal that ℵ ω is strong limit, 2 ℵ ω = ℵ ω +3 , and u ℵ ω +1 ≤ ℵ ω +2 . In therest of this Introduction we give an overview of the proof, and concludewith a guide to the structure of the paper.The Foreman-Woodin poset is a variation of Prikry forcing, whichadds a Prikry sequence κ i of inaccessible cardinals cofinal in κ , andin addition collapses all but finitely many cardinals between successivepoints on the Prikry sequence so that κ becomes ℵ ω . The only param-eter needed to define Prikry forcing is a normal measure U , but theForeman-Woodin forcing has an additional parameter F which is a fil-ter on the set of functions representing elements of a certain completeBoolean algebra in Ult( V, U ).We will start with a ground model V in which κ is a supercompactcardinal, which has been prepared so as to be indestructible under κ -directed closed forcing, and 2 κ = κ +3 . We will define an iterated forcingposet Q ∗ by iterating for κ +4 many steps with supports of size less than κ , forcing at each stage i with a poset Q i which is κ -directed closed and MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 has a strong form of κ + -cc. The cardinal κ will still be supercompactin V Q ∗ , and this will enable us to choose a normal measure U andfilter F , which can be used as parameters to define a Foreman-Woodinforcing P .The key idea is that the poset Q i will anticipate the results of forcingover V Q ∗ with P . To be more specific, at each stage i of the constructiona suitable form of diamond sequence will be used to produce “guesses” W i and F i at the final values of U and F , and there will be manystages i at which these guesses are correct (in the sense that W i and F i are the restrictions to V P i of U and F ).At stage i there is a poset P i which is computed from W i and F i in the same way that P is computed from U and F . If the guessesmade at stage i are correct then the final P -generic object will inducea P i -generic object. The poset Q i aims to add a P -name for a graph on κ + , whose interpretation absorbs all graphs in the extension of stage i by the induced P i -generic object.Our final model will be obtained by halting the construction at asuitable stage i ∗ of cofinality κ ++ , and forcing with P i ∗ . The pointhere (an idea which comes from [4]) is that we can read off a universalfamily of size κ ++ from a cofinal set of stages below i ∗ , and we are ina situation where 2 κ = κ +3 .We conclude this section with an overview of the paper and a coupleof remarks: • In Section 2 we discuss the filter F which is used in defining P and give an account of its main properties. • In Section 3 we construct the forcing P and prove various keyfacts about it using the properties of F . • In Section 4 we construct the “anticipation forcing” Q and provethat it has certain properties. Most notably Q is κ -compact andhas a strong form of the κ + -chain condition. • In Section 5 we describe the main iteration Q ∗ and prove a keytechnical fact by a master condition argument. • In Section 6 we prove the main theorem. • In Section 7 we discuss generalisations, related work and somenatural open problems.
Remark.
Foreman and Woodin’s paper [5] actually defines a supercom-pact Radin forcing with interleaved Cohen forcing, and its projectionto a Radin forcing with interleaved Cohen forcing controlled by certainfilters. Our forcing P here is a version of the projected forcing, withthe Cohen forcing replaced by collapsing forcing and the Radin forcingsimplified to the special case of Prikry forcing. P is also a close relative JAMES CUMMINGS, MIRNA DˇZAMONJA, AND CHARLES MORGAN of the forcing poset used by Woodin to obtain the failure of SCH at ℵ ω from optimal hypotheses, the difference being that in Woodin’s forcingposet the constraining filters are generic over the relevant ultrapowers.Our approach was dictated by the necessity to have the “approxima-tions” P i be well-behaved forcing posets, in a context where they canneither be obtained as projections of supercompact Prikry forcing withinterleaved collapsing nor constructed from filters which are genericover ultrapowers. Of course, all this work traces back ultimately toMagidor’s original model for the failure of SCH at ℵ ω [8].2. Constraints and filters
We start by assuming that 2 κ = κ + n for some n < ω and that κ is 2 κ -supercompact. We will fix U an ultrafilter on P κ κ + n witnessingthe 2 κ -supercompactness of κ , and let j : V → M = Ult( V, U ) be theassociated ultrapower map. We let U be the projection of U to anultrafilter on κ via the map x (cid:55)→ x ∩ κ . We remind the reader of somestandard facts.(1) U = { A ⊆ P κ κ + n : j “ κ + n ∈ j ( A ) } , and [ F ] U = j ( F )( j “ κ + n ) forevery function F with dom( F ) ∈ U .(2) U concentrates on the set of x ∈ P κ κ + n such that x ∩ κ is aninaccessible cardinal less than κ and ot( x ) = ( x ∩ κ ) + n . We willdenote this set by A good , and for x ∈ A good we let κ x = x ∩ κ and λ x = ot( x ).(3) U is a normal measure on κ , and U = { B ⊆ κ : κ ∈ j ( B ) } .We let j : V → M = Ult( V, U ) be the associated ultrapowermap, and note that [ f ] U = j ( f )( κ ) for every function f withdom( f ) ∈ U .(4) There is an elementary embedding k : M → M such that k ◦ j = j , which is given by the formula k : [ f ] U (cid:55)→ j ( f )( κ ).We now fix an integer m with n < m < ω , and define a family offorcing posets: for α and β inaccessible with α < β we let C ( α, β ) =Coll( α + m , < β ). We note that when α < β < γ we have that C ( α, β ) ⊆ C ( α, γ ) and the inclusion map is a complete embedding: in particular,if G is C ( α, γ )-generic over V then G ∩ C ( α, β ) is C ( α, β )-generic over V . Definition 2.1. A U -constraint is a function H such that dom( H ) ∈ U , dom( H ) ⊆ A good and H ( x ) ∈ C ( κ x , κ ) for all x ∈ dom( H ).It is easy to see that C M ( κ, j ( κ )) is the set of objects of the form[ H ] U for some U -constraint H . Definition 2.2.
Let H and H (cid:48) be U -constraints. MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 (1) H ≤ H (cid:48) if and only if dom( H ) ⊆ dom( H (cid:48) ) and H ( x ) ≤ H (cid:48) ( x )for all x ∈ dom( H ).(2) H ≤ U H (cid:48) if and only if { x : H ( x ) ≤ H (cid:48) ( x ) } ∈ U , or equivalently[ H ] U ≤ [ H (cid:48) ] U . Remark.
Since m > n , and κ + n M ⊆ M by the hypothesis that U wit-nesses the κ + n -supercompactness of κ , it is easy to see that C M ( κ, j ( κ ))is κ + n +1 -closed in V . It follows that any ≤ U -decreasing sequence of U -constraints of length less than κ + n +1 has a ≤ U -lower bound.We define the complete Boolean algebra B ( α, β ) to be the regularopen algebra of the forcing poset C ( α, β ), and then let B = B M ( κ, j ( κ ))and B = B M ( κ, j ( κ )). We note that for every α < κ the poset C ( α, κ ) is κ -cc and has cardinality κ , so that B ( α, κ ) has cardinality κ :by elementarity we see that B has cardinality j ( κ ) in M , so that in V we have | B | = 2 κ . Remark.
Officially elements of B ( α, κ ) are regular open subsets of theposet C ( α, κ ), so that B ( α, κ ) is not literally a subset of V κ . However,since C ( α, κ ) has the κ -chain condition, B ( α, κ ) is the direct limit ofthe sequence of algebras (cid:104) B ( α, γ ) : γ < κ (cid:105) , so that we may identify B ( α, κ ) with a subset of V κ . With this identification we may representelements of B in the form [ h ] U , where h is a function from κ to V κ .This becomes important later, when we use such functions h as com-ponents of forcing conditions in the poset P . When we move to a genericextension W with the same V κ but new subsets of κ , we will need toknow that h can still be interpreted as a function which returns anelement of B ( α, κ ) on argument α .Following Foreman and Woodin, we define a filter Fil( H ) on B fromeach U -constraint H . Definition 2.3.
Let H be a U -constraint and let A ∈ U . We define afunction b ( H, A ) as follows:dom( b ( H, A )) = { κ x : x ∈ dom( H ) ∩ A } , and b ( H, A )( α ) = (cid:95) { H ( x ) : x ∈ dom( H ) ∩ A and κ x = α } . In the definition of b ( H, A )( α ) we are forming the Boolean supremumof a nonempty subset of C ( α, κ ), thereby defining a nonzero elementof B ( α, κ ). Since { κ x : x ∈ dom( H ) ∩ A } ∈ U , the function b ( H, A )is defined on a U -large set and so represents a nonzero element of theBoolean algebra B in the ultrapower M . JAMES CUMMINGS, MIRNA DˇZAMONJA, AND CHARLES MORGAN
Lemma 2.4.
Let H be a U -constraint and let A , A ∈ U be such that A ⊆ A . Then dom( b ( H, A )) ⊆ dom( b ( H, A )) and b ( H, A )( α ) ≤ b ( H, A )( α ) for all α ∈ dom( b ( H, A )) .Proof. Straightforward. (cid:3)
It follows immediately that the set { [ b ( H, A )] U : A ∈ U } forms afilter base on B . Definition 2.5.
Let H be a U -constraint. Then Fil( H ) is the filtergenerated by { [ b ( H, A )] U : A ∈ U } . Lemma 2.6. If H ≤ U H then Fil( H ) ⊆ Fil( H ) .Proof. Straightforward. (cid:3)
Lemma 2.7.
For every U -constraint H and every Boolean value b in B , there is H (cid:48) ≤ U H such that either b ∈ Fil( H (cid:48) ) or ¬ b ∈ Fil( H (cid:48) ) .Proof. We may assume that b is non-zero. Let b = [ f ] U , where f ( α ) ∈ B ( α, κ ) and f ( α ) is non-zero for all α ∈ dom( f ). Let A = { x ∈ dom( H ) : κ x ∈ dom( f ) } and observe that A ∈ U .For each x in A , we may choose H ∗ ( x ) ≤ H ( x ) such that either H ∗ ( x ) ≤ f ( κ x ) or H ∗ ( x ) ≤ ¬ f ( κ x ). Let A = { x ∈ A : H ∗ ( x ) ≤ f ( κ x ) } . If A ∈ U then define H (cid:48) = H ∗ (cid:22) A , otherwise define H (cid:48) = H ∗ (cid:22) ( A − A ).If A ∈ U then consider the function b ( H (cid:48) , A ). For every relevant α we see that b ( H (cid:48) , A )( α ) is computed as a Boolean supremum ofvalues which are bounded by f ( α ), so that b ( H (cid:48) , A )( α ) ≤ f ( α ). Hence[ b ( H (cid:48) , A )] U ≤ [ f ] U , and accordingly b ∈ Fil( H (cid:48) ). Similarly if A / ∈ U then ¬ b ∈ Fil( H (cid:48) ). (cid:3) Lemma 2.8.
For every U -constraint H there is H (cid:48) ≤ U H such that Fil( H (cid:48) ) is an ultrafilter on B .Proof. This follows immediately from the preceding lemmas, the obser-vation that | B | = 2 κ , and the fact that any ≤ U -decreasing 2 κ -sequenceof U -constraints has a lower bound, (cid:3) Lemma 2.9.
Let H (cid:48) and H (cid:48)(cid:48) be U -constraints such that Fil( H (cid:48) ) is anultrafilter on B and H (cid:48)(cid:48) ≤ U H (cid:48) . Then Fil( H (cid:48) ) = Fil( H (cid:48)(cid:48) ) .Proof. Straightforward. (cid:3)
It will be convenient for the arguments of Section 5 to formulatethese ideas in a slightly different language. Recall that there is anelementary embedding k : M → M such that k ◦ j = j , given by theformula k : [ f ] U (cid:55)→ j ( f )( κ ). MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 Lemma 2.10.
For any U -constraint H , Fil( H ) = { b ∈ B : [ H ] U ≤ B k ( b ) } . Proof.
Let f be a typical function representing an element b of B ,that is to say dom( f ) ∈ U and f ( α ) ∈ B ( α, κ ) for all α . Now k ( b ) = j ( f )( κ ), and [ H ] U = j ( H )( j “ κ + n ), so that easily [ H ] U ≤ j ( f )( κ ) if andonly if { x ∈ dom( H ) : H ( x ) ≤ f ( κ x ) } ∈ U .If b ∈ Fil( H ) then by definition there is a set A ∈ U such that[ b ( H, A )] U ≤ [ f ] U , that is to say B = def { α : b ( H, A )( α ) ≤ f ( α ) } ∈ U . Now let A (cid:48) = A ∩ dom( H ) ∩ { x : κ x ∈ B } . Clearly A (cid:48) ∈ U ; fix x ∈ A (cid:48) and observe that H ( x ) ≤ b ( H, A )( κ x ) ≤ f ( κ x ) , where the firstinequality holds because x ∈ dom( H ) ∩ A and the second one holdsbecause κ x ∈ B . We have shown that { x ∈ dom( H ) : H ( x ) ≤ f ( κ x ) } ∈ U , so that [ H ] U ≤ B k ( b ).Conversely, if [ H ] U ≤ B k ( b ) we let A = { x ∈ dom( H ) : H ( x ) ≤ f ( κ x ) } . Then dom( b ( H, A )) = { κ x : x ∈ A } . For every α in this set wehave that b ( H, A )( α ) = (cid:95) { H ( x ) : x ∈ A and κ x = α } ≤ f ( α ) , where the second claim follows since (by the definition of A ) we areforming the Boolean supremum of a set of values which is bounded by f ( α ). (cid:3) We conclude this discussion of constraints and filters by collectingsome technical facts about filters of the form Fil( H ) which will be usefulwhen we define the forcing poset P . Definition 2.11. A U -constraint is a partial function h from κ to V κ such that dom( h ) ∈ U , dom( h ) is a set of inaccessible cardinals, and h ( α ) ∈ B ( α, κ ) for all α ∈ dom( h ).Clearly B is the set of objects of the form [ h ] U where h is a U -constraint. Definition 2.12.
Let h and h (cid:48) be U -constraints.(1) h ≤ h (cid:48) if and only if dom( h ) ⊆ dom( h (cid:48) ) and h ( α ) ≤ h (cid:48) ( α ) forall α ∈ dom( h ).(2) h ≤ U o h (cid:48) if and only if { α : h ( α ) ≤ h (cid:48) ( α ) } ∈ U or equivalently[ h ] U ≤ [ h (cid:48) ] U . Lemma 2.13.
Let h be a U -constraint and let H be a U -constraint.If [ h ] U ∈ Fil( H ) , then there is B ∈ U such that b ( H, B ) ≤ h . JAMES CUMMINGS, MIRNA DˇZAMONJA, AND CHARLES MORGAN
Proof.
Observe that by definition there is A ∈ U such that b ( H, A ) ≤ U h , and define B = { x ∈ A : b ( H, A )( κ x ) ≤ h ( κ x ) } . It is routine to check that this B works. (cid:3) We now record some crucial properties of filters of the form Fil( H ).In the sequel we will limit attention to the special case in which Fil( H )is an ultrafilter, but only Lemma 2.20 actually requires this assumption. Lemma 2.14 ( κ -completeness Lemma) . Let H be a U -constraint, let η < κ and let (cid:104) h i : i < η (cid:105) be a sequence of U -constraints such that [ h i ] ∈ Fil( H ) for all i . Then there exists a U -constraint h such that [ h ] ∈ Fil( H ) and h ≤ h i for all i .Proof. Appealing to Lemma 2.13 we choose for each i < η a set B i ∈ U such that b ( H, B i ) ≤ h i . Let B = (cid:84) i B i , then B ∈ U and it followsfrom Lemma 2.4 that b ( H, B ) ≤ b ( H, B i ) ≤ h i for all i < η . (cid:3) Definition 2.15.
Given a set s ∈ V κ and a U -constraint h , we define h (cid:25) s = h (cid:22) { α : s ∈ V α } . Lemma 2.16 (Normality lemma) . Let H be a U -constraint, let I ⊆ V κ and let (cid:104) h s : s ∈ I (cid:105) be an I -indexed family of U -constraints such that [ h s ] U ∈ Fil( H ) for all s . Then there exists a U -constraint h such that [ h ] U ∈ Fil( H ) and h (cid:25) s ≤ h s for all s .Proof. Choose for each s ∈ I a set A s ∈ U such that b ( H, A s ) ≤ h s . Bythe normality of U it follows that if we set A = { x ∈ dom( H ) : ∀ s ∈ I ∩ V κ x x ∈ A s } then A ∈ U . Let h = b ( H, A ).To show this works we fix α ∈ dom( h ) and s ∈ I ∩ V α . By definition h ( α ) = (cid:95) x ∈ A,κ x = α H ( x ) . For every x involved in this supremum we have s ∈ V κ x , so that x ∈ A s .Hence easily h ( α ) = b ( H, A )( α ) ≤ b ( H, A s )( α ) ≤ h s ( α ) . (cid:3) With a view towards the forcing construction of Section 3 we definethe notion of lower part . Definition 2.17. A lower part is a finite sequence( p , κ , p , . . . , κ k , p k )such that: MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 (1) k ≥ κ i is an inaccessible cardinal less than κ for 1 ≤ i ≤ k , and κ < κ < . . . < κ k .(3) With the convention that κ = 0 and κ k +1 = κ , p i ∈ C ( κ i , κ i +1 )for 0 ≤ i ≤ k .Given lower parts s and t with s = ( p , κ , . . . , κ i − , p i − ) , we say that t ≤ s if and only if t = ( q , κ , . . . , κ i − , q i − )and q i ≤ p i for all i . Definition 2.18.
A set X of lower parts is downwards closed if andonly if for all s ∈ X and all t ≤ s we have t ∈ X .Now let us fix H a U -constraint such that Fil( H ) is an ultrafilter. Definition 2.19. h is an upper part if and only if h is a U -constraintsuch that [ h ] ∈ Fil( H ).The fact that Fil( H ) is maximal is at the heart of the followingcrucial lemma. Lemma 2.20 (Capturing Lemma) . Let X be a downwards closed setof lower parts and let h be an upper part. Then there exists an upperpart h + ≤ h such that (1) For all α, β ∈ dom( h + ) with α < β , h + ( α ) ∈ C ( α, β ) . (2) For all lower parts s , exactly one of the two following statementsholds: (a) For all α ∈ dom( h + ) such that s ∈ V α , there exists q ≤ h + ( α ) such that s (cid:95) ( α, q ) ∈ X . (b) For all α ∈ dom( h + ) such that s ∈ V α , there does not exist q ≤ h + ( α ) such that s (cid:95) ( α, q ) ∈ X . (3) For all lower parts s , and all α, β ∈ dom( h + ) such that s ∈ V α with α < β , IF there is q ≤ h + ( α ) such that s (cid:95) ( α, q ) ∈ X THEN { q ∈ C ( α, β ) : s (cid:95) ( α, q ) ∈ X } is dense below h + ( α ) in C ( α, β ) .Proof. Strengthening h if necessary, we may assume that h = b ( H, A )for some A ∈ U . Fix for the moment a lower part s , and let A s ⊆ { x ∈ dom( H ) ∩ A : s ∈ V κ x } be such that A s ∈ U and one of the following statements holds: (Case One) For all x ∈ A s there is q ≤ H ( x ) such that s (cid:95) ( κ x , q ) ∈ X .(Case Two) For no x ∈ A s is there q ≤ H ( x ) such that s (cid:95) ( κ x , q ) ∈ X .We now choose H s ≤ H such that dom( H s ) = A s , and if s fallsin Case One then s (cid:95) ( κ x , H s ( x )) ∈ X for all x ∈ A s , and then let h s = b ( H s , A s ). By Lemma 2.9, Fil( H s ) = Fil( H ) and so h s is alegitimate upper part.Claim One: If there exist α ∈ dom( h s ) and p ≤ h s ( α ) such that s (cid:95) ( α, p ) ∈ X , then { r ∈ C ( α (cid:48) , κ ) : s (cid:95) ( α (cid:48) , r ) ∈ X } is dense below h s ( α (cid:48) ) in C ( α (cid:48) , κ ) for all α (cid:48) ∈ dom( h s ).Proof of Claim One: Fix some α and p ≤ h s ( α ) with s (cid:95) ( α, p ) ∈ X , andrecall that h s ( α ) = (cid:87) x ∈ A s ,κ x = α H s ( x ). It follows that there is x ∈ A s such that κ x = α and p is comparable with H s ( x ), and we may fix p (cid:48) ≤ p, H s ( x ). Since X is downwards closed, s (cid:95) ( α, p (cid:48) ) ∈ X . Since x ∈ A s and p (cid:48) ≤ H s ( x ) ≤ H ( x ), s falls in Case One above and so s (cid:95) ( κ x , H s ( x )) ∈ X for all x ∈ A s .Let α (cid:48) ∈ dom( h s ), let q ∈ C ( α, κ ) be arbitrary with q ≤ h s ( α (cid:48) ), andobserve that arguing as above there is x ∈ A s such that κ x = α (cid:48) and q is comparable with H s ( x ); if we now choose r ≤ q, H s ( x ) then itfollows from the downwards closedness of X and the definition of H s in Case One that s (cid:95) ( κ x , r ) ∈ X .Claim Two: For all α ∈ dom( h s ), if there is p ≤ h s ( α ) with s (cid:95) ( α, p ) ∈ X then there is an inaccessible cardinal β s ( α ) < κ such that h s ( α ) ∈ C ( α, β s ( α )) and { r ∈ C ( α, β ) : s (cid:95) ( α, r ) ∈ X } is dense below h s ( α ) in C ( α, β ) for all β ≥ β s ( α ).Proof of Claim Two: By the preceding claim, { r ∈ C ( α, κ ) : s (cid:95) ( α, r ) ∈ X } is dense below h s ( α ) in C ( α, κ ), and since X is downwards closed thisset is open. Choose a maximal antichain A below h s ( α ) consisting ofpoints in this set, and then appeal to the κ -chain condition to find β s ( α ) < κ such that A ⊆ C ( α, β s ( α )).Let B s = { β : ∀ α < β β s ( α ) < β } . Since U is normal, B s ∈ U .To finish the proof, use Lemma 2.16 to find an upper part h − suchthat h − (cid:25) s ≤ h s for all s , and let B = { β : ∀ s ∈ V β β ∈ B s and ∀ α < β h − ( α ) ∈ V β } . MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 By normality B ∈ U , and so we may define an upper part h + = h − (cid:22) B .Claim Three: h + is as required.Proof of Claim Three: It is immediate from the definitions that h + ≤ h ,and Clause 1) from the conclusion is satisfied.Towards showing Clauses 2) and 3), suppose that α ∈ dom( h + ), s ∈ V α , q ≤ h + ( α ) and s (cid:95) ( α, q ) ∈ X . By construction h + ( α ) ≤ h s ( α ).By Claim One above, { r ∈ C ( α (cid:48) , κ ) : s (cid:95) ( α (cid:48) , r ) ∈ X } is dense below h s ( α (cid:48) ) in C ( α (cid:48) , κ ) for all α (cid:48) ∈ dom( h s ). So if s ∈ V α (cid:48) and α (cid:48) ∈ dom( h + ), then since h + ( α (cid:48) ) ≤ h s ( α (cid:48) ) this same set is dense below h + ( α (cid:48) ) and so Clause 2) is satisfied.By Claim Two above, { r ∈ C ( α, β ) : s (cid:95) ( α, r ) ∈ X } is dense below h s ( α ) for all β ≥ β s ( α ). If β ∈ dom( h + ) with α < β then (since β ∈ B ) we have that h + ( α ) ∈ V β and also β ∈ B s , so that β s ( α ) < β and hence { r ∈ C ( α, β ) : s (cid:95) ( α, r ) ∈ X } is dense below h + ( α ). This shows that Clause 3) is satisfied. (cid:3) Definition 2.21. If X is a downwards closed set of lower parts and h + is an upper part satisfying the conclusion of the Capturing Lemmathen we say that h + captures X .3. The forcing P and its properties The filter.
In the last section we used the 2 κ supercompactnessof κ to show that there exists a U -constraint H such that Fil( H ) is anultrafilter. We then established that if F is an ultrafilter of the formFil( H ) then F has three properties:I. ( κ -completeness) Let η < κ and let (cid:104) h i : i < η (cid:105) be a sequenceof upper parts. Then there exists an upper part h such that h ≤ h i for all i .II. (normality) Let I be a set of lower parts and let (cid:104) h s : s ∈ I (cid:105) bean I -indexed family of upper parts. Then there exists an upperpart h such that h (cid:25) s ≤ h s for all s .III. (capturing) Let X be a downwards closed set of lower parts andlet h be an upper part. Then there exists an upper part h + ≤ h such that h + captures X . Remark.
In III above, the last part implies immediately that if thereis q ≤ h + ( α ) such that s (cid:95) ( α, q ) ∈ X then { q ∈ C ( α, κ ) : s (cid:95) ( α, q ) ∈ X } is dense below h + ( α ) in C ( α, κ ).For the rest of this section we will weaken our assumptions on κ , tobe precise we will assume only that:(1) κ is measurable, and U is a normal measure on κ , with associ-ated ultrapower map j : V −→ M = Ult( V, U ).(2) 2 κ = κ + n and n < m < ω .(3) F is an ultrafilter on B = C ( κ, j o ( κ )) M with properties I-III.3.2. The forcing.
We now fix a filter F satisfying properties I-IIIabove, and use F to define a forcing poset P . Conditions in P are pairs( s, h ) such that:(1) s is a lower part.(2) h is an upper part.When p = ( s, h ) we will refer to s as the stem or lower part of p , andto h as the upper part of p .Suppose that p = ( s, h ) and q = ( s (cid:48) , h (cid:48) ) are conditions where s =( p , α , p , . . . , α k , p k ) and s (cid:48) = ( q , β , q , . . . , β l , p l ). Then q ≤ p if andonly if(1) α i = β i and q i ≤ p i for 1 ≤ i ≤ k .(2) β i ∈ dom( h ) and q i ≤ h ( β i ) for k < i ≤ l .(3) h (cid:48) ≤ h . q is a direct extension of p if q ≤ p and in addition k = l . We write q ≤ ∗ p in this case.The generic object for P is a sequence f , κ , f , κ , f . . . where the κ i form an increasing and cofinal ω -sequence of inaccessiblecardinals less than κ (which will be generic for the Prikry forcing de-fined from U ), f i is C ( ω, κ )-generic and f i is C ( κ i , κ i +1 )-generic for i >
0. The condition ( s, h ) where s = ( p , α , p , . . . , α k , p k ) carries theinformation that(1) κ i = α i and p i ∈ f i for 1 ≤ i ≤ k .(2) κ i ∈ dom( h ) and h ( κ i ) ∈ f i for i > k . Lemma 3.1.
The forcing poset P has the κ + -cc. MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 Proof.
Let ( s, h ) and ( s, h (cid:48) ) be two conditions with the same stem s .Since [ h ] U , [ h (cid:48) ] U ∈ F and F is a filter, it is easy to find h (cid:48)(cid:48) suchthat [ h (cid:48)(cid:48) ] U ≤ [ h ] U , [ h (cid:48) ] U . h (cid:48)(cid:48) ≤ U h, h (cid:48) , and if we let B = { α : h (cid:48)(cid:48) ( α ) ≤ h ( α ) and h (cid:48)(cid:48) ( α ) ≤ h (cid:48) ( α ) } then h (cid:48)(cid:48) (cid:22) B ≤ h, h (cid:48) . The condition( s, h (cid:48)(cid:48) (cid:22) B ) is clearly a lower bound for ( s, h ) and ( s, h (cid:48) ). (cid:3) The following Lemma is straightforward.
Lemma 3.2.
Let p = ( s, h ) where s = ( p , α , p , . . . , α k , p k ) , and let ≤ i ≤ k . Then the forcing poset P ↓ p is isomorphic to D × ( P (cid:48) ↓ ( t, h )) , where D = C ( ω, α ) ↓ p × . . . × C ( α i − , α i ) ↓ p i − , P (cid:48) is defined just like P except that α i plays the role of ω , and t = ( p i , α i +1 , . . . , α k , p k ) . The Prikry Lemma.Lemma 3.3 (Prikry Lemma for P ) . Let Φ be a sentence in the forcinglanguage and let p ∈ P , then there is a direct extension q ≤ p whichdecides Φ .Proof. We begin the proof with a construction that is done uniformlyfor all conditions p .For each lower part t , if there is an upper part h such that ( t, h )decides Φ then we fix such an upper part h t . Appealing to PropertyII for F , we find h ≤ h such that h (cid:25) t ≤ h t for all relevant t . Sofor every t , if there exists any h such that ( t, h ) decides Φ then ( t, h )decides Φ.We now define two sets of lower parts: X + = { t : ( t, h ) (cid:13) Φ } , and X − = { t : ( t, h ) (cid:13) ¬ Φ } . It is clear that both X + and X − are downwards closed. By two appealsto Property III we obtain h ≤ h such that h captures both X + and X − .Now let p = ( s, h ). As in the proof of Lemma 3.1, we may find anupper part h ∗ such that h ∗ ≤ h, h . Let ( t, h ∗∗ ) ≤ ( s, h ∗ ) be a conditiondeciding Φ, with lh ( t ) chosen minimal among all such extensions of( s, h ∗ ). We will show that lh ( t ) = lh ( s ), establishing that ( t, h ∗∗ ) is adirect extension of ( s, h ) and thereby proving the Lemma. We will assume that ( t, h ∗∗ ) (cid:13) Φ, the proof in the case when it forces ¬ Φ is the same. Suppose for a contradiction that lh ( t ) > lh ( s ), and let t be the concatenation of a shorter lower part t − and a pair ( α, q ). Since t is longer than s , we have that α ∈ dom( h ∗ ) and q ≤ h ∗ ( α ) ≤ h ( α ).By the construction of h we have also that ( t, h ) (cid:13) Φ, so that t ∈ X + .We claim that ( t − , h ∗∗ ) (cid:13) Φ, which will contradict the hypothesisthat lh ( t ) was chosen minimal and establish the Lemma.Towards the claim we observe that, since h captures X + and q ≤ h ( α ), for every β, γ ∈ dom( h (cid:25) t − ) with β < γ the set { r : t − (cid:95) ( β, r ) ∈ X + } is dense below h ( β ) in C ( β, γ ). We will use this to show that theset of conditions which force Φ is dense below ( t − , h ∗∗ ) in P , establishingthe claim that ( t − , h ∗∗ ) (cid:13) Φ.It will suffice to show that any extension of ( t − , h ∗∗ ) with a properlylonger lower part can be extended to force Φ. Consider such an exten-sion of the form ( t (cid:48) (cid:95) ( γ , q ) (cid:95) . . . (cid:95) ( γ i , q i ) , h ∗∗∗ ), where t (cid:48) ≤ t − and with-out loss of generality i >
0. Since q ≤ h ( γ ) and q ∈ C ( γ , γ ), by theremarks in the preceding paragraph there is r ≤ q with r ∈ C ( γ , γ )such that ( t − (cid:95) ( γ , r ) , h ) (cid:13) Φ.It is now easy to verify that by strengthening q to r we obtain acondition ( t (cid:48) (cid:95) ( γ , r ) (cid:95) . . . (cid:95) ( γ i , q i ) , h ∗∗∗ ) which extends ( t − (cid:95) ( γ , q ) , h ),and so forces Φ. This concludes the proof. (cid:3) Remark.
The proof of the Prikry Lemma extends without any changeto the forcing poset P (cid:48) defined in Lemma 3.2.3.4. Analysing names for bounded subsets of κ . It is clear thatthe forcing poset P collapses all cardinals in the open intervals ( ω + m , κ )and ( κ mi , κ i +1 ) for i >
0. One of the main applications of the PrikryLemma is to show that no other cardinals are collapsed, so that κ becomes ℵ ω in the generic extension. Lemma 3.4.
Let G be P -generic and let f , κ , f , κ , f . . . be the generic sequence added by G . Let x ∈ V [ G ] be a bounded subsetof ( κ + mi ) V for some i > . Then x ∈ V [ f × . . . × f i − ] .Proof. Working below a suitable condition, we may use Lemma 3.2 toview V [ G ] as a two-step extension V [ G (cid:48) ][ g ] where g = f × . . . × f i − and G (cid:48) is generic for P (cid:48) , a version of P in which κ i plays the role of ω .Let x = i G ( ˙ x ), where ˙ x is a P -name for a subset of γ for some γ < κ + mi . We may view ˙ x as a P (cid:48) -name for a D -name for a subset of γ ,where D = C ( ω, κ ) × . . . × C ( κ i − , κ i ).Since P (cid:48) satisfies the Prikry Lemma, it is easy to see that the D -namedenoted by ˙ x lies in V , so that x ∈ V [ f × . . . × f i − ] as required. (cid:3) MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 Using Lemma 3.4, standard chain condition and closure argumentsimply that only the cardinals in the intervals ( ω + m , κ ) and ( κ mi , κ i +1 )are collapsed by P . For the purposes of some later arguments, we willprove a more refined version of Lemma 3.4. The point at stake hereis that a priori it seems that a name for a bounded subset of κ maydepend on an arbitrarily large initial segment of the generic object, andthis would cause major difficulties in the chain condition arguments ofSection 4.Given an increasing sequence (cid:126)α = (cid:104) α , . . . , α k (cid:105) of inaccessible cardi-nals less than κ , we define D ( (cid:126)α ) = C ( ω, α ) × . . . × C ( α k − , α k ) . Lemma 3.5.
Let µ, η < κ and let ˙ x be a P -name for a subset of µ .Let h be an upper part and let S be the set of increasing sequences (cid:104) α , . . . , α k (cid:105) where α i < η and α i is inaccessible.Then there exist an ordinal β with µ, η < β < κ , names (cid:104) ˙ y (cid:126)α : (cid:126)α ∈ S (cid:105) and an upper part h (cid:48) ≤ h with min(dom( h (cid:48) )) > β such that for every (cid:126)α = ( α , . . . , α k ) ∈ S : (1) ˙ y (cid:126)α is a D ( (cid:126)α (cid:95) β ) name for a subset of µ . (2) If t = ( ∅ , α , ∅ , . . . , α k , ∅ ) then ( t, h (cid:48) ) (cid:13) ˙ x = ˙ y (cid:126)α . That is to saythat if G is P -generic with ( t, h (cid:48) ) ∈ G , and f , α , f , α , f . . . is the corresponding generic sequence, then i G ( ˙ x ) = i f ( ˙ y (cid:126)α ) ,where f = f × . . . × f k − × ( f k (cid:22) β ) .Proof. As in the first step of the proof of the Prikry Lemma, we find h ≤ h such that for every lower part t = ( p , β , . . . , β k , p k ), if there arean upper part h (cid:48) and a D ( (cid:104) β , . . . , β k (cid:105) ) -name ˙ y such that ( t, h (cid:48) ) (cid:13) ˙ x = ˙ y then ( t, h ) (cid:13) ˙ x = ˙ y .For each (cid:126)α = ( α , . . . , α k ) ∈ S , each inaccessible δ with µ, η < δ < κ and each canonical D ( (cid:126)α (cid:95) δ )-name ˙ y for a subset of µ , let X ( (cid:126)α, δ, ˙ y ) bethe set of lower parts s such that s = ( q , α , q , . . . , α k , q k , γ, r )for some γ > δ , and ( s, h ) (cid:13) ˙ x = ˙ y . Since this is a downwardsclosed set of lower parts, we may find h (cid:126)α,δ, ˙ y ≤ h which captures it.Using Lemmas 2.14 and 2.16 we may then find an upper part h suchthat h (cid:25) δ ≤ h (cid:126)α,δ, ˙ y for all (cid:126)α, δ, ˙ y , and also min(dom( h )) > µ, η .By shrinking dom( h ) if necessary, we will also arrange that dom( h )consists of Mahlo cardinals. Fix for the moment a sequence (cid:126)α = (cid:104) α , . . . , α k (cid:105) ∈ S . Fix some γ ∈ dom( h ) and consider the condition(( ∅ , α , ∅ , . . . , ∅ , α k , ∅ , γ, h ( γ )) , h ) . Working as in the proof of Lemma 3.4, we may find r ≤ h ( γ ) and h ∗ ≤ h such that(( ∅ , α , ∅ , . . . , ∅ , α k , ∅ , γ, r ) , h ∗ ) (cid:13) ˙ x = ˙ y where ˙ y is a canonical D ( (cid:126)α (cid:95) γ )-name for a subset of µ . Since D ( (cid:126)α (cid:95) γ )has the γ -cc and γ is Mahlo, ˙ y is a canonical D ( (cid:126)α (cid:95) δ )-name for someinaccessible δ with µ, η < δ < γ .By construction r ≤ h ( γ ) ≤ h (cid:126)α,δ, ˙ y ( γ ). By the choice of h , we seethat (( ∅ , α , ∅ , . . . , ∅ , α k , ∅ , γ, r ) , h ) (cid:13) ˙ x = ˙ y. By the choice of h , for every γ , γ ∈ dom( h ) with δ < γ < γ theset of r ∗ ∈ C ( γ , γ ) such that(( ∅ , α , ∅ , . . . , ∅ , α k , ∅ , γ , r ∗ ) , h ) (cid:13) ˙ x = ˙ y is dense below h ( γ ). So for every γ ∈ dom( h ) with δ < γ (( ∅ , α , ∅ , . . . , ∅ , α k , ∅ , γ , h ( γ )) , h ) (cid:13) ˙ x = ˙ y, which implies that(( ∅ , α , ∅ , . . . , ∅ , α k , ∅ ) , h (cid:25) δ ) (cid:13) ˙ x = ˙ y. To record their dependence on (cid:126)α , we write δ (cid:126)α for δ and ˙ y (cid:126)α for ˙ y .Let β be the supremum of the δ (cid:126)α for (cid:126)α ∈ S , and let h (cid:48) = h (cid:25) β . It isnow easy to see that the ordinal β , upper part h (cid:48) and family of names (cid:104) ˙ y (cid:126)α : (cid:126)α ∈ S (cid:105) are as required. (cid:3) Characterisation of genericity.
We will need one more techni-cal fact about P , namely a characterisation of the generic object. Sim-ilar “geometric” characterisations for other Prikry-type forcing posetsappear at many places [9, 11, 1] in the literature. Lemma 3.6 (Genericity Lemma) . Let f , κ , f , . . . be such that (1) f i is C ( ω, κ ) -generic for i = 0 and C ( κ i , κ i +1 ) -generic for i > . (2) For all upper parts h there is an integer s such that κ t ∈ dom( h ) and h ( κ t ) ∈ f t for all t ≥ s .Then this is a generic sequence for P . MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 Proof.
For our later convenience we define C = C ( ω, κ ) and C i = C ( κ i , κ i +1 ) for i >
0. We make the remark that by an easy applicationof Easton’s Lemma the filters f , . . . , f n are mutually generic, that is f × . . . × f n is generic over V for C × . . . × C n .We now fix E a dense open set in P , with the ultimate goal of showingthat E meets the filter on P generated by f , κ , f , . . . To achieve this goal we need to “canonise” E in a sense to be madeprecise later.By a familiar diagonal intersection argument, there is an upper part h such that for every lower part s , ∃ h ( s, h ) ∈ E ⇐⇒ ( s, h (cid:25) s ) ∈ E. Since the set E is open, it is easy to see that if we let X = { s : ( s, h (cid:25) s ) ∈ E } then X is a downwards closed set of lower parts.Applying Property III repeatedly we construct downwards closedsets X n and upper parts h n such that:(1) h n +1 ≤ h n .(2) h n +1 captures X n .(3) X n +1 is the set of lower parts s such that for some (equivalently,for every) α ∈ dom( h n +1 ) such that s ∈ V α there is q ≤ h n +1 ( α )with s (cid:95) ( q, α ) ∈ X n .We appeal to Property I to find an upper part h ∞ such that h ∞ ≤ h n for all n . By the hypotheses, we find an integer k such that κ l dom( h ∞ )and h ∞ ( κ l ) ∈ f l for all l ≥ k . Claim. { ( q , . . . , q k − ) : ∃ j ( q , κ , . . . , κ k − , q k − ) ∈ X j } is dense in C × . . . × C k − . Proof.
Let ( p , . . . , p k − ) ∈ C × . . . × C k − , and consider the condition(( p , κ , . . . , κ k − , p k − , κ k , h ∞ ( κ k )) , h ∞ ) . Since E is dense there is an extension(( q , κ , . . . , κ k − , q k − , ¯ κ k , q k , . . . , ¯ κ k + j − , q k + j − ) , h ) ∈ E for some j >
0. Call the lower part of this extension s , and observethat by construction of h we have ( s, h (cid:25) s ) ∈ E so that s ∈ X . Now observe that ¯ κ k + j − ∈ dom( h ∞ ) and q k + j − ≤ h ∞ (¯ κ k + j − ) ≤ h (¯ κ k + j − ), so that( q , κ , . . . , κ k − , q k − , ¯ κ k , q k , . . . , ¯ κ k + j − , q k + j − ) ∈ X . Stepping backwards in the obvious way we eventually obtain that( q , κ , . . . , κ k − , q k − ) ∈ X j . (cid:3) Since f × . . . × f k − is generic, we obtain conditions q i ∈ f i for i < k such that t ∈ X j where t = ( q , κ , . . . , κ k − , q k − ). Since t ∈ X j , κ k , κ k +1 ∈ dom( h j ) and κ k < κ k +1 , { p ∈ C k : t (cid:95) ( κ k , p ) ∈ X j − } is dense below h j ( κ k ). Also h j ( κ k ) ∈ f k because h ∞ ( κ k ) ∈ f k and h ∞ ≤ h j . So we may find q ∗ j ∈ f j such that t (cid:95) ( κ j , q ∗ j ) ∈ X j − . Repeating this argument j times we construct q ∗ i ∈ f i for k ≤ i < k + j such that u = t (cid:95) ( κ j , q ∗ j , . . . , κ k + j − , q ∗ k + j − ) ∈ X , that is to say that ( u, h (cid:25) u ) ∈ E .But it is now easy to verify that ( u, h (cid:25) u ) is in the filter generatedby the sequence of ( f i ): simply observe that(1) q i ∈ f i for i < k .(2) q ∗ i ∈ f i for k ≤ i < k + j .(3) h ( κ i ) ∈ f i for i ≥ k + j .This concludes the proof of the Genericity Lemma. (cid:3) The forcing Q and its properties We work throughout with the same hypotheses as in Section 3. Inparticular F has properties I, II and III and P is the Prikry-type forcingdefined from F . Let 2 κ + = λ , and let T be a tree of height κ + suchthat T has at least λ branches and each level of T has size at most κ + . Let (cid:104) x β : β < λ (cid:105) enumerate a sequence of distinct branches, andenumerate Lev α ( T ) as (cid:104) t ( α, i ) : i < | Lev α ( T ) |(cid:105) for each α < κ + . Definition 4.1.
Let A be a function such that dom( A ) is a bounded setof inaccessible cardinals less than κ , and A ( α ) ∈ B ( α, κ ) with A ( α ) (cid:54) = 0for all α ∈ dom( A ). Let s = ( q , α , q , . . . , α k , q k ) be a lower part, andlet η < κ . Then s is harmonious with A past η if and only if for all j such that α j ≥ η we have α j ∈ dom( A ) and q j ≤ A ( α j ). MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 Let (cid:104) ˙ G α : α < λ (cid:105) enumerate all the canonical P -names for graphs on κ + . We define a forcing poset Q .Conditions in Q are quadruples ( A, B, t, f ) such that:(1) A is a function such that dom( A ) is a bounded set of inaccessiblecardinals less than κ , and A ( α ) ∈ B ( α, κ ) with A ( α ) (cid:54) = 0 for all α ∈ dom( A ).(2) B is an upper part.(3) t is a triple ( ρ, a, b ) where ρ < κ , a ∈ [ κ + ] <κ and b ∈ [ λ ] <κ .(4) f is a sequence (cid:104) f η,β : η < ρ, β ∈ b (cid:105) such that each function f η,β has domain a .(5) f η,β ( ζ ) ∈ { x β (cid:22) ζ } × κ for all η < ρ , β ∈ b and ζ ∈ a .(6) For every η ∈ dom( A ) ∩ ρ , every lower part s harmonious with A past η , and every β, γ ∈ b and ζ (cid:48) , ζ ∈ a such that f η,β ( ζ (cid:48) ) = f η,γ ( ζ (cid:48) ) (cid:54) = f η,β ( ζ ) = f η,γ ( ζ ),( s, B ) (cid:13) ζ (cid:48) ˙ G β ζ ⇐⇒ ζ (cid:48) ˙ G γ ζ. Remark.
In the last clause, if s is one of the relevant stems then allordinals appearing in s are less than ssup(dom( A )).Let q = ( A, B, t, f ) and q (cid:48) = ( A (cid:48) , B (cid:48) , t (cid:48) , f (cid:48) ) be two conditions in Q .Then q (cid:48) ≤ q if and only if:(1) dom( A ) is an initial segment of dom( A (cid:48) ), and A (cid:48) (cid:22) dom( A ) = A .(2) B (cid:48) ≤ B , that is dom( B (cid:48) ) ⊆ dom( B ) and B (cid:48) ( α ) ≤ B ( α ) for all α ∈ dom( B (cid:48) ).(3) For all α ∈ dom( A (cid:48) ) \ dom( A ), α ∈ dom( B ) and A (cid:48) ( α ) ≤ B ( α ).(4) If we let t = ( ρ, a, b ) and t (cid:48) = ( ρ (cid:48) , a (cid:48) , b (cid:48) ) then ρ ≤ ρ (cid:48) , a ⊆ a (cid:48) and b ⊆ b (cid:48) .(5) f (cid:48) η,β ( ζ ) = f η,β ( ζ ) for all η < ρ , β ∈ b and ζ ∈ a . Remark.
The forcing poset Q is intended to add (among other things)a generic function h from κ to V κ of the right general form to be anupper part. If we ultimately force with some version of P for whichthe generic function h is a legitimate upper part, then we will add ageneric sequence x which eventually obeys h but we do not know pastwhich point on x this will begin to happen. This motivates the notionof “harmonious past η ”, and also explains why each η gets its own setof functions f η,β . Lemma 4.2. If G Q is Q -generic then: (1) If we let h G Q = (cid:83) { A p : p ∈ G Q } then h G Q is a function, dom( h G Q ) is unbounded in κ , and for every upper part h wehave that α ∈ dom( h G Q ) and h G Q ( α ) ≤ h ( α ) for all large enough α ∈ dom( h ) . (2) For all η < κ and β < λ , if we let F G Q η,β = (cid:83) { f pη,β : p ∈ G Q } then F G Q η,β is a function with domain κ + .Proof. For the first claim, we suppose that ν < κ , h is an upper part,and q is an arbitrary condition. Let µ ∈ dom( B q ) with µ > ν , anddefine r = ( A r , B r , t r , f r ) as follows: A r = A q ∪ { ( µ, B q ( µ )) } , B r issome upper part such that B r ≤ B q , h and µ < min(dom( B r )), t r = t q and f r = f q .We must verify that r is a condition and r ≤ q . The only non-trivial point is to see that r satisfies Clause 6) in the definition ofconditionhood in Q . Let t be a lower part harmonious with A r past η . There are now two cases. If t is harmonious with A q past η then( t, B r ) ≤ ( t, B q ), and we are done by Clause 6) for q . Otherwise t = s (cid:95) (cid:104) µ, p (cid:105) for some p ≤ B q ( µ ) and s harmonious with A q past η , ( t, B r ) ≤ ( s, B q ), and again we are done by Clause 6) for q .For the second claim, we fix ζ, η, β and then find a ⊇ a q , b ⊇ b q and ρ ≥ ρ q such that η < ρ , ζ ∈ a , β ∈ b . We then define r =( A r , B r , t r , f r ) as follows: A r = A q , B r = B q , t r = ( ρ, a, b ) and f r is chosen to extend f q and to be such that the values f rη (cid:48) ,β (cid:48) ( ζ (cid:48) ) for( η (cid:48) , ζ (cid:48) , β (cid:48) ) ∈ ( ρ × a × b ) \ ( ρ q × a q × b q ) are all distinct from each otherand from any of the values f qη (cid:48) ,β (cid:48) ( ζ (cid:48) ) for ( η (cid:48) , ζ (cid:48) , β (cid:48) ) ∈ ρ q × a q × b q . Thischoice ensures that Clause 6) in the definition of conditionhood holds,so that r is a condition with r ≤ q . (cid:3) We recall that for a regular uncountable cardinal ν , a poset R is ν -compact if and only if the following condition holds: for every X ⊆ R with | X | < ν , if every finite subset of X has a lower bound then X hasa lower bound. Lemma 4.3. Q is κ -compact.Proof. Let µ < κ , and let { q i : i < µ } be a set of conditions in Q suchthat for any finite subset s of µ the set { q i : i ∈ s } has a lower bound.Let q i = ( A i , B i , t i , f i ), and choose for each finite s ⊆ µ a condition r s = ( A s , B s , t s , f s ) which is a lower bound for { q i : i ∈ s } .We will define r = ( A r , B r , t r , f r ) as follows: • A r = (cid:83) i<µ A i . • B r is some upper part such that ssup(dom( A r )) < dom( B r )and B r ≤ B s for all s . • t r = ( ρ r , a r , b r ) where ρ r = (cid:83) i<µ ρ i , a r = (cid:83) i<µ a i , b r = (cid:83) i<µ b i . • If there is some i such that ( η, ζ, β ) ∈ ρ i × a i × b i , then f rη,β ( ζ ) = f iη,β ( ζ ). As in the proof of Lemma 4.2, we choose the values of f rη,β ( ζ ) for ( η, ζ, β ) ∈ ρ r × a r × b r \ (cid:83) i<µ ( ρ i × a i × b i ) to be distinct MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 from each other and from all values in { f rη,β ( ζ ) : ( η, ζ, β ) ∈ (cid:83) i<µ ( ρ i × a i × b i ) } .We note that by our hypotheses the definition of f rη,β ( ζ ) yields a uniquevalue.As usual, the main issue is to verify that Clause 6) holds. This isstraightforward: if f rη,β ( ζ ) = f rη,β (cid:48) ( ζ ) (cid:54) = f rη,β ( ζ (cid:48) ) = f rη,β (cid:48) ( ζ (cid:48) ), then forsome finite s we have f sη,β ( ζ ) = f sη,β (cid:48) ( ζ ) (cid:54) = f sη,β ( ζ (cid:48) ) = f sη,β (cid:48) ( ζ (cid:48) ), and so weare done because B r ≤ B s . (cid:3) Corollary 4.4.
The poset Q is κ -directed closed and also has the fol-lowing property, which was dubbed “parallel countable closure” in [2] :if (cid:104) q i : i < ω (cid:105) and (cid:104) q i : i < ω (cid:105) are decreasing sequences of conditionssuch that q i and q i are compatible for all i , then there is q such that q ≤ q i , q i for all i . We recall that for an regular cardinal ν , a poset R is strongly ν + -cc if and only if the following condition holds: for every ν + -sequence (cid:104) r i : i < ν + (cid:105) of conditions in R , there exist a club set E ⊆ ν + and aregressive function f on E ∩ cof( ν ) such that for all i and j , if f ( i ) = f ( j ) then r i is compatible with r j . Lemma 4.5. Q is strongly κ + -cc.Proof. Let q i = ( A i , B i , t i , f i ) ∈ Q for i < κ + , and let t i = ( ρ i , a i , b i ).We recall that dom( f iη,β ) = a i for all η < ρ i and β ∈ b i . Let µ i =ot( a i ). Let ˙ x iβ be a P -name for the set of pairs ( ν, ν (cid:48) ) such that ζ ˙ G β ζ (cid:48) ,where ζ and ζ (cid:48) are respectively the ν th and ν (cid:48) th elements of a i .Appealing to Lemma 3.5 and Property I we may assume, shrinking B i if necessary, that for every β ∈ b i there exist an ordinal γ iβ < κ and names ˙ y iβ,(cid:126)α for every increasing finite sequence (cid:126)α of ordinals fromssup(dom( A i )), such that B i “reduces” ˙ x iβ to ˙ y iβ,(cid:126)α which is a name inthe product of collapses D ( (cid:126)α (cid:95) (cid:104) γ iβ (cid:105) ) for the edge set of a graph on thevertex set µ i .We will enumerate (cid:83) i<κ + b i as (cid:104) β : < κ + (cid:105) . To make the restof the proof more readable, we will observe the following notationalconventions:(1) The letter i and its typographic variations will denote indicesfor conditions in Q on the sequence (cid:104) q i : i < κ + } .(2) The letter ζ and its variations will denote elements of (cid:83) i<κ + a i ,and the letter β and its variations will denote elements of (cid:83) i<κ + b i .(3) The letter and its variations will denote indices for ordinalsless than λ on the sequence (cid:104) β : < κ + (cid:105) . (4) Given a set x ⊆ κ + with | x | < κ , the letter σ and its variationswill denote indices for elements of x , enumerated in increasingorder.(5) Given a set y ⊆ (cid:83) i<κ + b i with | y | < κ , the letter τ and itsvariations will denote indices for elements of { : β ∈ y } , againenumerated in increasing order. Note that variations of τ denoteindices (in κ ) for indices (in κ + ) for elements of λ .(6) The letter φ and its variations will denote indices for elements t ∈ Lev ζ ( T ) on the sequence (cid:104) t ( ζ, φ ) : φ < | Lev ζ ( T ) (cid:105) .(7) The letter ψ and its variations will denote the second entries inpairs drawn from T × κ .We define functions F n with domain κ + for n < F ( i ) = ( ρ i , ot( a i ) , ot( { : β ∈ b i } )).(2) F ( i ) = a i ∩ i .(3) F ( i ) = { < i : β ∈ b i } .(4) F ( i ) = A i .(5) F ( i ) is the set of 5-tuples ( η, σ, τ, φ, ψ ) where η < ρ i , σ < ot( a i ), τ < ot ( { : β ∈ b i } ), φ < i , ψ < κ , and if we let ζ be the σ th element of a i and β = β for the τ th element of { : β ∈ b i } then f iη,β ( ζ ) = ( t ( ζ, φ ) , ψ ).(6) F ( i ) is the set of 3-tuples ( τ, γ, Y ) where τ < ot ( { : β ∈ b i } ), γ < κ , Y ∈ V κ , and if we let β = β for the τ th element of { : β ∈ b i } then γ = γ iβ , and Y is the function specified bysetting Y ( (cid:126)α ) = ˙ y iβ,(cid:126)α for each increasing finite sequence (cid:126)α fromssup(dom( A i )). Remark. F ( i ) is best viewed as a partial function on triples ( η, σ, τ )which records a code for the value of f iη,β ( ζ ) when this is “permissible”.The criterion for permissibility is that (after decoding σ and τ to obtain ζ and β ) the first entry ( x β (cid:22) ζ ) in f iη,β ( ζ ) is enumerated before i in theenumeration of level ζ of the tree T . The point is that we are aimingultimately to define a regressive function so we can only record limitedinformation.In a similar vein, F is a total function which records values of γ iβ and ˙ y iβ,(cid:126)α .Now let F ( i ) = ( F ( i ) , F ( i ) , F ( i ) , F ( i ) , F ( i ) , F ( i )), so that F ( i ) ∈ κ × [ i ] <κ × [ i ] <κ × V κ × [ κ × i × κ ] <κ × [ κ × V κ ] <κ . We fix an injective map H from κ × [ κ + ] <κ × [ κ + ] <κ × V κ × [ κ × κ + × κ ] <κ × [ κ × V κ ] <κ MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 to κ + . Since κ <κ = κ , we may fix a club set E ⊆ κ + such that if i ∈ E ∩ cof( κ ) thenrge( H (cid:22) κ × [ i ] <κ × [ i ] <κ × V κ × [ κ × i × κ ] <κ × [ κ × V κ ] <κ ) ⊆ i, so that in particular H ◦ F is regressive on E ∩ cof ( κ ).Let E be the club subset of κ + consisting of those i such that forall i (cid:48) < i :(1) a i (cid:48) ⊆ i .(2) { : β ∈ b i (cid:48) } ⊆ i .(3) For all ζ ∈ a i (cid:48) and β ∈ b i (cid:48) , x β (cid:22) ζ = t ( ζ, φ ) for some φ < i .(4) For all β, β ∗ ∈ b i (cid:48) with β (cid:54) = β ∗ , x β (cid:22) i (cid:54) = x β ∗ (cid:22) i .We claim that the function H ◦ F and the club set E ∩ E serve asa witness to the strong κ + -cc for Q . To see this, let i (cid:48) < i be points in E ∩ E ∩ cof ( κ ) such that F ( i (cid:48) ) = F ( i ). We will show that q i (cid:48) and q i are compatible.We start by decoding the assertion that F n ( i (cid:48) ) = F n ( i ) for n < ρ i (cid:48) = ρ i = ρ ∗ say.(2) ot( a i (cid:48) ) = ot( a i ) = µ ∗ say.(3) ot( { : β ∈ b i (cid:48) } ) = ot( { : β ∈ b i } ) = (cid:15) ∗ say.(4) a i (cid:48) ∩ i (cid:48) = a i ∩ i = r say. Since a i (cid:48) ⊆ i by Clause 1) in thedefinition of E , a i (cid:48) \ i (cid:48) and a i \ i are disjoint and a i (cid:48) ∩ a i = r .(5) { < i (cid:48) : β ∈ b i (cid:48) } = { < i : β ∈ b i } = r say. As in the lastclaim, { ≥ i (cid:48) : β ∈ b i (cid:48) } and { ≥ i : β ∈ b i } are disjoint and { : β ∈ b i ∩ b i (cid:48) } = r .(6) A i (cid:48) = A i = A ∗ say. Claim.
When both sides are defined, f i (cid:48) η,β ( ζ ) = f iη,β ( ζ ). Proof.
We will use the fact that F ( i (cid:48) ) = F ( i ). Since both sides aredefined, η < ρ ∗ , ζ ∈ a i (cid:48) ∩ a i and β ∈ b i (cid:48) ∩ b i . By the remarks in thepreceding paragraph, ζ ∈ r and β = β for some ∈ r .Since r is the common initial segment of a i (cid:48) and a i , we have ot( a i (cid:48) ∩ ζ ) = ot( a i ∩ ζ ) = σ say. Similarly has the same index (say τ ) in theincreasing enumerations of { : β ∈ b i (cid:48) } and { : β ∈ b i } .Now let f i (cid:48) η,β ( ζ ) = ( x β (cid:22) ζ, ψ (cid:48) ), let f iη,β ( ζ ) = ( x β (cid:22) ζ, ψ ), and let x β (cid:22) ζ = t ( β, φ ). Since i ∈ E , i (cid:48) < i , β ∈ b i (cid:48) and ζ ∈ a i (cid:48) , it followsfrom Clause 3) in the definition of E that φ < i .By the definition of F , the set F ( i ) contains the tuple ( η, σ, τ, φ, ψ ).This is the unique tuple in F ( i ) which begins with ( η, σ, τ ), and since F ( i (cid:48) ) = F ( i ) this tuple also appears in F ( i (cid:48) ). It follows that ψ = ψ (cid:48) and so f i (cid:48) η,β ( ζ ) = f iη,β ( ζ ). (cid:3) We will now define q ∗ = ( A ∗ , B ∗ , t ∗ , f ∗ ), which will be a lower boundfor q i and q i (cid:48) . • Recall that ρ ∗ = ρ i (cid:48) = ρ i . We set a ∗ = a i (cid:48) ∪ a i , b ∗ = b i (cid:48) ∪ b i , t ∗ = ( ρ ∗ , a ∗ , b ∗ ). • Recall that A ∗ = A i = A i (cid:48) . • Let B ∗ be some upper part such that B ∗ ≤ B i (cid:48) , B i . • We define f ∗ η,β ( ζ ) for all η < ρ ∗ , ζ ∈ a ∗ and β ∈ b ∗ . Naturallywe set f ∗ η,β ( ζ ) = f i (cid:48) η,β ( ζ ) when ζ ∈ a i (cid:48) and η ∈ b i (cid:48) , and similarlywe set f ∗ η,β ( ζ ) = f iη,β ( ζ ) when ζ ∈ a i and η ∈ b i . As we arguedalready the sequences f i (cid:48) and f i agree sufficiently for this tomake sense.To define f ∗ η,β ( ζ ) for η < ρ ∗ and ( ζ, β ) ∈ a ∗ × b ∗ \ ( a i × b i ∪ a i (cid:48) × b i (cid:48) ), we will proceed as in the proof of Lemma 4.3. That isto say we will choose suitable values whose second coordinatesare distinct from each other, and also distinct from any valueappearing as a second coordinate of f ∗ η,β ( ζ ) for η < ρ ∗ and( ζ, β ) ∈ a i × b i ∪ a i (cid:48) × b i (cid:48) .It is routine to check that if q ∗ is a condition then it is a commonrefinement of q i (cid:48) and q i , and also that q ∗ satisfies all the clauses in thedefinition of Q except possibly for the final Clause 6). With a view toverifying this clause, suppose that η ∈ dom( A ∗ ) ∩ ρ ∗ , s is a lower partharmonious with A ∗ past η , and f ∗ η,β (cid:48) ( ζ (cid:48) ) = f ∗ η,β ( ζ (cid:48) ) (cid:54) = f ∗ η,β (cid:48) ( ζ ) = f ∗ η,β ( ζ ) . where ζ (cid:48) < ζ and β (cid:48) = β (cid:48) , β = β for some (cid:48) < .By the construction of f ∗ , it is immediate that all four of the pairsin { ζ (cid:48) , ζ } × { β (cid:48) , β } lie in the set a i (cid:48) × b i (cid:48) ∪ a i × b i .If all four pairs above lie in a i (cid:48) × b i (cid:48) , then we are done by Clause 6)for q i (cid:48) and the fact that B ∗ ≤ B i (cid:48) . A similar argument works if all fourpairs lie in a i × b i . From this point we assume that we are not in eitherof these cases.Now recall that a ∗ = a i (cid:48) ∪ a i = r ∪ ( a i (cid:48) \ r ) ∪ ( a i \ r ), where r < a i (cid:48) \ r < a i \ r . Similarly if we let s = { : β ∈ b i (cid:48) ∪ b i } , then s = r ∪ ( { : β ∈ b i (cid:48) } \ r ) ∪ ( { : β ∈ b i } \ r ), where r < { : β ∈ b i (cid:48) } \ r < { : β ∈ b i } \ r .An easy case analysis shows that there are only two possibilities: In figure 1, all pairs ( ζ ∗ , ∗ ) with ( ζ ∗ , β ∗ ) ∈ a i (cid:48) × b i (cid:48) lie in the region shadedwith forward-sloping diagonal lines, and all pairs ( ζ ∗ , ∗ ) with ( ζ ∗ , β ∗ ) ∈ a i × b i liein the region shaded with backward-sloping diagonal lines. Points in { ζ (cid:48) ζ } × { (cid:48) , } must all lie in the shaded region, and must not all lie in subregions shaded in asingle direction. MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:48) ¯ { j : β j ∈ b i − r }{ j : β j ∈ b i (cid:48) − r } r a i − r a i (cid:48) − r r ζζ (cid:48) Figure 1.
Subcase 2b: ¯ is the “clone” of . Case 1: (cid:48) and are both in r , ζ (cid:48) ∈ a i (cid:48) \ r , ζ ∈ a i \ r . Case 2: ζ (cid:48) and ζ are both in r , (cid:48) ∈ b i (cid:48) \ r , ∈ b i \ r .We will first show that Case 1 is not possible. To dismiss Case 1,assume that we are in this case and recall that f ∗ η,β (cid:48) ( ζ (cid:48) ) = f ∗ η,β ( ζ (cid:48) ) (cid:54) = f ∗ η,β (cid:48) ( ζ ) = f ∗ η,β ( ζ ) , from which it follows that x β (cid:48) (cid:22) ζ = x β (cid:22) ζ . Since i ∈ E , i (cid:48) < i and β (cid:48) , β ∈ b i (cid:48) , it follows from clause 4 in the definition of E that x β (cid:48) (cid:22) i (cid:54) = x β (cid:22) i . By the case assumption we have ζ ∈ a i \ r = a i \ ( a i ∩ i ),so that in particular ζ ≥ i . This is a contradiction, so Case 1 does notoccur.We now assume that we are in Case 2. In order to use the informationcoded in the equality of F ( i (cid:48) ) and F ( i ), we make some definitions: • σ (cid:48) is the index of ζ (cid:48) in the increasing enumeration of a i (cid:48) ∩ a i . • σ is the index of ζ in the increasing enumeration of a i (cid:48) ∩ a i . • τ (cid:48) is the index of (cid:48) in the increasing enumeration of { : β ∈ a i (cid:48) } . • τ is the index of in the increasing enumeration of { : β ∈ a i } .By definition, F ( i (cid:48) ) (which is equal to F ( i )) contains the tuples( τ (cid:48) , γ i (cid:48) β (cid:48) , Y (cid:48) ) and ( τ, γ iβ , Y ), where Y (cid:48) ( (cid:126)α ) = ˙ y i (cid:48) β (cid:48) ,(cid:126)α and Y ( (cid:126)α ) = ˙ y iβ,(cid:126)α for α any increasing finite sequence from ssup(dom( A )). Recall that s is a lower part harmonious with A ∗ past η , and η ∈ dom( A ∗ ) ∩ ρ ∗ . Let s = ( p , α , p , . . . , α k , p k ) , and let (cid:126)α = (cid:104) α , . . . , α k (cid:105) . By the choice of B i (cid:48) and the definitions ofthe names ˙ x i (cid:48) β (cid:48) and ˙ y i (cid:48) β (cid:48) ,(cid:126)α , ( s, B i (cid:48) ) reduces the truth value of ζ (cid:48) ˙ G β (cid:48) ζ (aBoolean value for P ) to the truth value of σ (cid:48) ˙ y i (cid:48) β (cid:48) ,(cid:126)α σ (a Boolean valuefor the product of collapses D ( (cid:126)α (cid:95) γ i (cid:48) β (cid:48) )). Similarly ( s, B i (cid:48) ) reduces thetruth value of ζ (cid:48) ˙ G β ζ to the truth value of σ (cid:48) ˙ y iβ,(cid:126)α σ . Subcase 2a: τ (cid:48) = τ .In this subcase γ i (cid:48) β (cid:48) = γ iβ = γ ∗ say, and Y (cid:48) = Y , so that in particular˙ y i (cid:48) β (cid:48) ,(cid:126)α = ˙ y iβ,(cid:126)α . It is then immediate from the preceding discussion that,since ( s, B ∗ ) is a common refinement of ( s, B i (cid:48) ) and ( s, B i ), ( s, B ∗ ) (cid:13) ζ (cid:48) ˙ G β (cid:48) ζ ⇐⇒ ζ (cid:48) ˙ G β ζ . Subcase 2b: τ (cid:48) (cid:54) = τ .In this subcase we will consider a “cloned” version ¯ β of β lying in b i (cid:48) ,which we define by setting ¯ β = β ¯ for ¯ the element with index τ in theincreasing enumeration of { : β ∈ b i (cid:48) } . The argument from subcase2a shows that ( s, B ∗ ) (cid:13) ζ (cid:48) ˙ G ¯ β ζ ⇐⇒ ζ (cid:48) ˙ G β ζ .Since β (cid:48) ∈ b i (cid:48) and ζ (cid:48) ∈ a i (cid:48) , and also i (cid:48) < i and i ∈ E , it followsfrom Clause 3 in the definition of E that x β (cid:48) (cid:22) ζ (cid:48) = t ( ζ (cid:48) , φ ) for some φ < i . Now since f i (cid:48) η,β (cid:48) ( ζ (cid:48) ) = f iη,β ( ζ (cid:48) ), x β (cid:48) (cid:22) ζ (cid:48) = x β (cid:22) ζ = t ( ζ (cid:48) , φ ), so thatthe set F ( i ) contains some tuple ( η, σ (cid:48) , τ, φ, ψ ) coding the statement“ f iη,β ( ζ (cid:48) ) = ( t ( ζ (cid:48) , φ ) , ψ )”. This tuple is also in F ( i (cid:48) ), and decoding itsmeaning we find that f i (cid:48) η, ¯ β ( ζ (cid:48) ) = ( t ( ζ (cid:48) , φ ) , ψ ) = f iη,β ( ζ (cid:48) ). Since ζ ∈ a i (cid:48) also, a similar argument shows that f i (cid:48) η, ¯ β ( ζ ) = f iη,β ( ζ (cid:48) ).So now we have f i (cid:48) η,β (cid:48) ( ζ (cid:48) ) = f i (cid:48) η, ¯ β ( ζ (cid:48) ) and f i (cid:48) η,β (cid:48) ( ζ (cid:48) ) = f i (cid:48) η, ¯ β ( ζ (cid:48) ), so that(by Clause 6) for the condition q i (cid:48) ) ( s, B i (cid:48) ) (cid:13) ζ (cid:48) ˙ G β (cid:48) ζ ⇐⇒ ζ (cid:48) ˙ G ¯ β ζ .So ( s, B ∗ ) (cid:13) ζ (cid:48) ˙ G β (cid:48) ζ ⇐⇒ ζ (cid:48) ˙ G β ζ . and we are done. (cid:3) The main construction
We will start with a model V in which GCH holds and κ is super-compact. In this model we define in the standard way [7] a “Laverpreparation” forcing L , and let V = V [ G ] where G is L -generic over V . Let A be the poset Add ( κ + , κ +3 ) V , let G be A -generic over V ,and let V = V [ G ]. MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 Let T be the complete binary tree of height κ + as defined in V .Clearly T has κ +3 branches and every level of T has size at most κ + .For use later we fix an enumeration (cid:104) x β : β < κ +3 (cid:105) of a set of dis-tinct branches, and enumerations (cid:104) t ( α, i ) : i < | Lev α ( T ) |(cid:105) of the levels Lev α ( T ) of T .Since 2 κ +3 = κ +4 in V , a theorem of Shelah [14] implies that in V we have ♦ κ +4 (cof( κ ++ )).Working in V , we will define a forcing iteration with < κ -supportsof length κ +4 . Each iterand Q i will either be trivial forcing or will be κ -closed, parallel countably closed (in the sense of Corollary 4.4) andstrongly κ + -cc. By a suitable adaptation of arguments of Shelah [12],this is sufficient to show that the whole iteration will be κ -directedclosed and strongly κ + -cc. We refer the reader to [2] for a detailedaccount of the chain condition proof, noting (for the experts) that theproperty “parallel countably closed” follows from the property “count-ably closed plus well-met” used in [12] and is sufficient to make theproof from that paper work.The cardinality of the final iteration Q ∗ will be κ +4 . We will have2 κ = κ +4 in V Q ∗ , while 2 κ = κ +3 in the intermediate models of the theiteration. We note that by the closure of Q ∗ , the terms “ V κ ” and “ P κ µ ”have the same meanings in V , V Q ∗ , and every intermediate model.As we build the iteration Q ∗ , we will also (using the diamond from V ) construct a sequence of names ˙ S i such that • ˙ S i is a Q ∗ (cid:22) i -name for every i < κ +4 . • ˙ S i names a pair ( W i , F i ) where W i ⊆ P ( κ ), F i is a family ofpartial functions from κ to V κ , and dom( H ) ∈ W i for all H ∈ F i . • If G ∗ is Q ∗ -generic, and ( W, F ) ∈ V [ G ∗ ] with W ⊆ P ( κ ) and F a family of functions from sets in W to V κ , then { i ∈ κ +4 ∩ cof( κ ++ ) : W ∩ V [ G ∗ (cid:22) i ] = W i and F ∩ V [ G ∗ (cid:22) i ] = F i } is stationary in V [ G ∗ ].This is possible because: • Pairs (
W, F ) as above in the extension by Q ∗ may be coded assubsets of κ +4 , and names for them may be coded as subsets of Q ∗ × κ +4 . • If we enumerate the conditions in Q ∗ as (cid:104) q j : j < κ +4 (cid:105) , then Q ∗ (cid:22) i = { q j : j < i } for almost all i ∈ κ +4 ∩ cof ( κ ++ ). • Q ∗ preserves stationary subsets of κ +4 , by virtue of being κ + -cc.We refer the reader to [2] for a more detailed discussion of this kind ofconstruction. We observe that by κ + -cc, if i < κ +4 with cf( i ) > κ then everysubset of V κ in the extension by Q ∗ (cid:22) i is in the extension by Q ∗ (cid:22) j forsome j < i . We observe also that each of the properties I-III can beformulated as ∀∃ assertions about the power set of V κ .The considerations in the last paragraph imply a crucial reflectionstatement for Properties I-III: If G ∗ is Q ∗ -generic, in V [ G ∗ ] we havea normal measure U and filter F with properties I-III, and we set F = { h : [ h ] U ∈ F } , then for almost all i with cf( i ) > κ we have that: • U ∩ V [ G ∗ (cid:22) i ] and F ∩ V [ G ∗ (cid:22) i ] are elements of V [ G ∗ (cid:22) i ]. • In V [ G ∗ (cid:22) i ], U ∩ V [ G ∗ (cid:22) i ] is a normal measure and the func-tions in F ∩ V [ G ∗ (cid:22) i ] represent a filter with properties I-III.After these preliminaries we can specify the iterands Q i of the iter-ation Q ∗ . We assume that G ∗ (cid:22) i is Q ∗ (cid:22) i -generic and that ( W i , F i ) isthe realisation of ˙ S i , and work in V [ G ∗ (cid:22) i ]. We will set Q i to be trivialforcing unless we have the conditions: • cf( i ) = κ ++ . • W i is a normal measure on κ . • { [ h ] W i : h ∈ F i } is an ultrafilter satisfying properties I-III.In this case we will let Q i be the forcing Q from Section 4, defined in V [ G ∗ (cid:22) i ] from the parameters W i , F i , (cid:104) x β : β < κ +3 (cid:105) , and a suitableenumeration (cid:104) ˙ G iβ : β < κ +3 (cid:105) of canonical names for graphs.We recall from the Introduction we will ultimately force over V [ G ∗ ]with a poset P of the type discussed in Section 3. The forcing P willbe defined from some normal measure U and ultrafilter F , and thepoint of the diamond machinery in the definition of Q i is to anticipatethe poset P (and in particular P -names for graphs on κ + ). To be abit more precise, we will actually anticipate U and F where F = { h :[ h ] U ∈ F } , or to put it another way F is the set of upper parts for theposet P .Recall further from Section 4 that in the case when Q i is not trivialforcing, part of the generic object for Q i will be a partial function h i from κ to V κ such that • dom( h i ) is an unbounded set of inaccessible cardinals • dom( h i ) is eventually contained in each measure one set for themeasure W i . • For all h ∈ F i , h i ( α ) ≤ h ( α ) for all large enough α ∈ dom( h ).The following Lemma will be used in Section 6 to show that oftenenough Q i does its job, by adding a P -name for a graph which willabsorb all graphs whose names lie in V [ G ∗ (cid:22) i ]. MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 Lemma 5.1.
Let G ∗ be Q ∗ -generic. Then in V [ G ∗ ] there is an ultrafil-ter U on P κ κ +4 such that if U is the projection of U to κ , and we per-form the construction of Section 2 to produce an ultrafilter F = F il ( H ) for some U -constraint H , then there are stationarily many i < κ +4 suchthat: (1) U ∩ V [ G ∗ (cid:22) i ] = W i . (2) F ∩ V [ G ∗ (cid:22) i ] = F i , where F = { h : [ h ] U ∈ F } . (3) h i ∈ F . Before starting the proof, we emphasise that the diamond propertyensures that there many i where the first two clauses are satisfied.What takes work is arranging that the the third clause is also satisfied. Proof.
We will construct U as in the standard proof of Laver’s inde-structibility result [7], with the proviso that we will be very carefulabout the construction of the master condition.We begin by falling back to the initial model V , where we will choosean embedding j : V → M with critical point κ witnessing that κ is µ -supercompact for some very large µ , and with the additional propertiesthat the forcing poset A ∗ Q ∗ is the iterand at stage κ in the iteration j ( L ), and that the least point greater than κ in the support of theiteration j ( L ) is greater than µ .Recall that V = V [ G ][ G ] where G is L -generic and G is A -generic. By standard arguments, for any choice of a generic object H tail for j ( L ) /G ∗ G ∗ G ∗ over the model V [ G ∗ ], we have j “ G ⊆ G ∗ G ∗ G ∗ ∗ H tail . We may therefore lift j to obtain a generic embedding j : V [ G ] → M [ G ∗ G ∗ G ∗ ∗ H tail ]. In order to lift further, we willneed to construct master conditions.We will now work in V [ G ∗ ] and perform a recursive construction oflength κ +4 , choosing a decreasing sequence of conditions ( r i , a i , q i ) with( r i , a i , q i ) ∈ j ( L ) / ( G ∗ G ∗ G ∗ ) ∗ j ( A ) ∗ j ( Q ∗ (cid:22) i ). We will arrange that r i (cid:13) a i ≤ j “ G , so that forcing below ( r i , a i ) we obtain H tail ∗ H such that j can belifted to j : V [ G ][ G ] → M [ G ∗ G ∗ G ∗ ∗ H tail ∗ H ]. Keeping this inmind, we will also arrange that( r i , a i ) (cid:13) q i ≤ j “( G ∗ (cid:22) i ) . Using the hypothesis that j witnesses µ -supercompactness and theremark that G ∈ M [ j ( G )], we may argue that for any choice of H tail we have j “ G ∈ M [ G ∗ G ∗ G ∗ ∗ H tail ]. Since j ( A ) is j ( κ + )-directed closed we may find a “strong master condition” a ∈ j ( A ) with a ≤ j “ G . We will therefore choose r to be the trivial condition in j ( L ) / ( G ∗ G ∗ G ∗ ), a to be (a name for) a condition a ∈ j ( A ) with a ≤ j “ G , and q as (a name for) the empty sequence.The limit stages are straightforward, since the choice of µ and j givesenough closure to take lower bounds. If Q i is trivial it is easy to definesuitable r i +1 , a i +1 and q i +1 . so we assume that Q i is non-trivial.Forcing below ( r i , a i , q i ) we can obtain a generic object H tail ∗ H ∗ H ∗ i such that there is a lifted embedding j : V [ G ∗ (cid:22) i ] → M [ j ( G ∗ G ) ∗ H ∗ i ],where j ( G ∗ G ) = G ∗ G ∗ G ∗ ∗ H tail ∗ H . Let g i be the Q i -genericfilter added at stage i by G ∗ , and let h i be the partial function from κ to V κ added by g i .To take the next step, we ask whether it is possible that the set { j ( h )( κ ) : h ∈ F i } has a non-zero lower bound: more formally, weask whether there is a condition extending ( r i , a i , q i ) which forces thisset to have a non-zero lower bound, and define ( r (cid:48) , a (cid:48) , q (cid:48) ) to be such acondition if it exists and to be ( r i , a i , q i ) otherwise. In the case that( r (cid:48) , a (cid:48) , q (cid:48) ) forces that { j ( h )( κ ) : h ∈ F i } has a non-zero lower bound,we let b name the Boolean greatest lower bound for this set. In eithercase we force below ( r (cid:48) , a (cid:48) , q (cid:48) ), lift j and work in M [ j ( G ∗ G ) ∗ H ∗ i ] todefine a condition in j ( Q i ).Let (cid:104) f iη,β : η < κ, β < κ +3 (cid:105) be the family of functions added by g i .We define Q = ( A Q , B Q , t Q , f Q ) as follows: • t Q = κ × j “ κ + × j “ κ +3 . • For all η < κ , α < κ + and β < κ +3 , f Qη,j ( β ) ( j ( α )) = j ( f iη,β ( α )). • If the Boolean value b is not defined, then:(1) A Q = h i .(2) B Q is some upper part such that κ ∩ dom( B Q ) = 0 and B Q ≤ j ( B ) for all upper parts B ∈ F i .If b is defined, then:(1) A Q = h i ∪ { ( κ, b ) } .(2) B Q is some upper part such that ( κ + 1) ∩ dom( B Q ) = 0and B Q ≤ j ( B ) for all upper parts B ∈ F i .In the case when the Boolean value b is not defined, it is routine tocheck that Q is a condition in j ( Q i ) and Q ≤ j “ g i . This is essentiallythe argument of Lemma 4.3 applied to the directed (hence linked) set j “ g i . In the case that b is defined, the definition of b ensures that wewill still have Q ≤ j “ g i so long as we can verify that Q is a condition.As usual the only issue is Clause 6) in the definition of conditionhood.So suppose that η ∈ dom( h i ) ∩ κ , f Qη,j ( β ) ( j ( ζ )) = f Qη,j ( β (cid:48) ) ( j ( ζ )) (cid:54) = f Qη,j ( β ) ( j ( ζ (cid:48) )) = f Qη,j ( β (cid:48) ) ( j ( ζ (cid:48) )) , MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 and y is a lower part which is harmonious with A Q past η . Let x bethe largest initial segment of y which lies in V κ , so that either y = x or y = x (cid:95) (cid:104) κ, p (cid:105) where p ≤ b .By elementarity and the definition of Q , f iη,β ( ζ ) = f iη,β (cid:48) ( ζ ) (cid:54) = f iη,β ( ζ (cid:48) ) = f iη,β (cid:48) ( ζ (cid:48) ) . We now choose q ∈ g i such that η < ρ q , ζ, ζ (cid:48) ∈ a q , β, β (cid:48) ∈ b q , anddom( A q ) contains every ordinal in [ η, κ ) which is mentioned in y . It iseasy to see that x is harmonious with A q past η , and hence (as q is acondition) ( x, B q ) (cid:13) ζ (cid:48) ˙ G β ζ ⇐⇒ ζ (cid:48) ˙ G β (cid:48) ζ .By elementarity ( x, j ( B q )) (cid:13) j ( ζ (cid:48) ) j ( ˙ G β ) j ( ζ ) ⇐⇒ j ( ζ (cid:48) ) j ( ˙ G β (cid:48) ) j ( ζ ).To finish we just observe that by definition (and the choice of b in thecase when it is defined, which ensures that b ≤ j ( B q )( κ ) ) ( y, B Q ) ≤ ( x, j ( B q )).Having chosen Q as above, we let r i +1 = r (cid:48) , a i +1 = a (cid:48) , and q i +1 bethe unique condition such that q i +1 (cid:22) j ( i ) = q (cid:48) and q i +1 ( j ( i )) = Q .At the end of the construction, we obtain ( r ∗ , a ∗ , q ∗ ) ∈ j ( L ) / ( G ∗ G ∗ G ∗ ) ∗ j ( A ) ∗ j ( Q ∗ ) such that r ∗ (cid:13) a ∗ ≤ j “ G , and ( r i , a i ) (cid:13) q ∗ ≤ j “ G ∗ . Forcing below ( r ∗ , a ∗ , q ∗ ) we obtain a generic object H tail ∗ H ∗ H ∗ anda lifted embedding j : V [ G ] → M [ j ( G ∗ G ) ∗ H ∗ ]. Following the ideaof the Laver construction we define U = { A ∈ ( P κ κ +4 ) V [ G ] : j “ κ +4 ∈ j ( A ) } . Since H tail ∗ H ∗ H ∗ is generic over V [ G ] for highly closedforcing we have U ∈ V [ G ], and so U is an ultrafilter witnessing the κ +4 supercompactness of κ in V [ G ]. By the results in Section 2, we mayuse U to define a U -constraint H such that Fil( H ) is an ultrafilter. Itis easy to check that if U is the projection of U to a normal measureon κ , and F is the set of upper parts associated with Fil( H ), then F = { h : h is a U -constraint and j ( h )( κ ) ≥ j ( H )( j “ κ +4 ) } . By the diamond property, there is a stationary set of i ∈ κ +4 ∩ cof( κ +2 ) such that U ∩ V [ G (cid:22) i ] = W i and F ∩ V [ G (cid:22) i ] = F i . For eachsuch i , we observe that for all hh ∈ F i = ⇒ h ∈ F = ⇒ j ( h )( κ ) ≥ j ( H )( j “ κ +4 ) . So { j ( h )( κ ) : h ∈ F i } has a nonzero lower bound, and by the TruthLemma there is a condition in H tail ∗ H ∗ H ∗ i which extends ( r i , a i , q i )and forces this. So when we chose q i +1 , we arranged that j ( h i )( κ ) isthe Boolean infimum of { j ( h )( κ ) : h ∈ F i } . Since j ( H )( j “ κ +4 ) is a lower bound for this set, j ( h i )( κ ) ≥ j ( H )( j “ κ +4 ), and hence h i ∈ F asrequired. (cid:3) Universal graphs
We are now ready to prove the main result. By the results of Section5, we will assume that we have in V [ G ] a measure U on κ , a filter F (with associated set of upper parts F ) and a stationary set S such thatfor every i ∈ S :(1) U ∩ V [ G (cid:22) i ] = W i .(2) F ∩ V [ G (cid:22) i ] = F i .(3) h i ∈ F .We now let P be the forcing poset defined from U and F as inSection 3, and force with P over V [ G ], obtaining a generic sequence x = f , κ , f , κ , f . . . By the characterisation of genericity from Lemma 3.6, we see that forevery i ∈ S x is P i -generic over V [ G (cid:22) i ], where P i is the forcing definedin V [ G (cid:22) i ] from W i and F i .We now define for each i ∈ S a graph U i ∈ V [ G ][ x ], which will embedevery graph on κ + in V [ G (cid:22) i ][ x ]. We begin by using the criterion forgenericity to choose some j such that κ k ∈ dom( h i ) and h i ( κ k ) ∈ f k for all k ≥ j . We set η = κ j .The underlying set of the graph U i is T × κ , and the edges are definedas follows:( z, δ ) U i ( z (cid:48) , δ (cid:48) ) if and only if there exist a lower part t and a condition q ∈ Q i such that(1) q ∈ g i .(2) ( t, B q ) is in the generic filter on P i corresponding to the genericsequence x .(3) t is harmonious with A q past η .(4) There exist β ∈ b q and distinct ζ, ζ (cid:48) ∈ a q such that:(a) f qη,β ( ζ ) = ( z, δ ), f qη,β ( ζ (cid:48) ) = ( z (cid:48) , δ (cid:48) ), and ( t, B q ) (cid:13) ζ ˙ G iβ ζ (cid:48) .Since (cid:104) ˙ G iβ : β < κ +3 (cid:105) enumerates all P i -names for graphs on κ + , itwill suffice to verify that the generic function f iη,β is an embedding of G iβ (the realisation of the name ˙ G iβ ) into the graph U i . One direction iseasy: if f iη,β ( ζ ) U i f iη,β ( ζ (cid:48) ) then by definition there is ( t, B q ) in the genericfilter on P i induced by x such that ( t, B q ) (cid:13) ζ ˙ G iβ ζ (cid:48) , and so by the TruthLemma ζ G iβ ζ (cid:48) . MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 For the converse direction, suppose that ζ G iβ ζ (cid:48) . We may find acondition ( s, B ) in the generic filter induced by x on P i , such that( s, B ) (cid:13) ζ ˙ G iβ ζ (cid:48) . Let s = q , κ , q , . . . , κ n , q n . Extending the condition( s, B ) if need be, we may assume that n ≥ j . Since ( s, B ) is in thegeneric filter induced by x , we have that q m ∈ f m for m ≤ n , while κ m ∈ dom( B ) and B ( κ m ) ∈ f m for m > n .By the properties of the forcing poset Q i , we may find a condition q in g i such that ssup(dom( A q )) > κ n , dom( B q ) ⊆ dom( B ), B q ( α ) ≤ B ( α )for all α ∈ dom( B ), β ∈ b q and ζ, ζ (cid:48) ∈ a q .Recall now that κ k ∈ dom( h i ) and h i ( κ k ) ∈ f k for all k ≥ j , andalso that η = κ j and n ≥ j . Let ¯ n be the largest k such that κ k < ssup(dom( A q )).Define a lower part t as follows: t = q (cid:48) , κ , q (cid:48) , . . . , κ ¯ n , q (cid:48) ¯ n where:(1) q (cid:48) k = q k for k < j .(2) q (cid:48) k = q k ∪ A q ( κ k ) for j ≤ k ≤ n .(3) q (cid:48) k = A q ( κ k ) ∪ B ( κ k ) for n < k ≤ ¯ n .We note that since q ∈ g i and h i is added by g i , A q is an initialsegment of h i and h i ( α ) ≤ B q ( α ) for all α ∈ dom( h i ) \ dom( A q ).For j ≤ k ≤ n we have that q k ∈ f k and A q ( κ k ) = h i ( κ k ) ∈ f k , sothat q k ∪ A q ( κ k ) is a condition and lies in f k . For n < k ≤ ¯ n , again A q ( κ k ) = h i ( κ k ) ∈ f k and also B ( κ k ) ∈ f k , so that A q ( κ k ) ∪ B ( κ k ) is acondition and lies in f k .We will verify that t is harmonious with A q past η , ( t, B q ) extends( s, B ), and ( t, B q ) is in the filter generated by x . This will suffice,since it will then be clear that t and q will serve as witnesses that f iη,β ( ζ ) U i f iη,β ( ζ (cid:48) ).The harmoniousness is immediate from the definitions. ( t, B q ) ex-tends ( s, B ) because q (cid:48) k ≤ q k for k ≤ n , q (cid:48) k ≤ B ( κ k ) for n < k ≤ ¯ n , and B q ≤ B . We already checked that q (cid:48) k ∈ f k for all k ≤ ¯ n , so to finishwe just need to see that B q ( κ k ) ∈ f k for all k > ¯ n ; this is immediatebecause h i ( κ k ) ≤ B q ( κ k ) for all such k , and also h i ( κ k ) ∈ f k for all k ≥ j .To finish the construction of a small family of universal graphs, wewill fix i ∗ ∈ S which is a limit of points of S , and an increasing κ ++ -sequence of points i η ∈ S which is cofinal in i ∗ . By routine chaincondition arguments, every P i ∗ -name for a graph on κ + may be viewedas a P i η -name for some η < κ ++ . We now consider the model V [ G (cid:22) i ∗ ][ x ]. The family of graphs {U i η : η < κ ++ } is universal in this model,where 2 κ = 2 κ + = κ +3 and of course κ = ℵ ω .We have proved: Theorem 6.1.
It is consistent from large cardinals that ℵ ω is stronglimit, ℵ ω = 2 ℵ ω +1 = ℵ ω +3 , and there is a family of size ℵ ω +2 of graphson ℵ ω +1 which is jointly universal for all such graphs. Afterword
There is some flexibility in the proof of Theorem 6.1, in particular itwould be straightforward to modify the construction so that in the finalmodel 2 ℵ ω = ℵ ω + k for an arbitrary k such that 3 ≤ k < ω . Larger valuescan probably be achieved but would require a substantial modificationto the construction.Theorem 6.1 leaves a number of natural questions open: • Can we have a failure of SCH at ℵ ω with u ℵ ω +1 = 1? • On a related topic, what is the exact value of u ℵ ω +1 in the modelof Theorem 6.1? • As far as the authors are aware, the only known results on thevalue of u κ + for κ singular strong limit and 2 κ > κ + are con-sistency results of the kind proved in this paper. In particular,we lack a forcing technique to show that u κ + can be arbitarilylarge.For κ regular adding Cohen subsets to κ makes u κ + arbitrarilylarge, is there an analogous result for κ singular? • The class of graphs is a very simple class of structures. Whatcan be done in more complex classes? • In the model of Theorem 6.1, GCH fails cofinally often below ℵ ω , and in fact 2 ℵ n = ℵ n +4 for unboundedly many n < ω . Isthe conclusion consistent if we demand that GCH holds below ℵ ω ?The authors’ joint paper with Magidor and Shelah [2] contains somerelated work, in which the final “Prikry type” forcing is a version ofRadin forcing and we obtain models where µ is singular strong limit ofuncountable cofinality, SCH fails at µ and u µ + < µ . References [1] James Cummings. A model in which GCH holds at successors but fails atlimits.
Transactions of the American Mathematical Society , 329(1):1–39, 1992.[2] James Cummings, Mirna Dˇzamonja, Menachem Magidor, Charles Morgan,and Saharon Shelah. A framework for forcing constructions at successors ofsingular cardinals. Submitted.
MALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω +1 [3] Mirna Dˇzamonja and Saharon Shelah. Universal graphs at the successor of asingular cardinal. Journal of Symbolic Logic , 68:366–387, 2003.[4] Mirna Dˇzamonja and Saharon Shelah. On the existence of universal models.
Archive for Mathematical Logic , 43(7):901–936, 2004.[5] Matthew Foreman and Hugh Woodin. The generalized continuum hypothesiscan fail everywhere.
Annals of Mathematics , 133(1):1–35, 1991.[6] Menachem Kojman and Saharon Shelah. Nonexistence of universal orders inmany cardinals.
The Journal of Symbolic Logic , 57(3):875–891, 1992.[7] Richard Laver. Making the supercompactness of κ indestructible under κ -directed closed forcing. Israel Journal of Mathematics , 29(4):385–388, 1978.[8] Menachem Magidor. On the singular cardinals problem. I.
Israel Journal ofMathematics , 28:1–31, 1977.[9] Adrian Mathias. Sequences generic in the sense of Prikry.
Journal of the Aus-tralian Mathematical Society , 15(4):409–414, 1973.[10] Alan Mekler. Universal structures in power ℵ . Journal of Symbolic Logic ,55(2):466–477, 1990.[11] William J. Mitchell. How weak is a closed unbounded ultrafilter? In
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Studies in Logic and the Foun-dations of Mathematics , pages 209–230. North-Holland, Amsterdam, 1982.[12] Saharon Shelah. A weak generalization of MA to higher cardinals.
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Proceedings of the American Mathematical Society ,138:2151–2161, 2010.
Department of Mathematical Sciences, Carnegie Mellon Univer-sity, Pittsburgh PA 15213-3890, USA
E-mail address : [email protected] School of Mathematics, University of East Anglia, Norwich, NR47TJ, UK
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Department of Mathematics, University College London, GowerStreet, London, WC1E 6BT, UK
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