A Framework for Forcing Constructions at Successors of Singular Cardinals

A FRAMEWORK FOR FORCING CONSTRUCTIONS AT SUCCESSORS OF SINGULAR CARDINALS

JAMES CUMMINGS, MIRNA DAMONJA, MENACHEM MAGIDOR, CHARLES MORGAN, AND SAHARON SHELAH

Abstract. We describe a framework for proving consistency results about singular cardinals of arbitrary conality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal κ of uncountable conality, while κ+ enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal κ of uncountable conality where SCH fails + and such that there is a collection of size less than 2κ of graphs on κ+ such that any graph on κ+ embeds into one of the graphs in the collection.

Introduction

The class of uncountable regular cardinals is naturally divided into three disjoint classes: the successors of regular cardinals, the successors of singular cardinals and the weakly inaccessible cardinals. When we

2010 Mathematics Subject Classication. Primary: 03E35, 03E55, 03E75. Key words and phrases. successor of singular, Radin forcing, forcing axiom, universal graph. Cummings thanks the National Science Foundation for their support through grant DMS-1101156. Cummings, Dºamonja and Morgan thank the Institut Henri Poincaré for their support through the Research in Paris program during the period 24-29 June 2013. Dºamonja, Magidor and Shelah thank the Mittag-Le er Institute for their support during the month of September 2009. Mirna Dºamonja thanks EPSRC for their support through their grants EP/G068720 and EP/I00498. Morgan thanks EPSRC for their support through grant EP/I00498. This publication is denoted [Sh963] in Saharon Shelah's list of publications. Shelah thanks the United States-Israel Binational Science Foundation (grant no. 2006108), which partially supported this research. We thank Jacob Davis for his attentive reading of and comments on this paper. 1 2 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH consider a combinatorial question about uncountable regular cardinals, typically these classes require separate treatment and very frequently the successors of singular cardinals present the hardest problems. In particular there are subtle constraints (for example in cardinal arith- metic) on the combinatorics of successors of singular cardinals, and consistency results in this area often involve large cardinals.

To give some context for our work, we review a standard strategy for proving consistency results about the successors of regular cardinals. This strategy involves iterating <κ-closed κ+-cc forcing with <κ-supports for some regular cardinal κ, with the plan that the whole iteration will also enjoy the κ+-chain condition. The κ+-chain condition of the iteration will of course ensure that all cardinals are preserved, and is also very helpful in the catch your tail arguments which frequently appear in iterated forcing constructions.

When κ = ω this proof strategy is completely straightforward as any nite support iteration of ccc forcing is ccc; for regular κ > ω we need to assume that κ<κ = κ and that the iterands have some strong form of κ+-cc and some other properties in order to ensure κ+-cc for the iteration (this issue is extensively discussed in 1 below). When κ is singular the strategy is no longer available, and this is one diculty among many in proving consistency results involving singular cardinals and their successors.

Dºamonja and Shelah [5] introduced a new idea, which we briey describe. Initially κ is a supercompact cardinal whose supercompactness is indestructible under <κ-directed closed forcing. The nal model is obtained by a two-step forcing. The rst step, P, is a <κ-directed closed and κ+-cc iteration, whilst the second step is Prikry forcing, P rU , dened from a normal measure U on κ in V [G], where G is a P- generic lter over V . The iteration P is designed to anticipate and deal + with P rU -names for subsets of κ , so that after forcing over V [G] with P rU we obtain the desired consistency result. Dºamonja and Shelah used this method to obtain the consistency, for κ singular strong limit of conality ω, of the existence of a family of fewer than 2κ+ many graphs on κ+ which are jointly universal for all graphs on κ+.

A well known early interaction between model theory and set theory gives one that if 2κ = κ+ there is a saturated graph (i.e., model of the theory of graphs) on κ+ and any such graph is universal. So the point of results such as that of Dºamonja and Shelah is to address FORCING CONSTRUCTIONS 3 the possibility of having small universal families of graphs on κ+ when 2κ > κ+. In this paper we build a similar framework in which the nal forcing step is a version of Radin forcing, and changes the conality of κ to become some uncountable cardinal less than κ. After building the framework, we prove a version of the result on universal graphs mentioned in the last paragraph. The need for large cardinals in the broad context of combinatorics at singular cardinals and their successors is at least partially explained by the theory of core models and covering lemmas. If there is no inner model with a Woodin cardinal then there is an inner model K with many strong combinatorial properties (for example GCH and square hold), and such that κ+ = (κ+)K for every singular cardinal κ. This resemblance between V and K in the absence of inner models with large enough cardinals exerts a strong inuence on the combinatorics of κ+ in , for example it implies that holds in under this hypothesis. V κ V In the instance of the results on universal graphs we obtain a model in which κ is a singular strong limit cardinal where 2κ > κ+, i.e. the singular cardinals hypothesis fails. It is known through work of Gitik, Mitchell and Woodin that the consistency strength of the singular cardinals hypothesis failing alone is exactly that of a measurable cardinal κ of Mitchell order o(κ) = κ++. Thus some large cardinal hypotheses are necessary for these results on universal graphs that we prove. This paper is in some sense a sequel to the work by Dºamonja and Shelah [5], and so we briey record some of the innovations introduced here.

• Prikry forcing at κ is homogeneous and adds no bounded subsets of κ, while Radin forcing has neither of these features. This entails major changes in the analysis of names for κ+ in the nal model, the proof of κ+-cc for the main iteration, and the proof that the nal model has a small jointly universal family.

• The arguments of [5] involve a complex iteration scheme which is used to build a Prikry forcing P rU , and would have become even more complex if we had used it to build a suitable Radin forcing. In this paper we use diamond sequences to achieve similar goals.

• One of the central points is that our main iteration enjoys a strong form of the κ+-chain condition. None of the standard 4 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH preservation theorems were quite suitable to show this, so we took a detour into iteration theory to formulate and prove a suitable preservation theorem. The paper is organized as follows: 1 shows that cardinals are preserved in <κ-support iterations of certain types of <κ-closed, κ+-stationary chain condition forcings (and as a spin-o we give a generalized Mar- tin's axiom for these forcings). 2 is on preservation of diamond through forcing iterations. 3 collects relevant material on Radin forcing. 4 describes the long Prikry/Mathias forcing and its variant for adding Radin names for universal graphs. 5 contains the proof of the stationary κ+-chain condition for the forcings of 4. 6 gives the main iteration. 7 gets a small family of universal graphs at a cardinal of uncountable singular conality. The problem of the existence of a small family of universal graphs has some independent interest, and is also a natural test question for our forcing framework. In a projected sequel to this paper we will consider other applications.

Notation. Card and On are the classes of cardinals and ordinals re- spectively. The size of a set A is denoted by either A or |A|. If X is a set and κ is a cardinal then P (X) is the set of all subsets of X and [X]κ is the set of all subsets of X of size κ. We dene [X]<κ, etc., in the obvious way. If X, Y are sets then X Y is the set of functions from X to Y . We designate names for the projection functions on cartesian products.

If X, Y are sets we dene π0 : X × Y −→ X and π1 : X × Y −→ Y by, if x ∈ X and y ∈ Y then π0(x, y) = x and π1(x, y) = y. We write f · g for the composition of the functions f and g. When f and g agree on dom(f) ∩ dom(g) we write f + g for the unique function h such that dom(h) = dom(f) ∪ dom(g), h dom(f) = f and h dom(g) = g. If X is a set of ordinals we write ssup(X) for the strong supremum of X, that is the least ordinal α such that X ⊆ α, and cl(X) for the closure of X in the order topology. One non-standard piece of notation which will be useful and which we dene here is the following. If y = hyτ | τ < lh(y)i is a sequence of sets and y0 ∈ Card we often write κy for y0, the rst entry in y. FORCING CONSTRUCTIONS 5

If , are regular cardinals with we write, on occasion, κ for µ κ µ < κ Sµ , and similarly for κ , κ and so on. If is a {ξ < κ | cf(ξ) = µ} S<µ S≥µ S stationary subset of κ we say that T ⊆ S is a club relative to S if there is some club C ⊆ κ such that T = S ∩ C. A function g : κ −→ κ is regressive on a set S ⊆ κ if for every ε ∈ S we have g(ε) < ε. A subset X of a partial order (P, ≤) is linearly ordered if for all p, q ∈ X either p ≤ q or q ≤ p, directed if every nite subset of X has a lower bound in X, and centred if every nite subset of X has a lower bound in P. A partial order (P, ≤) is <κ-closed if every linearly ordered subset of size <κ of P has a lower bound, <κ-directed closed if every directed subset of size <κ of P has a lower bound, and <κ-compact if every centred subset of size <κ of P has a lower bound. P is countably compact if and only if it is <ω1-compact. Clearly any <κ-compact partial order is <κ-directed closed, and any <κ-directed closed partial order is <κ-closed. If (P, ≤) is a partial order and q, r ∈ P we write q k r to mean that q and r are compatible in P, that is there is some p ∈ P such that p ≤ q, r. If (Q, ≤) is a sub-partial order of (P, ≤) and q, r ∈ Q we write q kQ r to mean q and r are compatible in Q, that there is some p ∈ Q such that p ≤ q, r. A partial order is splitting if every element has two incompatible extensions.

A partial order (P, ≤) is well-met if every compatible pair of elements has a greatest lower bound: i.e., for all p, q ∈ P if p k q then there is some r ∈ P such that r ≤ p, q and for all s ∈ P with s ≤ p, q we have s ≤ r. For a regular cardinal κ, a partial order P has the κ+-stationary chain + condition if and only if for every sequence hpi | i < κ i of conditions in there exist a club set + and a regressive function on κ+ P C ⊆ κ f C ∩ Sκ such that for all κ+ with the conditions α, β ∈ C ∩ Sκ f(α) = f(β) pα and pβ are compatible. We note that by an easy application of Fodor's lemma, this property implies that P enjoys the strengthened form of the κ+-Knaster property in which any κ+-sequence of conditions has a stationary subsequence of pairwise compatible conditions.

When forcing with a partial order (P, ≤) over a model V we take the notions of names and canonical names to be as in [8] writing x˙ for a P-name in V and yˆ for a standard P-name for y ∈ V . We will freely use ˙ the well-known Maximum Principle, which states that if ∃xφ(x, y,ˆ G) 6 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

˙ then there is a name x˙ such that φ(x, ˙ y,ˆ G). Usually our usage should be clear: as is customary we take the Boolean truth values 0, 1 to be identied with the (ordinal) elements of {0, 1}. Thus forcing names for truth values are names for ordinals < 2. In the context of forcing, if p, q ∈ P then p ≤ q means that p is stronger than q.

The Laver iteration L for a supercompact cardinal κ is a κ-cc forcing poset of cardinality κ which make the supercompactness of κ indestructible under <κ-directed closed forcing [9].

If G = (X,E) and H = (Y,F ) are graphs then f : G −→ H is an embedding of G into H as an induced subgraph if ∀x0 ∈ X ∀x1 ∈ X ( x0 E x1 ←→ f(x0) F f(x1)). Since this is the only kind of graph embedding which concerns us we will simply call them embeddings.

Let κ be a regular cardinal. We say that a family F of graphs on κ is jointly universal for graphs of size κ if for every graph G = (κ, E) there is some H = (κ, F ) ∈ F and some embedding f : G −→ H of G into H. We say that F is a small universal family if F < 2κ.

If κ+ is a successor cardinal a κ+-tree is a tree of size κ+, height κ+ and each level of size at most κ.A binary κ+-tree is a κ+-tree such that each point in the tree has exactly two successors in the tree order. If Υ is a cardinal greater than κ+ a κ+-Kurepa tree with Υ-many branches is a κ+-tree with Υ-many branches of length κ+. If there is such a tree there is one which is binary and a subtree of the complete binary tree <κ+ 2 (see, for example, [8], Chapter (2)). In a slight abuse of terminology, we refer here to binary κ+-Kurepa subtrees of <κ+ 2 as binary κ+-Kurepa trees.

The usual forcing to add a κ+-Kurepa tree with Υ-many branches, due to Solovay(?), is the forcing notion in which conditions are pairs (t, f) consisting of a binary sub-tree, t, of <κ+ 2 of successor height, say γ +1, < κ+, each level of size at most κ and each point in the tree having a successor of height γ, and a bijection f between a subset of Υ and the set of points of height γ. The ordering is that p = (tp, f p) ≤ q = (tq, f q) if tq is obtained by, in Kunen's vivid précis, `sawing o tp parallel to the ground,' dom(f q) ⊆ dom(f p) and f q(ξ)

1. Some iterated forcing theory

In the classical exposition of iterated forcing Baumgartner [1, 4] wrote The search for extensions of MA for larger cardinals has proved to be rather dicult. One reason for this is that for κ regular and uncountable, an iteration of <κ-closed and κ+-cc forcing posets with supports of size less than κ does not in general have the κ+-cc. For example, a construction due to Mitchell described in a paper of Laver and Shelah [10] shows that in L there is an iteration of length ω of countably closed ℵ2-cc forcing posets such that the inverse limit at stage ω does not have the ℵ2-cc. The literature contains several preservation theorems for iterations involving strengthened forms of closure and chain condition, along with corresponding forcing axioms. The rst results in this direction are in unpublished work by Laver [1, 4]. Baumgartner [1] proved that under CH an iteration with countable supports of countably compact

ℵ1-linked forcing posets is ℵ2-cc, and proved the consistency of some related forcing axioms. Shelah [13] proved that under CH an iteration with countable supports of posets which are countably closed and well- met and which enjoy the stationary ℵ2-cc also enjoys the stationary ℵ2-cc. Shelah also proved more general results for certain iterations of stationary κ+-cc posets with supports of size less than κ, and proved the consistency of a number of related forcing axioms. The main result of this section is a common generalisation of the results of Baumgartner and Shelah quoted above. In order to state the theorem we require a denition. Denition 1.1. Let (P, ≤) be a partial order.

• Two descending sequences hqi | i < ωi and hri | i < ωi from P are pointwise compatible if for each i < ω we have qi k ri. • (P, ≤) is countably parallel-closed if each pair of pointwise compatible descending ω-sequences has a common lower bound. Theorem 1.2. Let κ be an uncountable cardinal with κ<κ = κ. Every iteration of countably parallel-closed, <κ-closed and κ+-stationary cc forcing with supports of size less than κ has κ+-stationary cc.

˙ Proof. Let P = hhPξ | ξ ≤ χi, hQξ | ξ < χii be a <κ-support iteration ˙ of forcings such that for each ξ < χ it is forced by Pξ that Qξ is countably parallel-closed, <κ-closed and κ+-stationary-cc. We prove, 8 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH by induction on ξ, and following the proof of Lemma (1.3) of [13] closely, + that for each ξ ≤ χ, Pξ has the κ -stationary chain condition. + Suppose that Pε has the κ -stationary chain condition for all ε < ξ. + κ+ + Let hpi | i < κ i ∈ Pξ. We construct for each i < κ a decreasing -sequence n such that 0 and n for each . ω hpi | n < ωi pi = pi pi ∈ Pξ n < ω Induction step . Suppose we have already dened n + . n + 1 hpi | i < κ i For each + and we have n ˙ and n i < κ ε < ξ Pε pi (ε) ∈ Qε pi (ε) = 1q˙ε if n . As ˙ has the stationary +-cc and n ε∈ / supp(pi ) Pε Qε κ hpi (ε) | i < + names a +-sequence of elements of ˙ , we may nd -names ˙ n κ i κ Qε Pε Cε for a club subset of + and n for a regressive function on ˙ n witnessing κ g˙ε Cε the +-stationary cc for the sequence named by n + . κ hpi (ε) | i < κ i As has the +-chain condition, if ˙ is a club subset of Pε κ Pε D κ there is some club 0 such that 0 ˙ . So we may as D ∈ V Pε D ⊆ D well assume that ˙ n ˆn for some club n . Cε = Cε Cε ∈ V For each +, dealing with each n inductively and using i < κ ε ∈ supp(pi ) the -closure of , nd some n+1 n such that <κ Pξ pi ≤ pi n n+1 n n ∀ε ∈ supp(pi ) pi ε Pε g˙ε (i) = ρε (i) for some ordinal n . Let n for n . ρε (i) < i ρε (i) = 0 ε∈ / supp(pi ) Let , for some +, enumerate S n {εα | α < µ} µ ≤ κ {supp(pi ) | n < + . For each let T n and let ω & i < κ } ε < ξ Cε = n<ω Cε C = {i < + . κ | ∀α < i (i ∈ Cεα )} Using the hypothesis that κ<κ = κ, we can nd a club E ⊆ C and a regressive function on κ+ such that if , 0 κ+ , 0 g Sκ ∩E i i ∈ Sκ ∩E g(i) = g(i ) and i < i0 then [ n (1) {supp(pi ) | n < ω} ∩ {εγ | γ < i} = [ n 0 {supp(pi0 ) | n < ω} ∩ {εγ | γ < i } [ n 0 (2) {supp(pi ) | n < ω} ⊆ {εγ | γ < i } 0 n n n 0 If , and 0 then . (3) γ < i n < ω εγ ∈ supp(pi ) ρεγ (i) = ρεγ (i )

Note that, by (1), for every γ to which clause (3) applies we have γ < i and S n . εγ ∈ {supp(pi ) | n < ω} Suppose that 0 +, , 0 κ+ and 0 . We i < i < κ i i ∈ Sκ ∩ E g(i) = g(i ) now construct, by an induction of length ξ, a condition q ∈ Pξ which is a common renement of n and n for all , and hence of and pi pi0 n pi FORCING CONSTRUCTIONS 9

. The support of will be the union of S n and pi0 q {supp(pi ) | n < ω} S n . {supp(pi0 ) | n < ω} For S n S n , as σ ∈ {supp(pi ) | n < ω}\ {supp(pi0 ) | n < ω} ˙ is countably closed , take to be any such that Pσ Qε q(σ) r q n . Similarly, for S n σ Pσ ∀n < ω (r ≤ pi (σ)) σ ∈ {supp(pi0 ) | n < ω} S n , take to be any such that \ {supp(pi ) | n < ω} q(σ) r q σ Pσ n . ∀n < ω (r ≤ pi0 (σ)) Finally, if S n S n then σ ∈ {supp(pi ) | n < ω}∩ {supp(pi0 ) | n < ω} σ = εγ for some γ < i (by conditions (1) and (2) above), for each n < ω we have n n 0 (by (3) above), and , 0 (by the construction ρεγ (i) = ρεγ (i ) i i ∈ Cεγ of C and the choice of E). n n By construction, for each n < ω we have q σ p (σ) k p 0 (σ) . Pσ i Q˙ σ i By the fact that ˙ is countably parallel-closed , we can choose Pσ Qσ to be some such that n n . q(σ) r q σ Pσ ∀n < ω (r ≤ pi (σ), pi0 (σ)) Thus for all we have that n , n . n < ω q (σ + 1) ≤ pi (σ + 1) pj (σ + 1) N1.2 We remark that the previous theorem allows us to give `generalised Martin's axiom' forcing axioms similar to those formulated by Baum- gartner [1] and Shelah [13]. One example is given by the following theorem. Theorem 1.3. Let κ be an uncountable cardinal such that κ<κ = κ, and let λ > κ+ be a cardinal such that γ<κ < λ for every γ < λ. Then there is a <κ-closed and stationary κ+-cc forcing poset P such that if G is P-generic, then in V [G] we have 2κ = λ and the following forcing axiom holds: For every poset Q which is <κ-closed, countably parallel-closed and + stationary κ -cc, every γ < λ and every sequence hDi | i < γ i of dense subsets of Q there is a lter on Q which meets each set Di.

As in [1] and [13], there are variations with weaker hypotheses on λ, yielding weaker forcing axioms which only apply to posets Q of bounded size. The following straightforward lemma indicates some connections between the hypotheses used in [1] and [13], and the ones used here. Lemma 1.4. Let P be a forcing poset. (1) If P is well-met and countably closed then it is countably compact. 10 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

(2) If P is countably compact then it is countably parallel-closed. (3) If P is κ-linked then it is stationary κ+-cc.

2. Preserving diamond under forcing

In this section we give an account of how versions of the diamond principle at a regular cardinal χ are preserved by certain forcing posets. We are most interested in the situation where the forcing poset is an iteration Pχ of length χ, and we can nd a diamond sequence hSα | α < χi ∈ V [Gχ] such that Sα ∈ V [Gα] for all α. We will use the results of this section in 6 and 7. The following result is well-known. For the reader's convenience we will sketch a proof. Theorem 2.1. Let χ be a regular uncountable cardinal, let P be a forcing poset of cardinality χ which preserves the regularity of χ and let G be P-generic over V .

(1) If ♦χ holds in V and P preserves stationary subsets of χ then ♦χ holds in V [G]. (2) If ∗ (resp +) holds in then then ∗ (resp +) holds in ♦χ ♦χ V ♦χ ♦χ V [G].

Proof. (1). Enumerate the elements of P as pi for i < χ, x in V a diamond sequence hTα ⊆ α×α | α < χi which guesses subsets of χ×χ, and in V [G] dene Sα = {η < α | ∃i < α (pi ∈ G and (i, η) ∈ Tα)}. Now if ˙ G we let ˙ , and observe S = S ⊆ χ T = {(i, η) | pi ηˆ ∈ S } that {α | T ∩ α × α = Tα } is stationary in V (hence also in V [G]) and and ˙ is club in ; at E = {α | ∀η ∈ S ∩ α ∃i < α pi ∈ G pi ηˆ ∈ S } V [G] any point α ∈ E such that T ∩ α × α = Tα we have that S ∩ α = Sα. (2). The proofs for ∗ and + are very similar. The key dierence is ♦χ ♦χ that the hypothesis that the regularity of χ is preserved suces to see that E is club in V [G], and the stronger form of guessing occurring in the ∗ and + gives a club from with which to intersect it. ♦χ ♦χ V N2.1 Observation 2.2. For future reference we note that: (1) An entirely similar argument shows that if B is a stationary subset of χ and P preserves stationary subsets of B, then P preserves ♦χ(B). If P preserves the stationarity of B then P preserves ∗ and + . ♦χ(B) ♦χ (B) FORCING CONSTRUCTIONS 11

(2) In both parts of Theorem (2.1), the value of the diamond sequence in V [G] at α is computed in a uniform way from G α where . In fact the restriction of the G α = G ∩ {pi | i < α} diamond sequence to α + 1 can be computed in a uniform way from G α. (3) In the proof of part (1) of Theorem (2.1), let A be the stationary set in V [G] given by A = {α | S ∩ α = Sα }. For every α ∈ E, S ∩ α ∈ V [G α] and so easily E ∩ α ∈ V [G α]. Also, by the preceding remarks, A ∩ α ∈ V [G α]. If we now let H = A ∩ E then H is a stationary set on which S is guessed correctly and additionally H ∩ α ∈ V [G α] for all α in the closure of H. Proposition 2.3. Let κ, χ be regular cardinals with κ < χ. Let P = ˙ hhPξ | ξ ≤ χi, hQξ | ξ < χii be a forcing iteration with <κ-supports with Pξ < χ for each ξ < χ. Let G be P-generic over V . Suppose is a stationary subset of χ and holds. Then there A S≥κ ♦χ(A) ˙ is a sequence hSξ | ξ ∈ Ai such that ˙ (1) For all ξ ∈ A, Sξ is a Pξ-name for a subset of ξ. ˙ (2) If in V [G] we dene Sξ as the interpretation of Sξ by Gξ then the sequence hSξ | ξ ∈ Ai has the following strengthened form of the ♦χ(A)-property: for every S ⊆ χ there is a stationary set H ⊆ A such that S ∩ ξ = Sξ for all ξ ∈ H, and in addition H ∩ ξ ∈ V [Gξ] for all ξ in the closure of H.

Proof. We start by observing that by arguments as in Baumgartner's survey paper on iterated forcing [1, 2] the poset P is χ-cc. We sketch the proof briey: given a sequence hpi | i < χi of elements of P we apply Fodor's theorem and the bound on the size of initial segments to nd a stationary set χ and an ordinal such that U ⊆ Sκ η < χ supp(pi) ∩ i ⊆ η and is constant for , then nd such that pi η i ∈ U i, j ∈ U supp(pi) ⊆ j and argue that pi is compatible with pj.

Noting that |P| = χ, we enumerate P as hpξ | ξ < χi and let F be {ξ < χ | Pξ = {pε | ε < ξ }}. Since P is an iteration with <κ-support, is club relative to χ and hence is non-stationary. F S≥κ A \ F Since P is χ-cc it preserves stationary subsets of χ, and we may therefore appeal to Theorem (2.1) and Observation (2.2) to obtain a se- ˙ quence hSξ | ξ ∈ A ∩ F i such that ˙ (1) For all ξ ∈ A ∩ F , Sξ is a Pξ-name for a subset of ξ. 12 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

˙ (2) If in V [G] we dene Sξ as the realisation of Sξ by Gξ, then for every S ⊆ κ there is a stationary set H ⊆ A ∩ F such that S ∩ ξ = Sξ for all ξ ∈ H and additionally H ∩ ξ ∈ V [Gξ] for all ξ in the closure of H. ˙ ˆ To nish the proof we ll in the missing values by dening Sξ = ∅ for . ξ ∈ A \ F N2.3

3. Radin forcing

As we commented in the preamble, our proof involves Radin forcing. There are several accounts of this forcing in the literature (see [12], [11], [3], [6] and [7]), each subtly dierent from the others, and it turns out that it does matter which version of the forcing we use. Although the proofs of the various properties of the forcing are easiest, or at least slickest, for the versions given in Cummings-Woodin ([3]) and in Gitik ([6]), using either of these here creates technical diculties. Conse- quently, we shall dene and use a version of Radin forcing which, ex- cept for one small alteration to which we draw attention below, closely follows that of Mitchell in [11].

First of all, we give, by induction on κ ∈ Card, the denition of the set of ultralter sequences at κ.

Denition 3.1. Let κ be a cardinal and let Uκ be the collection of ultralter sequences at cardinals smaller than κ. A sequence u, = huτ | τ < lh(u)i, is an ultralter sequence at κ if lh(u) is a non-zero ordinal, u0 = κ, so that, using the notation of the introduction, κu = κ, and, for τ ∈ (0, lh(u)), each uτ is a κ-complete ultralter on Vκ with Uκ ∈ uτ and satises the following normality and coherence conditions (with respect to u): (normality) if and then f : Uκ −→ Vκ {w ∈ Uκ | f(w) ∈ Vκw } ∈ uτ there is some x ∈ Vκ such that {w ∈ Uκ | f(w) = x} ∈ uτ .

(coherence 1) if f : Uκ −→ κ and {w ∈ Uκ | f(w) < lh(w)} ∈ uτ there is such that . σ < τ uσ = {X ⊆ Vκ | {w ∈ Uκ | X ∩ Vκw ∈ wf(w) } ∈ uτ } (coherence 2) if and then σ < τ X ∈ uσ {w ∈ Uκ | ∃σ < lh(w) X∩Vκw ∈ wσ } ∈ uτ . Denition 3.2. The class U is the class of all ultralter sequences on any cardinal: S . U = κ∈Card Uκ FORCING CONSTRUCTIONS 13

Denition 3.3. For clarity we re-iterate the following special cases of notation of Denition (3.1): Uκ+ is the set of ultralter sequences at cardinals less than or equal to κ and Uκ+ \ Uκ is the set of ultralter sequences at κ. Observation 3.4. As Mitchell comments in [11], by the coherence properties, if u ∈ U and lh(u) ≤ κu then for τ ∈ (0, lh(u)) one has that concentrates on (i.e., uτ {w ∈ Uκu | lh(w) = τ } {w ∈ Uκu | lh(w) = τ } ∈ uτ ). Observation 3.5. We need some large cardinal assumption in order to construct non-trivial ultralter sequences. For the purposes of this paper we can use a construction due to Radin [12]. Let j : V → M witness that κ is 2κ-supercompact. Derive a sequence u by setting and then for . One can u0 = κ uα = {X ⊆ Vκ | u α ∈ j(X)} α > 0 verify that for every κ + we have . In fact we will α < (2 ) u α ∈ Uκ only need ultralter sequences of length less than κ in the sequel. Denition 3.6. If then T . u ∈ U F (u) = τ∈(0,lh(u)) uτ Denition 3.7. A pair is some (u, A) with u ∈ U and A ∈ F (u) if lh(u) > 1 and A = ∅ if lh(u) = 1.

(Note that our notation thus far is marginally, but inessentially, dierent from that of [11]).

Denition 3.8. Let w ∈ U. Rw, Radin forcing at w, has as conditions sequences of pairs h(u0,B0),..., (un,Bn)i such that un = w, and, writing κi for κu , (ui,Bi) ∈ κ and Bi+1 ∩ V + = ∅ for i < n. i U i+1 κi

Let p = h(u0,B0),..., (un,Bn)i and q = h(v0,D0),..., (vm,Dm)i ∈ Rw. Then q ≤ p (q renes p) if m ≥ n and

(i) For every i ≤ n there is j ≤ m such that ui = vj and Dj ⊆ Bi.

(ii) For every j ≤ m, either vj = ui for some i or for the least i such that , and . κvj < κui vj ∈ Bi Dj ⊆ Bi ∗ We also dene q ≤ p if hui | i ≤ ni = hvi | i ≤ ni and for each i ≤ n we have Di ⊆ Bi. We say that q is a (Radin-)direct extension of p. Thus q ≤∗ p implies q ≤ p.

Denition 3.9. Let p = h(u0,B0),..., (un,Bn)i be a condition in . A pair appears in if there is some such that Run (u, B) p i ≤ n (u, B) = (ui,Bi). Similarly an ultralter sequence u appears in p if there is some i ≤ n such that u = ui. 14 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

Thus clause (ii) in the denition of ≤ in Denition (3.8) reads: if vj does not appear in and is minimal such that then p i κvj < κui vj ∈ Bi and Dj ⊆ Bi. Observation 3.10. [11] omits the last clause in the denition of what it is to be a condition. However our conditions form a dense subset of the conditions as dened in [11] and the facts that we quote from [11] are also true of our forcing. This minor change is advantageous for technical reasons in order to make the proof below run smoothly.

To help orient the reader, we record a few remarks (without proof) about the nature of the generic object for the forcing poset Rw. This generic object is best viewed as a sequence where hui | i < δi ui ∈ Uκw and is increasing and continuous. When the hκui | i < δi lh(w) > 1 sequence is conal in , and we will view this sequence hκui | i < δi κw as enumerating a club set in κw which we call the Radin-generic club set. When i = 0 or i is a successor ordinal then lh(ui) = 1, otherwise lh(ui) > 1. The translation between the generic sequence and the generic lter is given by the following.

Denition 3.11. Let G be an Rw-generic lter over V . The sequence hui | i < δi is the corresponding generic sequence if it enumer- ates appears in , i.e., the set of which {u ∈ Uκw | ∃p ∈ G u p} u ∈ Uκw appear in some condition in G.

Lemma 3.12. Let G be an Rw-generic lter over V and hui | i < δi the corresponding generic sequence. Then appears in G = {p ∈ Rw | ∀u ∈ Uκw (u p −→ ∃i < δ u = ui)&

∀i < δ ∃q ≤ p (ui appears in q)}, i.e. is the set of conditions such that every which G p ∈ Rw u ∈ Uκw appears in p is among the sequences ui, and every sequence ui appears in some extension of . p N3.12

Denition 3.13. A lower part for Rw is a condition in Ru for some . u ∈ Uκw _ We note that any condition in Rw has the form y (w, B) where y is empty or a lower part for Rw, and B ∈ F (w). In the case when y is non-empty we will say that y is the lower part of y_(w, B). It is easy to see that any two conditions with the same lower part are compatible, so that Rw is the union of κw-many κw-complete lters and in particular it enjoys the +-chain condition. κw FORCING CONSTRUCTIONS 15

A key point is that below a condition of the form y_(w, B) with y non-empty, everything up to the last ultralter sequence appearing in y is controlled by pairs in y. More formally:

Denition 3.14. Let y, = h(ui,Bi) | i ≤ ni, be a lower part for Rw. Then and . κy = κun Ry = {q ∈ Run | q ≤ y} _ Lemma 3.15. Let p = y (w, B) be a condition in Rw with y non- empty. Then the subforcing {q ∈ Rw | q ≤ p} is isomorphic to the product R × {q ∈ R | q ≤ (w, B \ + )}. y w Uκy

It follows from these considerations that if hui | i < δi is a generic sequence for Rw and ζ < δ, then hui | i < ζ i is a generic sequence for . More generally, if there is a condition in the generic lter with Ruζ lower part y then the generic sequence induces an Ry-generic object in the natural way.

Denition 3.16. Let y be a lower part for Rw and let G be Rw-generic over V . We say that y conforms with G if and only if y is the lower part of some condition in G.

The following Lemma is a local version of Lemma (3.12).

Lemma 3.17. Let G be an Rw-generic lter over V and let hui | i < δi be the corresponding generic sequence. Let y = h(vk,Bk) | k ≤ ni be a lower part. Then the following are equivalent: (1) y conforms with G.

(2) For every k ≤ n the sequence vk appears among the sequences , and for every with the ultralter sequence ui i κui ≤ κvn ui appears in some extension of y in Ry. 0 Moreover, if i < δ then ui appears in some lower part y which conforms with and with 0 , and such that if _ G κy = max({κy, κui }) y (w, B) ∈ G there is some 0 such that 0_ 0 _ . B ∈ F (w) y (w, B ) ≤ y (w, B) N3.17

The following result by Radin captures a key property of Rw Theorem 3.18 (Radin). (The Prikry property for Radin forcing.) Let p ∈ Rw and let φ be a sentence in the forcing language. Then either there is some 0 ∗ such that 0 or there is some 0 ∗ such p ≤ p p Rw φ p ≤ p that 0 . p Rw ¬φ Combining Theorem (3.18) and Lemma (3.15), we obtain a lemma (due to Radin) which will be very important in the proof of κ+-cc for the 16 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH main iteration. Recall that by our conventions a name for a truth value is just a name for an ordinal which is either 0 (false) or 1 (true). _ Lemma 3.19 (Radin). Let y (w, B) be a condition in Rw with y non- ˙ empty, and let b be a Rw-name for a truth value. Then there exist C ⊆ B with C ∈ F (w) and a Ry name for a truth value c˙ such that _ 0 whenever G is Rw-generic with y (w, C) ∈ G and G is the induced ˙ G G0 generic object for Ry, then (b) = (˙c) .

Proof. Using the factorisation given by Lemma (3.15) and the Product ˙ ∗ Lemma, we may view b as an R -name for a Ry-name for a truth value ∗ where R = {q ∈ R | q ≤ (w, B \ + )}. Since |R | < κ and a name w Uκy y for a truth value amounts to an antichain in Ry, there are fewer than κ many Ry-names for truth values. Since F (w) is κ-complete we may ∗ appeal to the Prikry property for R , and shrink (w, B \ + ) to (w, C) Uκy in order to decide which -name is in question. Ry c˙ N3.19

In the situation of Lemma (3.19), we will sometimes say that the con- _ ˙ dition y (w, C) reduces the Rw-name b to the Ry-name c˙. Similar arguments (which we omit) about reducing names for sets of ordinals give another important result.

Theorem 3.20 (Radin). Let w ∈ U, let G be Rw-generic over V and let huj | j < δi be the corresponding generic sequence. For every , if is largest such that , then every subset of lying α < κw i κui ≤ α α in lies in the -generic extension given by . V [G] Rui huj | j < ii

It follows readily that forcing with Rw preserves all cardinals. However conalities may change. The general situation is slightly complicated, the main point for us is given by the following result.

Theorem 3.21 (Radin). Let w ∈ U, let G be Rw-generic over V and assume that lh(w) is a regular cardinal λ with λ < κw. Let G be Rw-generic. If λ is not a limit point of the Radin-generic club, then V [G] |= λ is regular and cf(κ) = λ.

We note that the condition on λ in this theorem is easy to arrange by working below a suitable condition in Rw. For example we may arrange that the least point of the generic club is greater than λ. We will also require a characterisation of Radin-genericity which is due to Mitchell. FORCING CONSTRUCTIONS 17

Theorem 3.22 (Mitchell). Let hui | i < δi be a sequence of ultralter sequences in some outer model of V . Then the following are equivalent:

(1) The sequence hui | i < δi is Rw-generic. (2) For every the sequence is -generic, and if j < δ hui | i < j i Ruj lh(w) > 1 then F (w) = {X ∈ V | ∃j < δ ∀i j < i < δ =⇒ ui ∈ X } i.e., if lh(w) > 1 then F (w) is the tail lter generated on the V -powerset of Vκ by the generic sequence. Denition 3.23. Sequences which satisfy the Mitchell criterion from Theorem (3.22) are sometimes called geometric sequences.

The following considerations will play a central rôle in the proof of Theorem (7.10).

Observation 3.24. Suppose that 0 with V V 0 (e.g., 0 V ⊆ V Vκ = Vκ V might be a generic extension of V by <κ-closed forcing). Suppose that in V 0 there is an ultralter sequence w0 such that lh(w) = lh(w0) and 0 w(α) = w (α) ∩ V for 0 < α < lh(w). Let hui | i < δi be a Rw0 -generic sequence over V 0. • By the Mitchell criterion for genericity and the observation that 0 F (w) ⊆ F (w ), the sequence hui | i < δi is also a Rw-generic sequence over V .

Lemma 3.25. Let G be an Rw-generic lter over V and let hui | i < δi be the corresponding generic sequence. Let y = h(vk,Bk) | k ≤ ni be a lower part. Then the following are equivalent: (1) y conforms with G.

(2) For every k the sequence vk appears among the sequences ui, and for every with either appears among the i κui ≤ κvn ui sequences or for the least such that . vk ui ∈ Bk k κui < κvk Proof. It is immediate that the rst condition implies the second. For the converse direction, by Lemma (3.17), we need to verify that every sequence with appears in some element of . This is ui κui ≤ κvn Ry immediate when ui appears among the vk, so we assume that it does not and let be least with . If then we may k κui < κvk lh(ui) = 1 extend y by shrinking Bk to Bk \ κ and inserting the pair (ui, ∅). If U u+ i lh(ui) > 1 then i is limit and by hypothesis uj ∈ Bk for all large j < i, so that (by the geometric condition from part (2) of Theorem (3.22)) B ∩ V ∈ (u ). In this case we may extend y by shrinking B to k κui F i k 18 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

Bk \ κ and inserting the pair (ui,E) where E = B0 ∩ Vκ when U u+ ui i k = 0, and E = B ∩ V when k > 0. k κui N3.25

A very similar argument gives:

Lemma 3.26. Let G be an Rw-generic lter over V and let hui | i < δi be the corresponding generic sequence. Let y_(w, B) be a condition. _ Then y (w, B) ∈ G if and only if y conforms with G and ui ∈ B for all such that . i κui > κy

For use later we make a technical denition which is motivated by the Mitchell criterion for genericity, and will be used in the denition of the main iteration.

Denition 3.27. For y = h(ui,Bi) | i < ni a lower part for Rw, A = ∗ a sequence of subsets of and we say is hAρ | ρ < ρ i Vκw η < κw y harmonious with A past or above η, if for each i with i < n one of the following conditions holds. . • κui < η , and . • κui = η ui = hηi Bi = ∅

, and S + . • κui > η {ui } ∪ Bi ⊆ ρ<ρ∗ Aρ \ Uη Observation 3.28. We record some remarks about the preceding definition.

• If a lower part y is harmonious with A past η it divides rather strictly into a part below η and a part above: there is no (ui,Bi) appearing in such that while for some . y η ≤ κui κv ≤ η v ∈ Bi Denition (3.27) depends only on the set S rather than • ρ<ρ∗ Aρ the sequence A itself. It is phrased in this way to avoid encum- bering later denitions with union signs.

The following lemma shows how to thin a lower part conforming with a Radin generic sequence to one which still conforms and which is also harmonious past some η with a sequence of sets whose union contains the interval of the generic sequence consisting of those measure sequences with critical point at least η and which it can `see'. This, also, is useful for the proof of Theorem (7.10).

Lemma 3.29. Let G be an Rw-generic lter over V and let hui | i < δi ∗ be the corresponding generic sequence. Let A = hAρ | ρ < ρ i be such FORCING CONSTRUCTIONS 19 that each Aρ is a set of measure sequences with a common critical point , let ∗ and let S . κρ < κ a = {κρ | ρ < ρ } D = ρ<ρ∗ Aρ Suppose that η is a successor point in the generic club set C such that for all such that . ui ∈ D i η ≤ κui < ssup(a)

Let y be a lower part conforming with G with κy < ssup(a), and suppose that hηi appears in y. Then there is a lower part y0 such that

0 (1) y is a direct extension of y in Ry, (2) y0 conforms with G, and (3) y0 is harmonious with A past η.

Proof. By the denition of conformity all sequences appearing in y have 0 the form ui for some i. We will obtain y by some judicious shrinking of the measure one sets appearing as the second entries of pairs in y.

Consider the pairs (v, B) appearing in y. If κv < η there is no problem, and if κv = η then (as η is a successor point in C) we have v = hηε i and B = ∅, so again there is no problem.

Suppose now that (v, B) appears in y and κv > η. By the denition of conditionhood, B contains no measure sequence u with κu ≤ η. Let j be such that v = uj, and recall that lh(v) = 1 and B = ∅ for j a successor, and lh(v) > 1 and B ∈ F (v) for j a limit. By hypothesis we have for all such that ; it follows by ui ∈ D i κui ≥ η the geometric criterion for genericity (see Theorem (3.22)) that if j is a limit then . So when for a limit, we may D ∩ Vκv ∈ F (v) v = vj j shrink B to obtain a set B0 = B ∩ D with B0 ∈ F (v). We dene y0 to be the resulting lower part. N3.29 Lemma 3.30. If y is a lower part which is harmonious with A past η 0 0 and y ∈ Ry, then y is harmonious with A past η.

Proof. We consider each pair (u, B) appearing in y0. If u already appears in y then we have that (u, B0) appears in y for some B0 such that B ⊆ B0, and it is easy to see that in all cases the harmoniousness conditions are satised. If u does not appear in y then let (v, C) be the unique pair appearing in y such that u ∈ C, and consider the various cases of the denition for the pair (v, C): if κv < η then κu < η and we are done, the case κv = η cannot occur as C 6= ∅, and if κv > η then S . {u} ∪ B ⊆ C ⊆ ρ<ρ∗ Aρ \ Uη+ N3.30 20 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

4. The definition of Q(w), and useful properties of it.

Suppose that U is a measure on a measurable cardinal κ. Recall that κ-Mathias forcing using U has as conditions pairs (s, S) with s ∈ [κ]<κ, S ∈ U and ssup(s) ≤ min(S) and ordering given by (t, T ) ≤ (s, S) if s = t ∩ ssup(s) and T ∪ (t \ s) ⊆ S. This forcing preserves cardinals as it is <κ-directed closed and has the κ+-chain condition. If G is a generic lter over V for the forcing and y = S{s | ∃S ∈ U (s, S) ∈ G} then for all S ∈ P (κ) we have that S ∈ U if and only if there is some α < κ such that y \ α ⊆ S. Consequently, the forcing is also known as the forcing to diagonalise U.1 We also recall Ellentuck's topologically inspired notation that if (s, S) is a condition in this forcing then [s, S] = {y ∈ [κ]κ | s ⊆ y ⊆ s∪S }. Using this notation we have that (t, T ) ≤ (s, S) if and only if [t, T ] ⊆ [s, S]. In this section we dene two forcing posets. The rst poset M(w) is the analogue of the κ-Mathias forcing for ultralter sequences w; it will diagonalise the lter F (w). The second poset Q(w) is the variant of M(w) which is tailored, in the style of the analogous forcing from [4], to deal with Radin forcing names for graphs, and binary relations more generally, on κ+. One can view κ-Mathias forcing as adding a subset y of κ which is potentially a member of U 0 for some normal measure U 0 extending U in some generic extension. In the same spirit, M(w) is designed to add a set which is potentially a member of 0 for some A ⊆ Vκw F (w ) 0 0 sequence w in a < κw-closed generic extension where lh(w ) = lh(w) and 0 for . wi ⊇ wi 0 < i < lh(w) For the rest of this section we x an ultralter sequence w with lh(w) > 1, and write κ for κw. Denition 4.1. M(w) is the forcing with conditions p = (A, B), with: p p • A = hAρ | ρ < ρ i, where ρ < κ, p • ∀ρ < ρ (Aρ ⊆ Uκ & Aρ 6= ∅ & ∃κρ < κ ∀u ∈ Aρ κu = κρ), p , • ∀ρ < ρ ∀u ∈ Aρ ∀τ ∈ (0, lh(u)) u τ ∈ Aρ p • hκρ | ρ < ρ i is strictly increasing, • B ∈ F (w), • ∀v ∈ B ∀τ ∈ (0, lh(v)) (v τ ∈ B), and 1It has also sometimes been called long Prikry forcing. FORCING CONSTRUCTIONS 21

p • ssup({κρ | ρ < ρ }) ≤ min({κv | v ∈ B }). If , then if p q p and q S q p, p q ∈ M(w) q ≤ p A = A ρ B ∪ ρ∈[ρ p,ρ q) Aρ ⊆ B while q ≤∗ p if q ≤ p and Aq = Ap. If p, q ∈ M(w) and q ≤∗ p we say that q is a direct extension of p. Denition 4.2. Set [Ap,Bp] to be

ρ ρ ρ {hD | ρ < κi | ∀ρ < κ ( D ⊆ Uκ & ∃κρ < κ ∀u ∈ D (κu = κρ)& ρ ρ is strictly ∀u ∈ D ∀τ ∈ (0, lh(u)) (u τ ∈ D ))& hκρ | ρ < κi [ increasing & hDρ | ρ < ρp i = Ap & {Dρ | ρ ∈ [ρp, κ)} ⊆ Bp }.

Then (Aq,Bq) ≤ (Ap,Bp) if and only if [Aq,Bq] ⊆ [Ap,Bp]. It is useful to have a name for the set of cardinals which are the rst elements of the various ultralter sequences appearing anywhere in Ap for p ∈ M(w). Accordingly we make the following denition.

p p Denition 4.3. If p = (A, B) ∈ M(w) let a = {κρ | ρ < ρ }.

Now we move on to the denition of the forcing Q(w). This is carried out under the following running combinatorial assumption. Setting 4.4. Suppose T is a binary κ+-Kurepa tree with Υ many branches.

Denition 4.5. Let hbα | α < Υi enumerate of a set of branches ˙ through T . Let hEα | α < Υi be a list of canonical Rw-names for + binary relations on κ . We will use the sequences hbα | α < Υi and ˙ hEα | α < Υi as parameters in the denition of the forcing Q(w). Q∗(w) is the forcing with conditions p = (A, B, t, f) satisfying the following four clauses. (1) (A, B) ∈ M(w) (see Denition (4.1)). We set a = a(A,B). (2) <κ and η . For t ∈ [(a ∩ sup(a)) × Υ] f = hfα | (η, α) ∈ ti η ∈ a ∩ sup(a), set tη = {α | (η, α) ∈ t}. (3) η η η + <κ. ∀η ∈ a ∩ sup(a) ∀α ∈ t dα = dom(fα) ∈ [κ ] (4) η η η . ∀η ∈ a ∩ sup(a) ∀α ∈ t ∀ζ ∈ dα ∃ν < κ fα(ζ) = (bα ζ, ν) If p, q ∈ Q∗(w) then q ≤ p if [Aq,Bq] ⊆ [Ap,Bp], tp ⊆ tq and p η p η q; and ∗ if and q p, q p ∀(η, α) ∈ t (fα) ⊆ (fα) q ≤ p q ≤ p A = A t = t and f q = f p. (If q ≤∗ p we say q is a direct extension of p.) 22 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

We write Q(w) for the suborder of Q∗(w) consisting of conditions which also satisfy: (5) for all η ∈ a ∩ sup(a), for all α, β ∈ tη, for every lower part y for harmonious with past , and for all , 0 η η we Rw A η ζ ζ ∈ dα ∩ dβ have:

η η η 0 η 0 fα(ζ) = fβ (ζ) 6=fα(ζ ) = fβ (ζ ) =⇒ _ ˙ 0 ˙ 0 y (w, B) Rw ζ Eα ζ ←→ ζ Eβ ζ .

Observation 4.6. Observe that in clause (5), if y = h(ui,Bi) | i ≤ ni and is harmonious with past for some , then . y A η η ∈ a κun ≤ sup(a) Observation 4.7. It is important to remember that Q(w) and Q∗(w) ˙ depend on the tree T and the sequences hbα | α < Υi and hEα | α < Υi even though their names do not make this dependence explicit.

The poset Q(w) is designed to add a sequence A∗ which diagonalises F (w) in a sense made precise in Corollary (4.9) below, together with various objects that can potentially be understood as Rw∗ -names where w∗ is an ultralter sequence existing in some <κ-directed closed forcing extension of , with ∗ , ∗ for V lh(w) = lh(w ) wτ ∩ V = wτ 0 < τ < lh(w) and S A∗ ∈ F (w∗).

The aim is that if hui | i < ηi is a Rw∗ -generic sequence (where we note that by the Mitchell criterion this sequence is also Rw-generic) and S ∗ for all such that (along with various other technical ui ∈ A i κui ≥ η conditions), then the η-coordinate in the poset Q(w) will add a Rw∗ name for a binary relation of size κ+ together with embeddings of all ˙ the relations named by the names Eα into this relation. We are particularly interested in the posets Q(w) for certain specic ˙ lists of relations, for example when hEα | α < Υi is a list of canonical names for all graphs on κ+. In this case the η-coordinate will add a name for a graph whose realization will be universal for the graphs named by the names in the list.

Lemma 4.8. For every , the set of such that q ε < κ q ∈ Q(w) ε < κρ for some ρ < ρq is dense and open.

Proof. It is immediate that this set is open, so we only need to verify that it is dense. Let p ∈ Q(w) and let ρ = ρp. Find v ∈ Bp such that p and note that p for . κv > max({ε, sup(a )}) v τ ∈ B 0 < τ ≤ lh(v) FORCING CONSTRUCTIONS 23

Now let q , q p, q , ρ = ρ + 1 A ρ = A Aρ = {v τ | 0 < τ ≤ lh(v)} q 0 p q p q p B = {v ∈ B | κv0 > κv }, t = t and f = f . We claim that q ∈ Q(w) and q ≤ p. The only non-trivial point is to check that q satises Clause (5) in the denition of Q(w). Let 0 be such that η η η 0 η 0 , and note η, α, β, ζ, ζ fα(ζ) = fβ (ζ) 6= fα(ζ ) = fβ (ζ ) that since (η, α) ∈ tq = tp we have that η ∈ ap ∩ sup(ap). q Let y be harmonious with A above η. If κy < κv then clearly y is harmonious with Ap above η, and we are done by Clause (5) for p and _ q _ p the remark that y (w, B ) ≤ y (w, B ). If κy = κv then we have y = 0_ , where , 0 is harmonious with p and also y (v τ, D) κy0 < κv y A D ⊆ S p; by the choice of to be p we see that is bounded ρ<ρp Aρ κv > sup(a ) D in and so cannot be of measure for any measure on , hence nec- Vκv 1 v essarily and , so that actually . It follows that τ = 1 D = ∅ v τ = hκv i y_(w, Bq) ≤ y0_(w, Bp), and again we are done by Clause (5) for p. N4.8 Corollary 4.9. If G is Q(w)-generic over V and A∗ is the union of the sequences Ap for p ∈ G, then A∗ is a κ-sequence and for every there is ∗ such that ∗ for ∗ . B ∈ F (w) ρ < κ Aρ ⊆ B ρ < ρ < κ The proof of Lemma (4.8) also gives the following result.

Lemma 4.10. The set of p ∈ Q(w) such that ρp is a successor ordinal is dense in Q(w). Notation 4.11. For , and + let η η < κ α < Υ ζ < κ Dζ,α = {p ∈ Q | α ∈ η p η p . (t ) & ζ ∈ (dα) )} Lemma 4.12. Suppose p ∈ Q(w), η ∈ ap. α < Υ and ζ < κ+. Then there is η with , q p, q p, η q η p q ∈ Dζ,α q ≤ p A = A B = B (t ) = (t ) ∪ {α} and η q η p . (Here we take η p if η q.) (dα) = (dα) ∪ {η} (dα) = ∅ α∈ / (t ) Proof. If η then take and there is nothing more to do. p ∈ Dζ, α q = p So suppose that η . If η p observe that p p p p 6∈ Dζ, α α 6∈ (t ) hA ,B , t ∪ {(η, α)}, f p_h∅ii is a condition in Q(w) and renes p. (The concate- nated 1-tuple consisting of the empty set records that the value of the new f-part of the condition at the co-ordinate (η, α) is the empty function.) Consequently we may assume without loss of generality that η p and η p. α ∈ (t ) ζ∈ / (dα) Pick S η p p , let η q η p ν ∈ κ\ {rge((π1 ·fβ ) ) | (η, β) ∈ t } (fα) = (fα) ∪{hζ, (bα and set η0 q η0 p for 0 p with 0 . ζ, ν)i} (fβ ) = (fβ ) (η , β) ∈ t (η , β) 6= (η, α) Then p p p η q p is clearly an element of q = hA ,B , t , h(fβ ) | (η, β) ∈ t ii 24 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

Q∗(w) and renes p. Moreover, q satises (5) of the denition of Q(w) since if q, 0 η q we have that was chosen specically so (η, β) ∈ t ζ ∈ (dβ) ν that η q η p 0 . (fα) (ζ) 6= (fβ ) (ζ ) N4.12 Corollary 4.13. For each η < κ, each α < Υ and each ζ ∈ κ+, the set η η p η p is dense and open in Dζ,α = {p ∈ Q | α ∈ (t ) & ζ ∈ (dα) )} {p ∈ Q(w) | η ∈ ap }.

Proof. Immediate from Lemma (4.12). N4.13 Corollary 4.14. If G is Q(w)-generic over V , η ∈ S{ap | p ∈ G} and then S p,η p is a function with domain +. α < Υ {fα | p ∈ G, (η, α) ∈ t } κ Lemma 4.15. Q(w) has splitting.

Proof. This is proved using an argument very similar to the one for Lemma (4.12). As observed there, given a condition p we can easily nd a condition in Q(w) rening it with a non-empty t-part, so without loss of generality we may as well assume tp 6= ∅. Let (η, α) ∈ tp. Now pick + η p and working exactly as before we can dene two exten- ζ ∈ κ \ (dα) η sions , of which are both in but satisfy η q0 η q1 , q0 q1 p Dζ,α (fα) (ζ) 6= (fα) (ζ) simply by choosing, at the ultimate stage of that argument, two distinct elements and of S η p , where previously we ν0 ν1 κ \ (η,β)∈tp rge((fβ ) ) merely picked one. Then and are incompatible. q0 q1 N4.15 Lemma 4.16. The forcing poset Q(w) is <κ-compact.

Proof. Let A = {qi | i < µ} be a centred subset of Q(w) for some µ < κ, and x for each nite set Y ⊆ µ some condition sY ∈ Q(w) such that ∗ ∗ ∗ ∗ sY ≤ qi for all i ∈ Y . We will dene a quadruple (A ,B , t , f ) of the appropriate type, and prove that it is a condition in Q(w) and forms a lower bound for A.

• Let ρ∗ = sup{ρqi | i < µ}. By the hypothesis it is easy to see ∗ ∗ that there is a unique sequence A = hAρ | ρ < ρ i such that ∗ qi qi for all . We write for the common value A ρ = A i < µ κρ ∗ ∗ of κv for sequences v ∈ Aρ, and a for {κρ | ρ < ρ }.

Let ∗ T sY . By the completeness of the lter • B = Y ∈[µ]<ω B F (w) we see that B∗ ∈ F (w). It is also clear from the denition of ∗ ∗ Q(w) that κρ < κv for every ρ < ρ and every v ∈ B .

Let ∗ S qi . Clearly ∗ ∗ ∗ and ∗ . • t = i<µ t t ⊆ (a ∩ sup(a )) × Υ |t | < κ FORCING CONSTRUCTIONS 25

• For each (η, α) ∈ t∗, let

∗,η [ qi,η qi dα = {dα | (η, α) ∈ t , i < µ}

• By the hypothesis it is easy to see that there is a unique sequence of functions ∗ ∗,η ∗ with ∗,η f = hfα | (η, α) ∈ t i dom(fα ) = ∗,η, and ∗,η qi,η for all qi,η, dα fα (ζ) = fα (ζ) ζ ∈ dα It should be clear that (A∗,B∗, t∗, f ∗) is a condition in Q∗(w), and that if (A∗,B∗, t∗, f ∗) is a condition in Q(w) then it forms a lower bound for A. So to nish the proof, we must verify clause (5) in the denition of Q(w). Let η ∈ a∗ ∩ sup(a∗) with η < sup(a∗), let α, β ∈ t∗,η, let y = ∗ h(ui,Bi) | i ≤ ni be a lower part harmonious with A past η, and let ζ, 0 ∗,η ∗,η be such that ∗,η ∗,η η,∗ 0 ∗,η 0 . ζ ∈ dα ∩ dβ fα (ζ) = fβ (ζ) 6= fα (ζ ) = fβ (ζ ) Choose a nite set Y large enough that

There is with qi0 , qi0 and qi0 . • i0 ∈ Y η ∈ a η < sup(a ) κun ≤ sup(a ) q ,η qi i1 • There is i1 ∈ Y with (η, α) ∈ t 1 , and ζ ∈ dα .

qi ,η There is with qi2 , and 2 . • i2 ∈ Y (η, β) ∈ t ζ ∈ dβ q ,η qi 0 i3 • There is i3 ∈ Y with (η, α) ∈ t 3 , and ζ ∈ dα . q ,η qi 0 i4 • There is i4 ∈ Y with (η, β) ∈ t 4 , and ζ ∈ dα .

Now consider the condition sY . By construction y is harmonious with sY A past η, and since sY ∈ Q(w) it follows easily that

_ sY ˙ 0 ˙ 0 y (w, B ) Rw ζ Eα ζ ←→ ζ Eβ ζ . By construction B∗ ⊆ BsY , and so y_(w, B∗) also forces this equivalence and we are done.

Corollary 4.17. Q(w) is both <κ-directed closed and countably compact.

The following corollary to the proof of Lemma (4.16) and subsequent lemma are useful in the proof that Q(w) has the κ+-stationary chain condition which appears in the next section.

Corollary to the proof 4.18. Descending sequences from Q(w) of length less than κ have greatest lower bounds. 26 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

Lemma 4.19. Let p = (A, B, t, f) ∈ Q(w). Let t = S{tη | η ∈ ap } and let S η p η . Then there is with d = {dα | η ∈ a & α ∈ t } q ∈ Q(w) , q p, q p, η for all p and η q for all q ≤ p A = A B = B t = t η ∈ a (dα) = d η ∈ ap and α ∈ tη.

Proof. By induction on t × d , using Lemma (4.12) at initial and successor stages and Corollary (4.18) at limits. N4.19 We now introduce the notion of a weakening of a condition. Denition 4.20. Let p be a condition in Q(w) such that ρp is a successor ordinal and let ρ∗ < ρp. We dene the ρ∗-weakening of p to be the quadruple r r r r where r p ∗, r p ∗ , r = (A ,B , t , f ) A = A ρ a = {κρ | ρ < ρ } r p S p ∗ p , r p r r B = B ∪ {Aρ | ρ ≤ ρ < ρ } t = {(η, α) ∈ t | η ∈ a ∩ sup(a )} and r p,η r . Then and . f = hfα | (η, α) ∈ t i r ∈ Q(w) p ≤ r Lemma 4.21. Let p ∈ Q(w) with ρp a successor ordinal, let ρ∗ < ρp and let r be the ρ∗-weakening of p. Then (1) r ∈ Q(w), (2) p ≤ r, (3) and for every η ∈ ar and every lower part y which is harmonious with r past , if p,η p,η p,η 0 p,η 0 , then A η fα (ζ) = fβ (ζ) 6= fα (ζ ) = fβ (ζ ) _ r ˙ 0 ˙ 0 y (w, B ) ζEαζ ⇐⇒ ζEβζ . Proof. It is easy to see that r ∈ Q∗(w), and that p renes r in Q∗(w). Since p,η r,η when r, Clause (3) in the conclusion implies fα = fα (η, α) ∈ t that r satises Clause (5) in Denition (4.5) (and hence that r is in Q(w)). It will therefore suce to verify Clause (3). Suppose for a contradiction that y_(w, Br) does not force the desired equivalence, then there is an extension y0_(w, B) forcing that the equivalence is false. Shrinking B if necessary we may assume that B ⊆ Bp. 0 _ _ We will break up y as y0 y1 y2, where y0 ∈ Ry, y1 consists of pairs such that is drawn from S p ∗ p , and consists (u, C) u {Aρ | ρ ≤ ρ < ρ } y2 of pairs (u, C) such that u is drawn from Bp. This is possible because, 0 by the denition of extension in Rw, all pairs (u, C) in y with κu > κy have u ∈ Br. _ p We claim that y0 y1 is harmonious with A above η. An appeal to r Lemma (3.30) shows that y0 is harmonious with A above η, which handles the pairs appearing in y0. Let (u, C) be a pair appearing in y1, FORCING CONSTRUCTIONS 27 and observe that r p ∗ , while p ∗ p , η ∈ a = {κρ | ρ < ρ } u ∈ {Aρ | ρ ≤ ρ < ρ } so that η < κρ∗ ≤ κu. By the denition of the ordering of Rw we have r, and since for some p in fact S p ∗ C ⊆ B κu = κρ ρ < ρ C ⊆ {Aρ | ρ ≤ p ρ < ρ }. In particular as η < κρ∗ we have that C ∩ Vη+1 = ∅.

We claim that by shrinking measure one sets appearing in y2 (if nec- 0_ _ _ p essary) we may assume that y (w, B) ≤ y0 y1 (w, B ). For pairs p r p (u, C) appearing in y2 we have that u ∈ B and C ⊆ B = B ∪ S p ∗ p ; since p is a successor ordinal we have that {Aρ | ρ ≤ ρ < ρ } ρ for all S p ∗ p , so that we may κv ≤ κρp−1 < κu v ∈ {Aρ | ρ ≤ ρ < ρ } shrink C to obtain a pair (u, D) with D ⊆ Bp. _ p Since y0 y1 is harmonious with A above η and p is a condition, _ _ p ˙ 0 ˙ 0 y0 y1 (w, B ) forces the equivalence ζEαζ ⇐⇒ ζEβζ . This is a con- 0_ _ _ p tradiction as y (w, B) is an extension of y0 y1 (w, B ) and forces that the same equivalence fails. N4.21

Lemma 4.22. Let p and q be conditions in Q(w) with p ≤ q and ρp a successor ordinal. Let ρq ≤ ρ∗ < ρp and let r be the ρ∗-weakening of p. Then p ≤ r ≤ q.

Proof. By Lemma (4.21) we already know that p ≤ r so we need only q q p q q to check that r ≤ q. It is routine to check that A ρ = A ρ = A , r q, r,η p,η q,η for all q, and nally r S r q t ⊇ t fα = fα ⊇ fα (η, α) ∈ t B ∪ {Aρ | ρ ≤ r p S p q p q. ρ < ρ } = B ∪ {Aρ | ρ ≤ ρ < ρ } ⊆ B N4.22

5. has the stationary +chain condition Q(w) κw

For the duration of this section we x an ultralter sequence w with + lh(w) > 1, and let κ = κw. As in the previous section, let T be a κ - Kurepa tree, let hbα | α < Υi be an enumeration of a set of branches ˙ through T and let hEα | α < Υi be a list of canonical Rw-names for binary relations on κ+. Let Q = Q(w). + Notation 5.1. For each α < κ let Tα = {x ∈ T | lh(x) = α} and let eα : Tα −→ Tα be a bijection enumerating Tα.

+ + As T is a κ -Kurepa tree one has for α < κ that Tα ≤ κ.

Notation 5.2. For α < β < Υ let ∆(α, β) be the least ζ such that . bα ζ 6= bβ ζ Notation 5.3. If p = (A, B0, t, f) ∈ Q and there are t ∈ [Υ]<κ and + <κ such that η for all and η for all d ∈ [κ ] t = t η ∈ a d = dom(fα) 28 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

η ∈ a and α ∈ t, let Zp be the set of all tuples (η, y, x, ζ, ζ0, δ, δ0) where η ∈ a ∩ sup(a), y is a lower part harmonious with A past η, x ∈ T , ζ < ζ0 < κ+ and ζ, ζ0 ∈ d and δ, δ0 < κ. Lemma 5.4. Suppose p0 = (A, B0, t, f) ∈ Q and there are t ∈ [Υ]<κ and + <κ such that η for all and η for all d ∈ [κ ] t = t η ∈ a d = dom(fα) η ∈ a and α ∈ t. Then there is some B ∈ F (w) with B ⊆ B0, so that p = (A, B, t, f) ≤ p0, and such that whenever z = (η, y, x, ζ, ζ0, δ, δ0) ∈ Zp and α ∈ t is such that 0, η and η 0 0 there is an x = bα ζ fα(ζ) = (x ζ, δ) fα(ζ ) = (x, δ ) _ Ry-name σ˙ z such that y (w, B) forces that σ˙ z is the truth value of the ˙ 0 assertion that ζEαζ .

Proof. First of all notice that Zp = Zp0 . For each z ∈ Zp we may dene a -name such that for all with η,i Rw τ˙z α ∈ t fα (ζ) = (x ζ, δ) and η,i 0 0 , _ i forces that is the truth value of fα (ζ ) = (x, δ ) y (w, B ) τ˙z ˙ 0 the assertion that ζEαζ . (Condition (5) in the denition of being a condition shows τ˙z does not depend on α.) p By Lemma (3.19), for each z ∈ Z there is a set Bz ⊆ B such that _ p y (w, Bz) reduces τ˙z to an Ry-name σ˙ z. As Z < κ any B ∈ F (w) such that T suces. B ⊆ z∈Zp Bz N5.4 Proposition 5.5. Q has the κ+-stationary chain condition.

Proof. Let {p0i | i < κ+ } ∈ [Q]κ+ . For each i < κ+ let p0i = (Ai,B0i, t0i, f 0i), write ai for api and ρi for ρpi , and for each ρ < ρi write i for pi . κρ (κρ) We start by tidying up, for each i < κ+, the collection of domains of constituents of the f 0i. We want to emphasize that this step in the argument is not strictly necessary. At the cost of a more elaborate case analysis below, proofs of the stationary chain condition can be given which do not rely on the forcing having the property of having greatest lower bounds for descending sequences of length less than κ, a property which we do use in carrying out this tidying up. By applying Lemma (4.19) and then Lemma (5.4) we can nd {pi | i < κ+ } ∈ [Q]κ+ such that for each i < κ+ we have pi = (Ai,Bi, ti, f i) ≤ p0i and there is some t i ∈ [Υ]<κ and some d i ∈ [κ+]<κ such that • t i = tη,i for all η ∈ ai, i η,i for all i and i, and • d = dom(fα ) η ∈ a α ∈ t FORCING CONSTRUCTIONS 29

• Bi has the property stated in the conclusion of Lemma (5.4). Now we make a plethora of auxiliary denitions for each i, set out in the table below.

∗ S i + Let {αγ | γ < γ } be an enumeration of {t | i < κ }, for some ∗ +. Let i i be the increasing enumeration of i, for γ ≤ κ {αγ | γ < γ } t some γi < κ, for each i < κ+. Next, for each i < κ+ and η ∈ ai let i i , i i , • θ0 = ssup({γ | αγ ∈ t }) θ1 = ssup(d ) • Γi = {∆(α, α0) | α, α0 ∈ t i }, νi = ssup(Γi), i i i i i i • T = {γ < i | αγ ∈ t }, ∆ = Γ ∩ i, D = d ∩ i, • Y η,i = {y | y is a lower part harmonious with Ai past η}. η,i η,i η,i for i and i. • g (ζ, γ) = (eζ (π0(fαγ (ζ))), π1(fαγ (ζ))) γ ∈ T ζ ∈ D η,i 0 η η,i i 0 i • X = {hy, ε, ζ, ζ ,Fε , σ˙ i | y ∈ Y & ε < γ & ζ, ζ ∈ D & 0 on setting i one has η i is ζ < ζ & α = αε Fε : D −→ κ × κ given by η ∗ η,i ∗ η,i ∗ for ∗ i Fε (ζ ) = (eζ∗ (π0(fα (ζ ))), π1(fα (ζ ))) ζ ∈ D letting 0, η,i and η,i 0 0 , & x = bα ζ fα (ζ) = (x ζ, δ) fα (ζ ) = (x, δ ) we have i . σ˙ =σ ˙ (y,x,η,ζ,ζ0,δ,δ0) }

Recall that πl is projection onto the lth co-ordinate, so in the denition of η,i, e.g., and , while g π0(bαγ ζ, ν) = bαγ ζ π1(bαγ ζ, ν) = ν eζ is the function enumerating the ζth level of the κ+-Kurepa tree T , so that . Thus η,i i i . eζ (bαγ ζ) ∈ κ g : D × T −→ κ × κ Notice, also, that the various η, , , and 0 appearing in the de- Fε σ˙ x δ δ nition of Xη,i are uniquely determined by i, η and the hy, ε, ζ, ζ0 i. For the convenience of the reader we record the types of the objects which we have just dened. for ∗ +, i for αγ ∈ Υ γ < γ ≤ κ αγ ∈ Υ i , i +, i + <κ, i +, i +, i <κ, γ < γ < κ θ0 < κ Γ ∈ [κ ] ν < κ θ1 < κ T ∈ [i] i <κ i <κ η,i η,i ∆ ∈ [i] , D ∈ [i] , Y ∈ Vκ, and g is a partial function of size less than κ from i × i to κ × κ.

∗ + <κ + + Let h be an injection from Vκ × ([κ ] × Fn(κ × κ , κ × κ, κ) × + <κ + <κ + + + <κ [κ ] × [κ ] × κ × (Vκ × κ × κ × κ × Vκ × Fn(κ , κ × κ, κ)) ) × κ into κ+. Let C be the set of closure points of h∗. Let

+ i i i Ce = {j < κ | ∀i < j (θ0, θ1, ν < j)} ∩ C. So and are club subsets relative to κ+ . C Ce Sκ 30 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

Let h(i) = h∗(Ai, h(T i, gη,i,Di, ∆i, otp(t i),Xη,i) | η ∈ ai i, κi) for i ∈ κ+ and otherwise. Clearly for all nonzero Ce ∩ Sκ h(i) = 0 h(i) < i i < κ+. Now suppose that κ+ , , and . In particular i, j ∈ Ce ∩ Sκ i < j h(i) = h(j) we have that i j (and hence i j) and i j, that η,i, η,i, A = A a = a κ = κ θ0 θ1 νη,i < j, and nally for all η ∈ ai hT i, gη,i,Di, ∆i, otp(t i),Xη,i i = hT j, gη,j,Dj, ∆j, otp(t j),Xη,j i. We prove a series of lemmas which together describe the common parts of pi and pj. Let A be the common value of Ai and Aj and a that of ai and aj.

j i i j Lemma 5.6. (a) (t \ t ) ∩ {αγ | i ≤ γ < j } = ∅, and (b) t ∩ t ⊆ {αγ | γ < i}.

j j j i i Proof. Suppose αγ ∈ t . If γ < j then γ ∈ T . But T = T , so γ ∈ T . Hence and i, proving (a). If i then i . γ < i αγ ∈ t αγ ∈ t γ < θ0 < j Thus if i j we have (b). αγ ∈ t ∩ t N5.6 Lemma 5.7. d i ∩ d j ⊆ i.

Proof. If i j then i . So j j i i . ζ ∈ d ∩d ζ < θ1 < j ζ ∈ d ∩j = D = D = d ∩i N5.7 Lemma 5.8. If and i j then η,i i j η,j i j. η ∈ a α ∈ t ∩t fα d ∩d = fα d ∩d

Proof. Let η ∈ a. By Lemma (5.6) and Lemma (5.7), if α = αγ then d i ∩ d j ⊆ Di = Dj = dom(gη,i(., γ)) = dom(gη,j(., γ)). As α ∈ t i ∩ t j, if i j we have η,i η,j and, as is a ζ ∈ d ∩ d t(π0(fα (ζ))) = t(π0(fα (ζ))) t bijection, η,i η,j . Hence we have the claimed π0(fα (ζ)) = π0(fα (ζ)) = bα ζ agreement. N5.8

Recall that f i + f j is the unique function f such that dom(f) = i j i i j j dom(f ) ∪ dom(f ), f dom(f ) = f and f dom(f ) = f . Let q = (A, Bi ∩ Bj, ti ∪ tj, f i + f j). Lemma 5.9. The quadruple q is a condition in Q∗.

Proof. Recall that we dened A = Ai = Aj. Lemma (5.8) shows that i and j are compatible. f f N5.9 Lemma 5.10. Γi ∩ Γj ⊆ i. FORCING CONSTRUCTIONS 31

Proof. If ζ ∈ Γi ∩ Γj then ζ < νi < j. So ζ ∈ ∆j. But ∆j = ∆i, so . ζ < i N5.10 Notation 5.11. For i j dene q i j if i j, α ∈ t ∪ t dα = d ∪ d α ∈ t ∩ t q i if i j, and q j if j i. dα = d α ∈ t \ t dα = d α ∈ t \ t Lemma 5.12. For i j and we have q η,q . α ∈ t ∪ t η ∈ a dα = dom(fα )

Proof. Immediate from the denitions of and the q . q dα N5.12 Lemma 5.13. Suppose , i j, , 0 q q , 0, α β ∈ t ∩ t ζ ζ ∈ dα ∩ dβ ζ < ζ η ∈ a and η,q η,q η,q 0 η,q 0 . Then , 0 i j. fα (ζ) = fβ (ζ) 6= fα (ζ ) = fβ (ζ ) ζ ζ ∈ d ∩ d

0 Proof. Let α = αγ and β = αγ0 . Then γ, γ < i by Lemma (5.6) and ∆(α, β) ∈ Γi ∩ Γj ⊆ i, by Lemma (5.10). Hence ζ, ζ0 < i as η,q η,q and similarly for 0. Thus bα ζ = π0(fα (ζ)) = π0(fβ (ζ)) = bβ ζ ζ 0 j i i. Similarly j. ζ ∈ D = D ⊆ d ζ ∈ d N5.13

Finally, we can now show that q satises (5) of the denition of Q. Lemma 5.14. Suppose , i j, , 0 q q , and α β ∈ t ∪ t ζ ζ ∈ dα ∩ dβ η ∈ a η,q η,q η,q 0 η,q 0 and is harmonious with q fα (ζ) = fβ (ζ) 6= fα (ζ ) = fβ (ζ ) y A past . Then _ q ˙ 0 ˙ 0 . η y (w, B ) Rw ζEαζ ⇐⇒ ζEβζ

Proof. If for some k ∈ {i, j } we have α, β ∈ t k then, using Lemma i j 0 k k _ k (5.13) if α, β ∈ t ∩ t , ζ, ζ ∈ d . As p ∈ Q we have y (w, B ) ˙ 0 ˙ 0 q i j k _ q ζEαζ ⇐⇒ ζEβζ . But B = B ∩ B ⊆ B , so y (w, B ) ≤ _ k and _ q ˙ 0 ˙ 0 . y (w, B ) y (w, B ) ζEαζ ⇐⇒ ζEβζ Otherwise, either we have α ∈ t i \ t j, β ∈ t j \ t i and ζ, ζ0 ∈ d i ∩ d j, or we have the symmetric case with the rôles of i and j exchanged. We treat the former; for the symmetric case exchange i and j throughout. Suppose β is the ε-th element of t j. As otp(ti) = otp(tj), we can dene 0 to be the -th element of i. As η,i η,j we have that the ` η' β ε t X = X Fε for the tuple that starts hy, ε, ζ, ζ0,...i is the same for both i and j. Hence η,i 0 η,j 0 . As is a bijection and is eζ0 · π0 · fβ0 (ζ ) = eζ0 · π0 · fβ (ζ ) eζ0 π0 projection onto the rst co-ordinate, this gives 0 0. Let bβ ζ = bβ0 ζ 0 0, η,i η,j , η,i 0 η,j 0 x = bβ ζ = bβ0 ζ fβ0 (ζ) = fβ (ζ) = (x ζ, δ) fβ0 (ζ ) = fβ (ζ ) = (x, δ0), and, nally, z = (η, y, x, ζ, ζ0, δ, δ0). Then pi pj , and i j say, where is an -name for z ∈ Z ∩ Z σ˙ z =σ ˙ z =σ ˙ σ˙ Ry a truth value. Since Bq = Bi ∩ Bj, y_(w, Bq) simultaneously reduces ˙ 0 ˙ 0 the truth values of the statements ζ Eβ ζ and and ζ Eβ0 ζ to σ˙ , so 32 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

_ q ˙ 0 ˙ 0 . If 0 we are done. Otherwise, y (w, B ) ζ Eβ0 ζ ⇐⇒ ζ Eβ ζ β = α as i we also have that _ i ˙ 0 ˙ 0 , and p ∈ Q y (w, B ) ζ Eα ζ ⇐⇒ ζ Eβ0 ζ hence _ q ˙ 0 ˙ 0 , as required. y (w, B ) ζ Eα ζ ⇐⇒ ζ Eβ ζ N5.14

6. The main iteration

Let V be a model in which κ is supercompact, λ is a regular cardinal less than κ, and χ is a successor cardinal with predecessor χ− such that − ++. Suppose also that GCH holds in and χ holds. cf(χ ) ≥ κ V ♦χ(Sκ+ ) Let L be the Laver iteration ([9]), as dened in V , making the supercompactness of κ indestructible under <κ-directed closed forcing. Let be a canonical -name such that is a bijection. e˙ L L e˙ : κ −→ Vκ Using Theorem (1.2), Lemma (1.4), Lemma (4.16) and Proposition (5.5), we will dene an -name ˙ so that ˙ is an iteration of L Pχ L Pχ ˙ + length χ consisting of Q0, a name for the usual forcing to add a κ - Kurepa tree with χ−-many branches followed by a <κ-support iteration of <κ-directed closed, countably parallel-closed stationary κ+-cc forcings each of size less than χ. The constituents of the iteration will − ˙ depend on a xed sequence hx˙ α | α < χ i of L ∗ Q0-names for distinct branches through the κ+-Kurepa tree. The iteration will add χ-many subsets of κ, but at each intermediate stage 2κ = χ−. As we dene the ˙ iteration we also build an enumeration of L ∗ Pχ as hpξ | ξ < χi. As χ is a regular cardinal there will be a club set relative to χ of S≥κ ξ < χ ˙ such that L ∗ Pξ = {pε | ε < ξ }. We simultaneously inductively dene ˙ ˙ ˙ ξ L ∗ Pξ-names Sξ as in 2 and derive L ∗ Pξ-names u˙ . ˙ As each Sξ is a canonical name for a subset of ξ, when ξ = κ.λ.ξ (ordinal multiplication) we can easily convert it into a name for a set of order type λ of sequences of ξ many subsets of κ. In order to do this, for each , set ξ to be the name derived from ˙ for τ < λ u˙ 1+τ Sξ ˙ {e˙“{η < κ | κλε + κτ + η ∈ Sξ ∩ [κλε + κτ,\ κλε + κ(τ + 1))} | ε < ξ } and set ξ _ ξ . u˙ =κ ˆ hu˙ 1+τ | 1 + τ < λi We use the sequence u˙ ξ to help dene the next stage in the iteration. ˙ ξ ˙ Let U be a L ∗ Pξ-name for the class of all ultralter sequences. (As for each we have ˙ is -directed closed , we will have ξ < χ L Pξ <κ for all ξ < ξ0 ≤ χ that ˙ ξ = ˙ ξ0 .) L∗Pξ0 Uκ Uκ FORCING CONSTRUCTIONS 33

If +, ˙ , and ξ ˙ ξ and cf(ξ) = κ L ∗ Pξ = {pε | ε < ξ } ξ = κλξ L∗Pξ u˙ ∈ U let ˙ ξ − enumerate the canonical ˙ ˙ -names κ˙ u˙ ξ = κ hEα | α < χ i L ∗ Pξ ∗ Ru˙ ξ + ˙ ˙ ξ ˙ ξ ˙ for graphs on κ and let Qξ = Q(u ˙ ), where Q(u ˙ ) is an L ∗ Pξ-name for the forcing dened from ξ, − and ˙ ξ − as in u˙ hx˙ α | α < χ i hEα | α < χ i ˙ 4. Otherwise let Qξ name the trivial forcing. Fix which is -generic over and which is ˙ G-generic over . G L V H Pχ V [G] For let be the restriction of to ˙ G, let ˙ G∗Hξ , and let ξ < χ Hξ H Pξ Qξ = Qξ uξ = (u ˙ ξ)G∗Hξ . If ξ , and ξ , let V [G][Hξ] |= u ∈ U κuξ = κ lh(u ) = λ ξ • Kξ be the Q(u )-generic over V [G][Hξ] induced by H, ξ p • A = hAρ | ρ < κ & ∃p ∈ Kξ Aρ = (Aρ) i, ξ S p • and a = {a | p ∈ Kξ }. ˙ ˙ Fix an enumeration hDξ | ξ < χi of the L ∗ Pχ-names for subsets of Vκ ˙ ˙ such that each Dξ is a L ∗ Pξ-name. Let j : V −→ M witness that κ is 2χ-supercompact, such that in ˙ the iteration j(L) we force with Pχ at stage κ and then do the trivial χ + ˙ ˙ forcing at all stages between κ and (2 ) . Let j(L) = L ∗ Pχ ∗ L, ˙ ˙ where L is a L ∗ Pχ-name and we note that by the choice of j we have ˙ is (2χ)+-closed. . L∗P˙ χ L < We now carry out, in V [G][H], an inductive construction of length χ ˙ G∗H ˙ in which we build a chain of conditions (rξ, q˙ξ) ∈ L ∗ j(Pχ) for ξ < χ. Note that forcing with L˙ G∗H over V [G][H] always adds a generic embedding j : V [G] −→ M[j(G)] see [2, 9] so that the ˙ name j(Pχ) is well dened. The construction includes arranging for each that ˙ and that for all we have ξ < χ L˙ G∗H q˙ξ ∈ j(Pξ) p ∈ Hξ that rξ L˙ G∗H q˙ξ ≤ j(p). V [G][H] Dene U = ({v ∈ U | κv = κ & lh(v) ≤ λ}) . Then V [G][H] |= U ≤ ((22κ )λ) = 2χ. ˙ G∗H ˙ Inductive case: ξ = 0 or limit ξ. By the <χ-closure of L ∗ j(Pχ), just choose some such that ˙ , and for all (rξ, q˙ξ) L˙ G∗H q˙ξ ∈ j(Pξ) 0 ξ < ξ we have (rξ, q˙ξ) ≤ (rξ0 , q˙ξ0 ). Then for each p ∈ Hξ we have rξ L˙ G∗H q˙ξ ≤ j(p).

Inductive case: ξ + 1. If ξ = 0 then Q0 is the usual forcing to add a κ+-Kurepa tree with Υ-many branches as computed in V [G], and we 34 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH set r1 to be the trivial condition and q˙1 to be a name for the greatest lower bound of the union of the pointwise image of the Q0-generic lter. We now assume that ξ > 0. If ξ or ξ but or ξ and V [G][Hξ] |= u ∈/ U u ∈ U κuξ 6= κ u ∈ U κuξ = κ ξ but lh(u ) 6= λ there is nothing to do and we can take (rξ+1, q˙ξ+1) = ξ _˙ . So assume otherwise: , ξ (rξ, q˙ξ 1j(Qξ)) V [G][Hξ] |= u ∈ U κu = κ and lh(uξ) = λ.

By the construction so far (r , q˙ ) ˙ G∗H ˙ there is a lifting j : ξ ξ L ∗j(Pξ) V [G][Hξ] −→ M[j(G)][j(Hξ)] of j. χ ˙ G∗H ˙ As 2 is less than the closure of L ∗ j(Pχ) we may, by shrinking if necessary, assume that there is some 0 0 such that (rξ, q˙ξ) ≤ (rξ, q˙ξ) 0 0 ˙ for every . (rξ, q˙ξ) k “v ∈ j(Dξ) v ∈ U Denition 6.1. If v ∈ UV [G][H], lh(v) = λ and for all τ < λ we have ξ we say lls out ξ. vτ ∩ V [G][Hξ] = uτ v u

Note that if v lls out uξ then, a priori, v ∈ U. T q q q q q Lemma 6.2. Set B = {j(B ) | (A ,B , t , f ) ∈ Kξ }. Suppose there is some v ∈ UV [G][H] which lls out uξ and for all τ < λ we have . Then there is a master condition for , for v τ ∈ B j(Qξ) j and V [G][Hξ], such that on forcing below the condition we can lift j to a map j : V [G][Hξ][Kξ] −→ M[j(G)][j(Hξ)][j(Kξ)] and have that for every lling out ξ and every we have ξ . v ∈ U ∩ B u τ < λ v τ ∈ j(A )κ Proof. We construct a suitable master condition p∗ = (A∗,B∗, t∗, f ∗). ∗ ξ_ lls out ξ . • A = A {v τ | τ < λ & v ∈ B∩(Uκ+ \Uκ)& v u } ∗ • B = B \ Uκ+ . • t∗ = κ × j“Υ. For each ξ and , ∗,η S p,η . • η ∈ a α < Υ fj(α) = {j(fα ) | p ∈ Kξ } We note that since p,η is a partial function of size less than , and we fα κ have the density lemmas Lemma (4.8) and Lemma (4.13), it is easy to see that ∗,η ∗,η + and ∗,η p,η for any dj(α) = dom(fj(α)) = j“κ fj(α)(j(ζ)) = j(fα (ζ)) p p∗ ξ p ∈ Kξ such that (η, α) ∈ t . We also note that a = a ∪ {κ}. ∗ ∗ ∗ It is routine to verify that p ∈ j(Q ) and that p ≤ ∗ j(q) for q ∈ K . ξ j(Qξ ) ξ ∗ If we can verify that p is a condition in j(Qξ), then it is immediate from its denition that p∗ is a master condition of the type required. FORCING CONSTRUCTIONS 35

So, in order to complete the proof, we must show that p∗ satises (5) in the denition of j(Qξ). Let ξ, and suppose that ∗,η ∗,η ∗,η 0 η ∈ a fj(α)(j(ζ)) = fj(β)(j(ζ)) 6= fj(α)(j(ζ )) = ∗,η 0 for some and 0 +. Let be harmonious fj(β)(j(ζ )) α, β ∈ Υ ζ, ζ ∈ κ y ∗ 0 0 ∗ with A past η. If κy < κ let y = y. Otherwise let y , τ < λ, v ∈ B ξ 0_ which lls out u , and B ∈ F (v τ) be such that y = y (v τ, B). ∗ Using Lemma (4.10) and the denition of p we may nd p ∈ Kξ with ρp a successor ordinal such that • η ∈ ap with η < max(ap),

p • κy0 < max(a ), • (η, α), (η, β) ∈ tp, 0 p,η p,η, • ζ, ζ ∈ dα ∩ dβ p,η p,η p,η 0 p,η 0 , • fα (ζ) = fβ (ζ) 6= fα (ζ ) = fβ (ζ ) • and y0 is harmonious with Ap past η. If y = y0 we have that y is harmonious with Ap past η and y_(j(uξ),B∗) ≤ y_(j(uξ), j(Bp)) = j(y_(uξ,Bp)).

As is a condition in we have _ ξ p ˙ξ 0 ˙ξ 0 , p Qξ y (u ,B ) ζEαζ ←→ ζEβζ and we are done by elementarity.

0_ So we may assume that we are in the other case, that is y = y (v ∗ ξ τ, B) for some v ∈ B which lls out u . We note that v τ ∈ B, since each of the sets Bq is closed under taking initial segments. Let ρ∗ < ρp be minimal such that max({η, κy0 }) < κρ∗ . ∗ ∗ Since y was chosen to be harmonious with A past η, B ∩ Vκ = ∅, and , we see that S ξ and for all η < κ = κv B ⊆ {Aρ | η < κρ, ρ < κ} κv > η . Since 0_ is a condition, we also have that v ∈ B y (v τ, B) κv > κy0 for all . So by the choice of ∗, S ξ ∗ . v ∈ B ρ B ⊆ {Aρ | ρ ≤ ρ < κ} Since the condition , p for ∗ p and p for p ∈ Kξ Aσ = Aσ ρ ≤ σ < ρ Aσ ⊆ B p . It follows that S p ∗ p p. ρ ≤ σ < κ B ⊆ {Aσ | ρ ≤ σ < ρ } ∪ B We now appeal to Lemma (4.21) to obtain the ρ∗-weakening of p, that q ∗ q p ∗ is to say the condition q ∈ Q such that ρ = ρ , p ≤ q, A = A ρ , q S ∗ p p q 0 0 p 0 q q B = {Aσ | ρ ≤ σ < ρ } ∪ B , t = {(η , α ) ∈ t | η ∈ a ∩ sup(a )} and q p,η0 0 0 q . Since we have , and so f = hfα0 | (η , α ) ∈ t i p ≤ q q ∈ Kξ B∗ ⊆ j(Bq). By the choice of ρ∗, y0 is harmonious with Aq past η. 36 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

By Lemma (4.21) the condition q enjoys a strengthened form of condition (5) in the statement of Q(w), which implies in this case that

0_ ξ q ˙ξ 0 ˙ξ 0 y (u ,B ) ζEαζ ←→ ζEβζ .

We claim that

y_(j(uξ),B∗) ≤ y0_(j(uξ), j(Bq)) = j(y0_(uξ,Bq)).

The only non-trivial point is that the pair (v τ, B) can be added to the 0_ ξ q q condition y (j(u ), j(B )). This holds because v τ ∈ B ⊆ j(B ), and S p ∗ p p q . As in the case B ⊆ {Aσ | ρ ≤ σ < ρ } ∪ B = B = j(Bq) ∩ Vκ when 0, we are now done by elementarity. y = y N6.2

Having established the preceding Lemma, we can now complete the inductive step.

Case (i). There is some 0 0 with ˙ such that (r, q˙) ≤ (rξ, q˙ξ) L˙ G∗H q˙ ∈ j(Pξ)

ξ (r, q˙) ˙ G∗H ˙ ∃v ∈ ( v lls out u & ∀τ < λ L ∗j(Pξ) U T q q q q q v τ ∈ {j(B ) | (A ,B , t , f ) ∈ Kξ } ) .

Let rξ+1 = r and use Lemma (6.2) to choose q˙ξ+1 such that , is a lower bound for q˙ξ+1 ξ =q ˙ (rξ+1, q˙ξ+1 ξ) q˙ξ+1(ξ) j“Kξ and for every which lls out ξ and every (rξ+1, q˙ξ+1) v ∈ U u τ < λ ξ we have v τ ∈ j(A ). ˙ G∗H ˙ Case (ii). Otherwise. Again, by the closure of L ∗ j(Pχ), let 0 0 be such that is a lower (rξ+1, q˙ξ+1) ≤ (rξ, q˙ξ) (rξ+1, q˙ξ+1 ξ) q˙ξ+1(ξ) bound for since is trivial. j“Kξ Qξ+1 NInductive construction ˙ G∗H ˙ When the construction is complete use the χ-closure of L ∗ j(Pχ) ∗ ∗ again and take a lower bound (r , q˙ ) for h(rξ, q˙ξ) | ξ < χi such that for ∗ ∗ ∗ ∗ all q ∈ H we have r q˙ ≤ j(q). Thus (r , q˙ ) forces that j can be lifted to some j : V [G][H] −→ M[j(G)][j(H)].

Claim 6.3. If we generate u = huτ | τ < λi from such a lifting of j to V [G][H] in the usual way, by inductively, for τ < λ, setting , then in fact we have uτ = {D ∈ V [G][H] | u τ ∈ j(D)} u ∈ V [G][H] (and not merely u ∈ V [G][H][j(G)][j(H)]) and u ∈ UV [G][H].

Proof. This is true by closure and the fact that for all D we have D ∈ uτ ∗ ∗ ˙ if and only if (r , q˙ ) u τ ∈ j(D). FORCING CONSTRUCTIONS 37

We actually have that u is dened by induction on λ (all in V [G][H]) as follows. First of all, set u0 = κ. Then, suppose that for some τ < λ we have constructed uσ for all σ < τ and V [G][H] |= huσ | σ < τ i ∈ U. ∗ ∗ ˙ We made sure for each ξ < χ that (r , q˙ ) k huσ | σ < τ i ∈ j(Dξ) . G∗H ∗ ∗ ˙ So if u = {D | (r , q˙ ) ˙ G∗H hu | σ < τ i ∈ j(D)} then τ L ∗j(Pχ) σ V [G][H] |= uτ is a normal measure on Vκ.

We check that huσ | σ < τ + 1i ∈ U. First of all, we need to check that ∗ ∗ ∈ u , that is that (r , q˙ ) ˙ G∗H hu | σ < τ i ∈ j( ). Uκ τ L ∗j(Pχ) σ Uκ This is clear because χM[j(G)][j(H)] ⊆ M[j(G)][j(H)] by the Laver construction, and so the sequence is in M[j(G)][j(H)]. We also need that normality and the two coherence conditions from

Denition (3.1) hold for uτ .

Normality is true since if f : Uκ −→ Vκ is a function in V [G][H] and ∗ ∗ (r , q˙ ) ˙ G∗H j(f)(hu | σ < τ i) ∈ V then there is some z ∈ V L ∗j(Pχ) σ κ κ ∗ ∗ such that (r , q˙ ) ˙ G∗H j(f)(hu | σ < τ i) = z . For coherence: L ∗j(Pχ) σ ∗ ∗ if f : −→ κ and (r , q˙ ) ˙ G∗H j(f)(hu | σ < τ i) < τ then Uκ L ∗j(Pχ) σ ∗ ∗ for some σ < τ we have z ∈ u if and only if (r , q˙ ) ˙ G∗H σ L ∗j(Pχ) j(f)(huσ | σ<τ i) (j(z) ∩ Vκ) ∈ u which is true as j(z) ∩ Vκ = z; and if 0 ∗ ∗ τ < τ and z ∈ u then (r , q˙ ) ˙ G∗H ∃σ < τ j(z) ∩ V ∈ u . σ L ∗j(Pχ) κ σ N6.3 Claim 6.4. In there is a stationary set χ such that V [G][H] S ⊆ Sκ+ • For every ξ ∈ S one has lh(uξ) = λ and for every τ < λ that ξ . uτ ∩ V [G][Hξ] = uτ

• For every ξ in the closure of S, S ∩ ξ ∈ V [G][Hξ].

For every , ξ V [G][Hξ]. • ξ ∈ S u ∈ Uκ+ For every , ξ . • ξ ∈ S {v | ∃ρ < κ v ∈ Aρ } ∈ F (u)

Proof. Working in V [G][H], dene a set T ⊆ χ corresponding to u by enumerating each uτ in order type χ as hyτξ | ξ < χi and setting T ∩[κ.τ.ξ, κ.τ.(ξ +1)) = {κ.τ.ξ +η | η ∈ yτξ }. Appealing to Proposition (2.3), we may nd a stationary set χ such that satises the S ⊆ Sκ+ S rst two clauses of the claim. For and , we have that ξ is a measure on in ξ ∈ S 0 < τ < λ uτ Vκ . Since V [G] V [G][H] it is clear that each ξ concentrates V [G][Hξ] Uκ = Uκ uτ on Uκ. 38 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

V [G][Hξ] Ad normality. Suppose f : Uκ −→ Vκ and {w ∈ Uκ | f(w) ∈ ξ . As ξ we can apply normality for to get some Vκw } ∈ uτ uτ ⊆ uτ u x ∈ Vκ such that {w ∈ Uκ | f(w) = x} ∈ uτ . But x, f ∈ V [G][Hξ], hence V [G][Hξ] ξ . {w ∈ Uκ | f(w) = x} ∈ uτ ∩ Vκ = uτ

Ad coherence (i). Suppose f : Uκ −→ κ with f ∈ V [G][Hξ] and ξ . Again as ξ we can apply the {w ∈ Uκ | f(w) < lh(w)} ∈ uτ uτ ⊆ uτ rst coherence condition for u to get some σ < τ with x ∈ uσ if and only if . If , so that ξ , {w ∈ Uκ | x ∩ Vκw ∈ wf(w) } ∈ uτ x ∈ V [G][Hξ] x ∈ uσ then, recalling that , we have f ∈ V [G][Hξ] {w ∈ Uκ | x∩Vκw ∈ wf(w) } ∈ , and hence ξ . Conversely, if V [G][Hξ] {w ∈ Uκ | x ∩ Vκw ∈ wf(w) } ∈ uτ ξ then, as ξ , we have . {w ∈ Uκ | x ∩ Vκw ∈ wf(w) } ∈ uτ uτ ⊆ uτ x ∈ uσ Ad coherence (ii). If 0 and ξ then τ < τ x ∈ uτ 0 {w ∈ Uκ | ∃σ < lh(w) x ∩ and the set is clearly in as is, and hence is Vκw ∈ wσ } ∈ uτ V [G][Hξ] x an element of ξ . uτ Finally, at each ξ ∈ S when we did the inductive construction of (rξ+1, q˙ξ+1) we must have been in `Case (i)' and used Lemma (6.2) ∗ ∗ because (r , q˙ j(ξ)) would be an appropriate witness. Consequently for all we have that ξ , and hence ξ . τ < λ u τ ∈ j(A ) A ∈ F (u) N6.4 Proposition 6.5. Let S ∈ V [G][H] be as given by Claim (6.4). Let ξ be a limit of elements of the set S of conality at least κ+. For each let S ε . Then is an ultralter τ < λ vτ = {uτ | ε ∈ S ∩ ξ } V [G][Hξ] |= v sequence and ∀ε ∈ S ∩ ξ Aε ∈ F (v) .

Proof. By Claim (6.4) we have that S ∩ ξ ∈ V [G][Hξ], thus v ∈ V [G][Hξ]. It is clear that v is a sequence of measures concentrating V [G][Hξ] on Uκ . In order to see that the normality and coherence conditions hold it is enough to observe that since cf(ξ) ≥ κ+ we have that for each z ∈ P (κ) and f : Uκ −→ Vκ if z, f ∈ V [G][Hξ] there is some ε ∈ S ∩ ξ such that z, f ∈ V [G][Hε]. Again by Claim (6.4) we have that ε . A ∈ F (v) N6.5

7. Proof that we do get small universal families

Let S ∈ V [G][H] be as given by Claim (6.4). As in Proposition (6.5), ++ we choose ξ a limit point of S with cf(ξ) = κ . Let G ∗ Hξ be the ˙ L ∗ Pξ-generic lter over V induced by G ∗ H, and dene an ultralter sequence by setting S ε for . v ∈ V [G∗Hξ] vτ = {uτ | ε ∈ S ∩ξ } 0 < τ < λ ∗ Let g be Rv-generic over V [G ∗ Hξ]. As we discussed in 3, by forcing below a suitable condition we may arrange that FORCING CONSTRUCTIONS 39

∗ • the generic object induced by g is a λ-sequence hui | i < λi of ultralter sequences, dening for , the set is a club • κi = κui i < λ C = {κi | i < λ} subset of κ,

∗ • min(C) > λ, and so cf(κ) = λ in V [G ∗ Hξ ∗ g ].

Notation 7.1. For ε ∈ S ∩ ξ set Hε to be the induced Pε-generic lter over , ˙ G∗Hε , to be the generic lter over V [G] Qε = Qε Kε Qε V [G][Hε] induced by , ∗ε p p G ∗ Hξ A = hAρ | ρ < κ & ∃p ∈ Kε (ρ < ρ & Aρ = Aρ)i, ε to be the common value of for ∗ε, and ∗ε ε . κρ κw w ∈ Aρ a = {κρ | ρ < κ} As per Observation(3.24), the characterisation of genericity for Radin forcing implies that for every ε ∈ S ∩ξ, the sequence hui | i < λi is Ruε - ε generic over V [G][Hε]. Let g be the Ruε -generic lter over V [G][Hε] ε ∗ induced by this sequence, so that easily g = g ∩ Ruε .

Note that, by the characterisation of Radin-genericity from Theorem (3.22), for each ε ∈ S ∩ ξ, since Aε ∈ F (v) by Proposition (6.5), we have that S ∗ε for all large . Let be the uj ∈ {Aρ | ρ < κ} j < λ iε < λ least successor ordinal such that S ∗ε for , and uj ∈ {Aρ | ρ < κ} j ≥ iε let . Since is a successor point of the generic club , the ηε = κui ηε C sequence ε . uiε = hηε i We note that since S ∗ε , we may dene as uiε = hηε i ∈ {Aρ | ρ < κ} σε the unique such that ∗ε, and by denition we have ρ hηε i ∈ Aρ κuiε = ε . It follows that for any with q we have that ηε = κσε q ∈ Kε σε < ρ q and q. hηε i ∈ Aσε ηε ∈ a

ε Denition 7.2. For let ε ˙ ε G∗Hε∗g , and, for −, ε ∈ S ∩ ξ Eα = (Eα) α < χ let ε S ηε q q,ηε fα = {(fα ) | q ∈ Kε & α ∈ t }.

As we proved in Section 4, ε is a function with domain + such that fα κ ε for every +. fα(ζ) ∈ {bα ζ } × κ ζ < κ Denition 7.3. For each and − dene ε on ε ε ∈ S ∩ ξ α < χ Eα rge(fα) by ε 0 if and only if ε 0 . That is, ε ε ε. z Eα z ht(π0(z)) Eα ht(π0(z )) Eα = fα“Eα

We now prove a short technical lemma which will allow us to give an equivalent characterization of ε, which in turn facilitates the proof Eα that ε and ε are coherent for . Eα Eβ α 6= β Lemma 7.4. Let ε ∈ S ∩ ξ. Suppose y_(uε,D) ∈ gε and q = q q q q (A ,B , t , f ) ∈ Kε are such that 40 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

• ρq is a successor ordinal.

q • ηε, κy ≤ sup(a ). • Bq ⊆ D. Then there is a lower part y0 such that • y0_(uε,Bq) ≤ y_(uε,D) • y0_(uε,Bq) ∈ gε.

0 q • y is harmonious with A past ηε.

Proof. Let j < λ be the largest ordinal such that κ ≤ max(aq). For 0 uj0 all with we have that S ∗ε and q , j j0 < j < λ uj ∈ σ<κ Aσ κuj > max(a ) q so that uj ∈ B . Appealing to the second part of Lemma (3.17) and to Lemma (3.26), we may now extend y_(uε,D) to a condition y_(uε,Bq) ∈ gε by rst adding in (as necessary) the pair (hηε i, ∅) and a pair with rst entry u , then shrinking D to Bq. Note that κ = max({η , κ , κ }) ≤ j0 y ε y uj0 max(aq).

0 Appealing to Lemma (3.29), there is y directly extending y in Ry such that 0 conforms with ε, and 0 is harmonious with q past . y g y A ηε N7.4 Proposition 7.5. For each , − and , 0 ε we ε ∈ S ∩ ξ α < χ z z ∈ rge(fα) have ε 0 if and only if there exist a condition and a lower z Eα z p ∈ Kε p _ ε p part y harmonious with A past ηε such that p and y (u ,B ) witness that ε 0, i.e., letting and 0 0 , such that z Eα z ζ = lh(π0(z)) ζ = lh(π0(z ))

p,ηε , 0 p,ηε , p,ηε and p,ηε 0 0. • α ∈ t ζ, ζ ∈ dα fα (ζ) = z fα (ζ ) = z p • y is harmonious with A past ηε. • y_(uε,Bp) ∈ gε. • y_(uε,Bp) V [G][Hε] ζE˙ ε ζ0. Ruε α

Proof. If p and y are as in the equivalent, then y_(uε,Bp) ∈ gε. and y_(uε,Bp) V [G][Hε] ζE˙ ε ζ0, so that ζ E ε ζ0. Ruε α α For the converse direction, suppose that ε 0. Choose a condition ζ Eα ζ y_(uε,D) ∈ gε such that y_(uε,D) V [G][Hε] ζE˙ ε ζ0 and hη i appears Ruε α ε in y. By the choice of ηε, the sequence hηε i and all subsequent sequences appearing in are members of S ∗ε. y ρ<κ Aρ FORCING CONSTRUCTIONS 41

ε q q Choose a condition q ∈ K such that ηε, κy ∈ a , ρ is a successor ordinal, and Bq ⊆ D. Appealing to Lemma (7.4) we nd a condition y0_(uε,Bq) which is exactly of the kind need to form a witness (together with ) that ε 0. q z Eα z N7.5

Next we show that ε and ε cohere for . Eα Eβ α 6= β Lemma 7.6. Let , and and 0 ε ε . ε ∈ S ∩ ξ α 6= β z z ∈ rge(fα) ∩ rge(fβ) Then ε 0 if and only ε 0. z Eα z z Eβ z

Proof. Choose r ∈ Kε and y a lower part witnessing the equivalent conditions listed in Proposition (7.5) that ε 0. Choose z Eα z p ≤ r such that p ∈ Kε, ρp is a successor ordinal, and p contains enough information to verify that , 0 ε , that is to say that p,ηε , z z ∈ rge(fα) β ∈ t 0 p,ηε , p,ηε and p,ηε 0 0. ζ, ζ ∈ dβ fβ (ζ) = z fβ (ζ ) = z We now recall the notion of ρ∗-weakening from Denition (4.20). Let q r q p r r be the ρ -weakening of p, and note that by denition A = A ρ = A and aq = ar. It follows from Lemmas (4.21) and (4.22) that p ≤ q ≤ r, q r in particular q ∈ Kε and B ⊆ B . We claim that and q will serve as witnesses that ε 0, that is y B z Eβ z that _ ε q ε and that _ ε q ˙ ε 0. y (u ,B ) ∈ g y (u ,B ) ζEβζ By Lemma (3.26), to check that y_(uε,Bq) ∈ gε we must show that y conforms with g and u ∈ Bq for all u appearing on the generic _ r sequence with κy < κu. Since y (u ,B ) ∈ g , we see that the lower part y conforms with g and also that u ∈ Br for all u appearing on the generic sequence with κy < κu. Fix such a sequence u. Since r is a r q r condition and u ∈ B it follows that ssup(a ) = ssup(a ) ≤ κu. By the choice of and , we have S ∗ε , and since we ηε y u ∈ {Aρ | ρ < κ} q ∈ Kε have that u ∈ Bq. Since Bq ⊆ Br, we see that y_(u,Bq) is a renement of y_(u,Br), and in particular _ q ˙ ε 0. Since q r and is a con- y (u ,B ) ζEαζ A = A q _ q dition, it follows from Clause (5) in Denition (4.5) that y (u ,B ) ˙ ε 0. ζEβζ N7.6 ∗ Denition 7.7. Working in the model V [G][Hξ][g ], we dene for each a relation ε on the set S + − ε ∈ S ∩ ξ E {{bα ζ | ζ < κ } × κ | α < χ } by ε 0 − ε 0 z E z ←→ ∃α < χ z Eα z . ∗ ε + Proposition 7.8. V [G][Hξ][g ] |= E is of size κ and is universal for ε − . {Eα | α < χ } 42 CUMMINGS, DAMONJA, MAGIDOR, MORGAN, AND SHELAH

ε + − − Proof. E has size κ by the choice of {bα | α < χ }, as for α < χ and + we have that , where is the +-Kurepa tree ζ < κ bα ζ ∈ T T κ added by Q0. The universality follows from the denitions of ε , ε and ε, and Eα Eα E Lemma (7.6) which ensures that each ε is an embedding of ε into fα Eα ε. E N7.8 ∗ ∗ Proposition 7.9. In V [G][Hξ][g ] suppose Ξ (∈ V [G][Hξ][g ]) is conal in and for all we have that ˙ ε − is a list of S ∩ ξ ε ∈ Ξ {Eα | α < χ } canonical names for all graphs on κ+, Then {E ε | ε ∈ Ξ} is a universal family in the collection of graphs on κ+.

+ ∗ ˙ Proof. Let E be a graph on κ in V [G][Hξ][g ] and let E be a canonical ∗ ˙ G∗Hξ∗g + L ∗ Pξ ∗ Rv-name such that E = E . As cf(ξ) > κ there is some ε ˙ ˙ G∗Hε∗g ε ∈ Ξ such that E is a L ∗ Pε ∗ Ruε -name and hence E = E . By Proposition (7.8) we thus have that embeds into ε. E E N7.9 Theorem 7.10. Suppose κ is a supercompact cardinal, λ < κ is a regular cardinal and Θ is a cardinal with cf(Θ) ≥ κ++ and κ+3 ≤ Θ. There is a forcing extension in which conally many cardinals below κ, κ itself and all cardinals greater than κ are preserved, cf(κ) = λ, 2κ = 2κ+ = Θ and there is a universal family of graphs on κ+ of size κ++.

Proof. Let χ = Θ+. It is standard that if κ is supercompact there is a forcing extension in which κ remains supercompact and GCH and χ hold. The conclusion is now immediate from Proposition (7.9). ♦χ(Sκ+ ) N7.10

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Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213, USA E-mail address: [email protected]

School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK E-mail address: [email protected]

Institute of Mathematics, Hebrew University of Jerusalem, 91904 Gi- vat Ram, Israel E-mail address: [email protected]

School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK E-mail address: [email protected]

Institute of Mathematics, Hebrew University of Jerusalem, 91904 Gi- vat Ram, Israel and Rutgers University, New Brunswick, NJ, USA E-mail address: [email protected]