aa r X i v : . [ m a t h . L O ] J a n Coherent forests
Monroe Eskew
Abstract
A forest is a generalization of a tree, and here we consider the Aron-szajn and Suslin properties for forests. We focus on those forests satisfyingcoherence, a local smallness property. We show that coherent Aronszajnforests can be constructed within ZFC. We give several ways of obtainingcoherent Suslin forests by forcing, one of which generalizes the well-knownargument of Todorˇcevi´c that a Cohen real adds a Suslin tree. Another usesa strong combinatorial principle that plays a similar role to diamond. Weshow that, starting from a large cardinal, this principle can be obtainedby a forcing that is small relative to the forest it constructs.
We consider a type of structure called a forest, a generalization of a tree.Forests contain many trees, but can be much wider than a single tree. ThomasJech had previously studied the same type of object under the name “mess” [5].The nicer choice of terminology is due to Christoph Weiß [10]. In contrast tothe work of Weiß, we will focus on forests that do not contain long branches.The notions of being Aronszajn and Suslin carry over from trees to forests.In this paper, we explore several ways of obtaining large Aronszajn and Suslinforests that also satisfy a certain local smallness property called coherence. Weshow that large coherent Aronszajn forests can be constructed within ZFC andby forcing. Next, we explore a constraint imposed by the P-ideal dichotomy thatshows the optimality of some of these results. Finally, we give three ways offorcing large coherent Suslin forests. The first is a modification of Jech’s methodof forcing by local approximations. The second generalizes the argument ofTodorˇcevi´c that a Cohen real adds a Suslin tree. Here, we compose a Cohen-generic function with certain kind of coherent Aronszajn forest, and show thatwhile the structure remains non-trivial, all large antichains destroyed. Thethird method uses a guessing principle that plays a similar role to diamond inthe construction of Suslin trees. We show that this principle can be obtainedfrom a Mahlo cardinal κ using a forcing of size κ , yet results in a coherent Suslinforest of size 2 κ . In other work [2], this last result is applied to the study ofsaturated ideals. Definition
A ( κ, X, µ )- forest is a collection of functions F satisfying:(1) { dom( f ) : f ∈ F } = P κ ( X ).(2) ( ∀ f ∈ F ) ran( f ) ⊆ µ . 13) For z ∈ P κ ( X ), let F z = { f ∈ F : dom( f ) = z } . A forest must satisfy thatfor z ⊆ z in P κ ( X ), F z = { f ↾ z : f ∈ F z } .Forests are full of trees. If F is a ( κ, X, µ )-forest, and S = { x α : α < κ } is anenumeration of distinct elements of X , then T S = { f ∈ F : ( ∃ β < κ ) dom( f ) = { x α : α < β }} forms a tree of height κ under the subset ordering.A ( κ, X, µ )-forest F is called thin if for all z ∈ P κ ( X ), | F z | < κ . A collectionof functions F is called κ - coherent if for all f, g ∈ F , |{ x ∈ dom( f ) ∩ dom( g ) : f ( x ) = g ( x ) }| < κ . If F is a ( κ + , X, µ )-forest we say it is coherent if it is κ -coherent. Clearly, if µ ≤ κ = κ <κ , then any coherent ( κ + , X, µ )-forest is thin.A chain in a forest is a subset which is linearly ordered under ⊆ . Two el-ements f, g in a forest F are said to be compatible when they have a commonextension h ∈ F . An antichain in a forest is a subset of pairwise incompat-ible elements. We say that a ( κ, X, µ )-forest F is Aronszajn if it contains nowell-ordered chain of length κ . We say it is Suslin if it contains no antichain ofcardinality κ . If F is a ( κ, X, µ )-forest with µ ≥
2, closed under finite modifi-cations, then F is Suslin only if it is Aronszajn. This is because we can “splitoff” from any chain of length κ to get an antichain of size κ . Proposition 0.1 If F is a ( κ, X, µ ) -forest, then for any z ∈ P κ ( X ) , F z is amaximal antichain. Proof
Let f ∈ F , z ∈ P κ ( X ). By clause (3) of the definition of forests, thereis g ∈ F such that f ⊆ g and dom( g ) = dom( f ) ∪ z . Then g ↾ z ∈ F z , so g is acommon extension of f and something in F z . (cid:3) The following lemma will be useful in several constructions:
Lemma 0.2
Suppose F is a coherent ( κ + , X, µ ) -forest, and F is closed under < κ modifications. Then two functions in F have a common extension in F ifand only if they agree on their common domain. Proof
Let f, g ∈ F agree on dom( f ) ∩ dom( g ). Let h ∈ F be such thatdom( h ) = dom( f ) ∪ dom( g ). By coherence, we can change the values of h on a set of size < κ to get h ′ : dom( h ) → µ with h ′ ↾ dom( f ) = f , and h ′ ↾ dom( g ) = g . By the closure of F , h ′ ∈ F . (cid:3) The first theorem of this section generalizes of an argument of Koszmider [6].
Lemma 1.1
Let κ be a regular cardinal, and suppose F = { f α : α < κ } is a κ -coherent set of partial functions from κ to µ .(a) There is a function f : κ → µ such that { f } ∪ F is κ -coherent.(b) If µ = κ and each f α is < κ to 1, then there is a < κ to 1 function f : κ → κ such that { f } ∪ F is κ -coherent. roof For each α , let D α = dom( f α ) \ S β<α dom( f β ). Let E = κ \ S α D α . Forthe first claim, choose any function g : E → µ , and let f ( β ) = ( f α ( β ) if β ∈ D α g ( β ) if β ∈ E For any α , { β : f ( β ) = f α ( β ) } = S γ<α { β ∈ D γ ∩ dom( f α ) : f γ ( β ) = f α ( β ) } .This is a union of < κ sets of size < κ , so has size < κ .For the second claim, choose any < κ to 1 function g : E → κ , and let f ( β ) = ( max( α, f α ( β )) if β ∈ D α g ( β ) if β ∈ E For any α , { β : f ( β ) = f α ( β ) } ⊆ S γ ≤ α { β ∈ D γ : f γ ( β ) < γ or f γ ( β ) = f α ( β ) } .By the hypotheses, this set has size < κ . For each α , f − ( α ) ⊆ g − ( α ) ∪ S { f − γ ( β ) : γ, β ≤ α } , so f is < κ to 1. (cid:3) Theorem 1.2
Let κ be a regular cardinal. For every ζ < κ , there is a coherent ( κ + , κ + ζ , κ ) -forest consisting of < κ to 1 functions. Proof
We will prove by induction the following stronger statement: For every ζ < κ and every sequence h ( X α , F α ) : α < κ i such that:(1) each X α ⊆ κ + ζ ,(2) each F α is a ( κ + , X α , κ )-forest of < κ to 1 functions,(3) S α F α is κ -coherent,there is a coherent ( κ + , κ + ζ , κ )-forest F ⊇ S α F α consisting of < κ to 1 func-tions.For ζ = 0, pick a collection { f α : α < κ } such that for each α , f α ∈ F α , anddom( f α ) = X α . By Lemma 1.1(b), there is a < κ to 1 function f : κ → κ thatcoheres with each f α , and we can take F = { g : dom( g ) ⊆ κ and |{ x : f ( x ) = g ( x ) }| < κ } .Assume ζ = η + 1 and the statement holds for η . For each β < κ + ζ , let F βα = S α { f ↾ β : f ∈ F α } . We will construct F ⊇ S F α as the union of a ⊆ -increasing sequence h G β : β < κ + ζ i such that for each β , G β is a coherent( κ + , β, κ )-forest of < κ to 1 functions containing S α F βα . Let G = {∅} . Given G β , let G β +1 = { f : dom( f ) ⊆ ( β + 1), ran( f ) ⊆ κ , and f ↾ β ∈ G β } .Suppose β is a limit ordinal of cofinality ≤ κ , and let h γ i : i < δ ≤ κ i be cofinal in β . The collection S i<δ G γ i ∪ S α<κ F βα is κ -coherent, because( ∀ α < κ )( ∀ f ∈ F βα )( ∀ i < δ )( f ↾ γ i ∈ F γ i α ⊆ G γ i ). Since β has cardinality ≤ κ + η , the inductive assumption implies that we can extend to a forest G β with the desired properties.Suppose β is a limit ordinal of cofinality > κ . Let G β = S γ<β G γ . Then G β is a forest with the desired properties because S α<κ F βα = S γ<β ( S α<κ F γα ).Finally, we let F = S β<κ + ζ G β . 3ow assume ζ is a limit ordinal of cofinality < κ , and the statement holdsfor all η < ζ . Let h γ i : i < δ = cf( ζ ) i be an increasing cofinal sequence in ζ .Like above, recursively build an increasing sequence h G i : i < δ i such that each G i is a ( κ + , κ + γ i , κ )-forest of < κ to 1 functions extending S α F γ i α . This is doneby applying the inductive hypothesis for κ + γ i to S α F γ i α ∪ S j
Koszmider showed that in the case κ = ω , if λ is a singular cardinalof cofinality ω , and (cid:3) λ and λ ω = λ + hold, then the induction can push through λ as well. The argument generalizes almost verbatim to show for any regular κ ,the induction can go forward at λ of cofinality κ , under the assumptions (cid:3) λ and λ κ = λ + . (The reader may want to verify this.) As a consequence, we get thatin L , for every regular κ and every λ ≥ κ , there is a coherent, ( κ + , λ, κ )-forestof < κ to 1 functions.Recall that a partial order P is called κ -Knaster if for any A ⊆ P of size κ ,there is B ⊆ A of size κ that consists of pairwise compatible elements. Corollary 1.3
For every regular cardinal κ and every ζ < κ , there is a coherent ( κ + , κ + ζ , κ ) -forest, which is Aronszajn, does not have the <κ or the κ + chaincondition, but is (2 κ ) + -Knaster. If ζ is finite or <κ < κ + ω , then the forest is (2 <κ · κ + ) + -Knaster. Proof
Let F be given by Theorem 1.2. We may assume F is closed under < κ modifications. To see the failure of the 2 <κ chain condition, note that for any z ⊆ κ + ζ of size κ , F z is an antichain of size 2 <κ .Let { α β : β < κ + } be any enumeration of distinct ordinals in κ + ζ , and foreach γ < κ + , let f γ ∈ F have domain { α β : β < γ } . Since each f ∈ F mapsinto κ , there is a ξ < κ and a stationary subset S ⊆ { γ < κ + : cf( γ ) = κ } suchthat for all γ ∈ S , f γ +1 ( α γ ) = ξ . Since each f ∈ F is < κ to 1, each set { β <γ : f γ +1 ( α β ) = ξ } is bounded below γ when cf( γ ) = κ . Thus there is an η < κ + and a stationary S ⊆ S such that for all γ ∈ S , { β < γ : f γ +1 ( α β ) = ξ } ⊆ η .Therefore, for any γ < γ in S \ η , f γ +1 ( α γ ) = f γ +1 ( α γ ). This shows that F does not have the κ + chain condition.It also shows that F is Aronszajn. For otherwise, let h f α : α < κ + i be astrictly increasing ⊆ -chain in F . Let { ξ β : β < κ + } = S α dom( f α ), and for4ach γ let g γ = ( S α f α ) ↾ { ξ β : β < γ } . Then h g γ : γ < κ + i is a strictlyincreasing chain, but by the above paragraph, it contains an antichain of size κ + , contradiction.To show the (2 κ ) + -Knaster property, let { f α : α < (2 κ ) + } ⊆ F . Let T ⊆ (2 κ ) + have size (2 κ ) + and be such that { dom( f α ) : α ∈ T } forms a delta-system with root r . Let T ⊆ T have size (2 κ ) + and be such that for a fixed g , f α ↾ r = g for all α ∈ T . The union of any two of these is in F .For the case where ζ < ω or 2 <κ < κ + ω , let θ = (2 <κ · κ + ) + . First notethat it is easy to see by induction that for every n < ω , P κ + ( κ + n ) has a cofinalsubset of size κ + n . Let A = { f α : α < θ } ⊆ F , and let S = S α dom( f α ).Suppose first that | S | < θ . There is an R ⊆ P κ + ( S ) that covers { dom( f α ) : α < θ } and has cardinality | S | . Therefore, by the coherence of F , there is a G ⊆ F of cardinality ≤ | S | · <κ < θ such that for all α < θ , there is g ∈ G with f α ⊆ g . Therefore there is a g ∈ G which is a common lower bound to θ many f α . Now suppose that | S | = θ . Since θ is regular and θ > κ + , we can use thedelta-system argument to get an S ⊆ S of cardinality less than θ and a T ⊆ θ of cardinality θ such that for all α , α ∈ T , dom( f α ) ∩ dom( f α ) ⊆ S . Bythe above paragraph, there is a T ⊆ T of cardinality θ such that for any α , α ∈ T , f α and f α agree on their common domain contained in S . (cid:3) One may ask whether the condition “ < κ to 1” can be strengthened to “1 to1” in Theorem 1.2. But this cannot always be achieved:
Proposition 1.4
If there is a coherent ( κ + , λ, κ ) -forest consisting of injectivefunctions, then there are λ many almost disjoint subsets of κ . Proof
Let F be such a forest, and for each z ∈ P κ + ( λ ), choose f z ∈ F withdomain z . Let S be a collection of λ many pairwise disjoint subsets of λ , eachof cardinality κ . For x = y in S , ran( f x ) is almost disjoint from ran( f y ). Thisis because the sets A = ran( f x ∪ y ↾ x ) and B = ran( f x ∪ y ↾ y ) are disjoint, and | A △ ran( f x ) | < κ , and | B △ ran( f y ) | < κ . (cid:3) A positive answer in the following special case is well-known (see [7], ChapterII, Theorem 5.9 and exercise 37):
Theorem 1.5
Let κ be a regular cardinal. There is a κ -coherent collection offunctions { f α : α < κ + } , such that each f α is an injection from α to κ . A more general positive answer can be forced:
Theorem 1.6
Assume κ is a regular cardinal with <κ = κ , and λ ≥ κ . Thereis a κ -closed, κ + -c.c. partial order that adds a coherent ( κ + , λ, κ ) -forest ofinjective functions. Remark
Such a forest will be Aronszajn because a chain of length κ + wouldgive an injection from κ + to κ . Unlike the forests of Theorem 1.2, it will neverhave the λ chain condition. 5 roof Let P be the collection partial functions p that assign to < κ many z ⊆ λ of size ≤ κ , a partial injective function from z to κ defined at < κ many points.Let p ≤ q when:(a) dom( p ) ⊇ dom( q ).(b) For all z ∈ dom( q ), p ( z ) ⊇ q ( z ).(c) If z , z ∈ dom( q ), α ∈ z ∩ z \ (dom( q ( z )) ∪ dom( q ( z )), and α ∈ dom( p ( z )), then α ∈ dom( p ( z )) and p ( z )( α ) = p ( z )( α ).It is easy to check that ≤ is transitive and that h P , ≤i is κ -closed. To checkthe chain condition, let A ⊆ P have size κ + . Since κ <κ = κ , we can find a B ⊆ A of size κ + such that { dom( p ) : p ∈ B } forms a delta-system with root R .Again since κ <κ = κ , there is a C ⊆ B of size κ + and a collection of functions { f z : z ∈ R } such that ∀ p ∈ C , ∀ z ∈ R , p ( z ) = f z . If p, q ∈ C , then p ∪ q is acommon extension.If G ⊆ P is generic, then for all z ∈ P κ + ( λ ) V , G gives an injective function f z : z → κ as S { p ( z ) : z ∈ p ∈ G } . For z , z ∈ P κ + ( λ ) V , there is some p ∈ G such that z , z ∈ dom( p ). p forces that f z and f z agree outside dom( p ( z )) ∪ dom( p ( z )). Finally, by the κ + -c.c., P κ + ( λ ) V is cofinal in P κ + ( λ ) V [ G ] . So wecan define a ( κ + , λ, κ )-forest F as { f : f is an injection into κ , ( ∃ z ) dom( f ) ⊆ z ∈ P κ + ( λ ) V , and f disagrees with f z at < κ many points } . (cid:3) In the previous section, we saw that coherent, Aronszajn ( ω , ω n , ω )-forests canbe constructed in ZFC for every natural number n . Here we show that the thirdcoordinate is optimal, in the sense that for n < ω and λ ≥ ω , ZFC cannotprove the existence of a coherent, Aronszajn ( ω , λ, n )-forest. Let us recall therelevant notions: Definition
An ideal I ⊆ P ( X ) is a P-ideal when P ω ( X ) ⊆ I ⊆ P ω ( X ), andfor any { z n : n < ω } ⊆ I , there is z ∈ I such that z n \ z is finite for all n . Definition
The
P-ideal dichotomy (PID) is the statement that for any P-ideal I on a set X , either(1) there is an uncountable Y ⊆ X such that P ω ( Y ) ⊆ I , or(2) there is a partition of X into { X n : n < ω } such that for all n and all z ∈ I , z ∩ X n is finite.PID is a consequence of the Proper Forcing Axiom, and is also known tobe consistent with ZFC+GCH relative to a supercompact cardinal [9]. Therestriction of PID to ideals on sets of size ω is known to be consistent withoutthe use of large cardinals, both with and without GCH [1].6sing a coherent, Aronszajn ( ω , ω , ω )-forest F , we can obtain a coherent,Aronszajn ω -tree T of binary functions by taking the collection of characteristicfunctions of members of F whose domain is an ordinal, considering the functionsas subsets of α × ω for α < ω . A cofinal branch would be a function g : ω × ω → g ↾ ( α × ω ) ∈ T for all α < ω , and this would code an uncountable well-ordered chain in F . Further, using a regressive function argument, we can seethat the closure of T under finite modifications remains Aronszajn. On the otherhand, forests are more flexible. If we take such a tree T , close it under subsetsto get a forest F , then it may be that there is an uncountable well-ordered chain C ⊆ F , but with dom( S C ) a proper subset of ω × ω . This is what happensunder PID. Theorem 2.1
Assume PID, and let F be a coherent ( ω , λ, n ) -forest closedunder finite modifications, for some λ ≥ ω , n < ω . Then F is not Aronszajn. Proof
First we prove this for n = 2. Let F be a coherent ( ω , λ, I be the collection of z ⊆ λ such that for some f ∈ F , z ⊆ { α : f ( α ) = 1 } .We claim I is a P-ideal. Let { z n : n < ω } ⊆ I , and for each n , choose f n ∈ F witnessing z n ∈ I . Let f ∈ F have domain S n dom( f n ), and let z = { α : f ( α ) = 1 } . For any n , f disagrees with f n on a finite set, so there canonly be finitely many α ∈ z n \ z .Assume that alternative (1) of PID holds, and let Y ⊆ λ be uncountablesuch that P ω ( Y ) ⊆ I . Enumerate Y as h y α : α < ω i . For each α < ω , let f α be the function that has f α ( y β ) = 1 for β < α , and is undefined elsewhere.Since F is closed under subsets, each f α ∈ F , and these form an uncountablewell-ordered chain.Assume alternative (2) of PID holds. Let X n ⊆ λ be uncountable such thatfor all z ∈ I , X n ∩ z is finite. Let g have constant value 0 on X n . If f ∈ F anddom( f ) ⊆ X n , then { α : f ( α ) = 1 } is finite. Thus for any countable z ⊆ X n , g ↾ z ∈ F , so again we have an uncountable well-ordered chain.Now assume the result holds for n , and let F be a coherent ( ω , λ, n + 1)-forest. Let r ( k ) = 0 for k < n , and r ( n ) = 1. Consider the forest G = { r ◦ f : f ∈ F } , and let g , g be the functions on λ with constant value 0 and1 respectively. By the above argument, there is some uncountable Y ⊆ λ suchthat either g ↾ z ∈ G for all countable z ⊆ Y , or likewise for g . The lattercase shows that F is not Aronszajn. In the former case, we have that for allcountable z ⊆ Y , there is a function f z ∈ F with domain z that only takesvalues below n . If H = { g : ( ∃ z ∈ P ω ( Y )) g : z → n and { α : g ( α ) = f z ( α ) } isfinite } , then H is a coherent ( ω , Y, n )-forest contained in F . By induction, H contains an uncountable well-ordered chain. (cid:3) Lemma 3.1
Let κ be a regular cardinal. All Suslin ( κ, λ, µ ) -forests are κ -distributive. roof Let F be a Suslin ( κ, λ, µ )-forest, and let h A α : α < δ < κ i be a sequenceof maximal antichains contained in F . By the Suslin property, each A α has size < κ , so if z = S α { dom( f ) : f ∈ A α } , | z | < κ . By maximality, for every α < δ and every g ∈ F z , there is an f ∈ A α such that g is compatible with f . Butsince dom( f ) ⊆ dom( g ), this means f ⊆ g . Thus F z refines each A α . (cid:3) The boolean completion of a Suslin ( κ, λ, µ )-forest is a κ -Suslin algebra,which is a complete boolean algebra with that is both κ -c.c. and κ -distributive.The cardinality of this algebra is at least λ . Therefore the existence of varietiesSuslin forests is constrained by the following (see [4], Theorem 30.20): Theorem 3.2 (Solovay) If B is a κ -Suslin algebra, then | B | ≤ κ . Large Suslin forests can be obtained by forcing. In [5], Jech defined a class ofpartial orders P λ such that under CH, P λ is countably closed, ω -c.c., and addsa Suslin ( ω , λ, Theorem 3.3
Assume κ is a regular cardinal, <κ = κ , and κ = κ + . Thenfor all λ > κ , there is a κ + -closed, κ ++ -c.c. forcing of size λ <κ that adds acoherent, Suslin ( κ + , λ, -forest. Proof (sketch)
Let P be the set of all partial functions f from λ to 2 of size ≤ κ , and say f ≤ g when dom( f ) ⊇ dom( g ) and |{ α : f ( α ) = g ( α ) }| < κ . κ + -closure follows from Lemma 1.1(a), and the κ ++ -c.c. follows from a delta-systemargument. If G is P -generic over V , in V [ G ] let F = { f : ( ∃ g ∈ G ) dom( g ) =dom( f ) and |{ α : f ( α ) = g ( α ) }| < κ } . Clearly F is coherent. The argumentthat F is Suslin in V [ G ] proceeds as in [5]. (cid:3) By adapting an argument of Todorˇcevi´c that appears in [8], we can obtainlarge Suslin forests in a different way:
Theorem 3.4
Assume κ is a regular cardinal, <κ = κ , and there is a coherent ( κ + , λ, κ ) -forest of injective functions. Then adding a Cohen subset of κ adds acoherent, Suslin ( κ + , λ, -forest. Proof
Let F be a coherent ( κ + , λ, κ )-forest of injections closed under < κ modifications to other injections. Let g : κ → Add ( κ ) generic functionover V . Consider the family G = { g ◦ f : f ∈ F } . Since Add ( κ ) is κ + -c.c., P κ + ( λ ) V is cofinal in P κ + ( λ ) V [ g ] , so G generates a forest G when we closeunder subsets. G inherits coherence from F . We claim G is Suslin.First we note that G is closed under < κ modifications. If f ∈ F , then bythe argument for Proposition 1.4, κ \ ran( f ) has size κ . By a density argument, { α ∈ κ \ ran( f ) : g ( α ) = i } has size κ for both i = 0 ,
1. So if g ◦ f ∈ G , and x ⊆ dom f has size < κ , we can switch values of g ◦ f on x by choosing distinctordinals { α i : i ∈ x } ⊆ κ \ ran( f ) such that g ( α i ) = g ( f ( i )) + 1 mod 2. If f ′ = f except that f ′ ( i ) = α i for i ∈ x , then f ′ ∈ V by κ -closure, so g ◦ f ′ ∈ G .8o by Lemma 0.2, members of G have a common extension when they agree ontheir common domain.Towards a contradiction, suppose A = { g ◦ f α : α < κ + } is an antichain in G , and let p ∈ Add ( κ ) force this. Since | Add ( κ ) | = κ , there is some p ≤ p such that p (cid:13) ˙ g ◦ ˇ f ∈ ˙ A for κ + many f ∈ F . Let A = { f : p (cid:13) ˙ g ◦ ˇ f ∈ ˙ A } ,and let Z = S { dom( f ) : f ∈ A } .Case 1: | Z | ≤ κ . Let h ∈ F be such that dom( h ) = Z . There are at most κ many < κ modifications of h , so there are f , f ∈ A such that both agree withthe same modification of h . But p forces that g ◦ f and g ◦ f are compatible,contradiction.Case 2: | Z | = κ + . Let h α i : i < κ + i be an enumeration of Z . Let β =sup(dom( p )) + 1, and for each f ∈ A , let X f = { α : f ( α ) < β } . Since each f is injective, each | X f | < κ . For each X f , let h X f ( i ) : i < β f i be an enumerationof X f that agrees in order with the above enumeration of Z .Case 2a: There is no i < κ such that |{ X f ( i ) : f ∈ A }| = κ + . Then thereis a γ < κ + such that for all f ∈ A , { i : α i ∈ X f } ⊆ γ . Since κ <κ = κ , wemay choose some A ⊆ A such that for all f ∈ A , X f is the same set S , andfurther that f ↾ S is the same for all f ∈ A .Let f , f ∈ A , and let D = { α ∈ dom( f ) ∩ dom( f ) : f ( α ) = f ( α ) } . | D | < κ , D ∩ S = ∅ , and if α ∈ D , then f ( α ) , f ( α ) ≥ β . Thus we can definea q ≤ p such that for all α ∈ D , q ◦ f ( α ) = q ◦ f ( α ) = 0. q forces that g ◦ f and g ◦ f are compatible, contradiction.Case 2b: There is some i < κ such that |{ X f ( i ) : f ∈ A }| = κ + . Let i be the least such ordinal. We choose a sequence h f α : α < κ + i . Let f ∈ A be arbitrary. Let f be such that X f ( i ) has index in the enumeration of Z above { i : α i ∈ dom( f ) } . Keep going in this fashion such that for β < γ < κ + , X f γ ( i ) has index greater than sup { i : α i ∈ dom( f β ) } . By the minimality of i , there is C ⊆ κ + of size κ + and a set S ⊆ Z such that for all α ∈ C , { X f α ( i ) : i < i } = S , and f α ↾ S is the same.Now let β < γ be in C , and let D = { α ∈ dom( f β ) ∩ dom( f γ ) : f β ( α ) = f γ ( α ) } . As before, | D | < κ and D ∩ S = ∅ . If α ∈ D , then f γ ( α ) ≥ β ,because X f γ ∩ dom( f β ) = S . We construct q ≤ p such that for all α ∈ D , q ◦ f γ ( α ) = q ◦ f β ( α ). Let D = { α ∈ D : f β ( α ) ∈ dom( p ) } , and let q = p ∪ {h f γ ( α ) , p ◦ f β ( α ) i : α ∈ D } . We are free to do this because f γ is injectiveand f γ ( α ) / ∈ dom( p ) for α ∈ D .Note that for all α ∈ D , q is defined at f γ ( α ), only if it is defined at f β ( α ).But it may be that for some α ∈ D and some α ′ ∈ D \ D , f γ ( α ) = f β ( α ′ ).Assume we have a sequence q ≥ ... ≥ q n such that:(1) for all k ≤ n , D ∩ f − γ [dom( q k )] ⊆ D ∩ f − β [dom( q k )],(2) for all k ≤ n , q k ◦ f γ ↾ ( D ∩ f − γ [dom( q k )]) = q k ◦ f β ↾ ( D ∩ f − γ [dom( q k )]),(3) if k + 1 ≤ n , then D ∩ f − γ [dom( q k +1 )] = D ∩ f − β [dom( q k )].If D ∩ f − γ [dom( q n )] = D ∩ f − β [dom( q n )], let q n +1 = q n . Otherwise, let D n +1 =9 ∩ f − β [dom( q n )], and let q n +1 = q n ∪ {h f γ ( α ) , q n ◦ f β ( α ) i : α ∈ D n +1 } . Clearlythe induction hypotheses are preserved for n + 1.Put q ω = S q n . (Note in the case κ = ω , D is finite, so q ω = q n for some n .)By (1) and (3), D ∩ f − β [dom( q ω )] = D ∩ f − γ [dom( q ω )], so call this set D ω . Let q = q ω ∪ {h f β ( α ) , i : α ∈ D \ D ω } ∪ {h f γ ( α ) , i : α ∈ D \ D ω } . This q forcesthat g ◦ f β and g ◦ f γ are compatible, again in contradiction to the assumptionabout p . (cid:3) Corollary 3.5
Assume κ is a regular cardinal, <κ = κ , and λ > κ . Then thereis a κ -closed, κ + -c.c. forcing that adds a coherent, Suslin ( κ + , λ, -forest. Proof
Apply Theorems 1.6 and 3.4. (cid:3)
Large Suslin forests can also be obtained from combinatorial principles ratherthan forcing. As reported by Jech [3] [4] [5], Laver proved in unpublished workthat the existence of Suslin ( ω , ω , W and ♦ , both of which hold in L . Unfortunately, Laver’s proof seems to be lostto history. In trying to reconstruct it, we encountered technical issues that ledto the development of a new combinatorial principle, which we prove consistentfrom a Mahlo cardinal, that can be used to construct large Suslin forests. Themain appeal for us is that, unlike the above forcing constructions, it allows aSuslin ( κ, κ + , κ rather than κ + .Let us establish some notation concerning trees. Suppose T is a κ -tree and α < κ . T α is the set of nodes at level α . If b is a cofinal branch in T , π α ( b ) isthe node at level α in b . If β < α , and x ∈ T α , π α,β ( x ) is the node in T β below x . Definition W κ ( λ ) is the statement that there is a κ -tree T , a set of cofinalbranches B , and a sequence h W α : α < κ i with the following properties:(1) | B | = λ .(2) For each α , | W α | < κ , and W α ⊆ P ( T α ).(3) For every z ∈ P κ ( B ), there is an α < κ such that for all β ≥ α , π β [ z ] ∈ W β .Let T , B , h W α : α < κ i be as above. If z ∈ P κ ( B ), say “ z is captured at α ”when for all β ≥ α , π β [ z ] ∈ W β and π β ↾ z is injective. If z ∈ W α and γ < α ,say “ z is captured at γ ” when for all β such that γ ≤ β < α , π α,β [ z ] ∈ W β and π α,β ↾ z is injective. Definition W ∗ κ ( λ ) asserts W κ ( λ ), and that there exists a stationary S ⊆ κ anda sequence h A α : α < κ i with each A α ⊆ W α , such that the following additionalclauses hold:(4) κ = µ + for a regular cardinal µ , and each W α is a µ -complete subalgebraof P ( T α ) containing all singletons.105) For all α ∈ S , { z ∈ W α : z is captured below α } is closed under arbitrary < µ sized unions and taking subsets which are in W α .(6) If f : κ → P κ ( B ) is such that | S α<κ f ( α ) ∪ f ( α ) | = κ , let h b α : α < κ i enumerate the elements of S α<κ f ( α ) ∪ f ( α ). The set of α ∈ S with thefollowing properties is stationary:(a) { b β : β < α } is captured at α .(b) If z ⊆ { π α ( b β ) : β < α } is captured below α , then sup { β : π α ( b β ) ∈ z } < α .(c) {h π α [ f ( β )] , π α [ f ( β )] i : β < α } = A α . Remark
It is easy to see that W κ ( λ ) implies 2 <κ = κ , and in fact W κ ( κ ) isequivalent to 2 <κ = κ . If κ = µ + and S forms part of the witness to W ∗ κ ( λ ),then clause (4) implies µ <µ = µ , and clause (6) can be used to show ♦ κ ( S ).On the other hand, it follows from the next theorem that W ∗ κ ( λ ) prescribes novalue for 2 κ , besides that λ ≤ κ . Theorem 3.6
Suppose κ is a Mahlo cardinal and µ < κ is regular. If G ∗ H ⊆ Col ( µ, < κ ) ∗ Add ( κ ) is generic, then V [ G ∗ H ] satisfies W ∗ κ (2 κ ) . Proof In V , let T be the complete binary tree on κ , and let B be the set of allbranches. For α < κ , let G α = G ∩ Col ( µ, < α ), and let W α = P ( T α ) V [ G α ] . Let S = { α < κ : α is inaccessible in V } . In V [ G ], fix enumerations h s αβ : β < µ i of the W α , and in V [ G ∗ H ], let A α = { s αβ : H ( α + β ) = 1 } . Let us check eachcondition.(1) (2 κ ) V = (2 κ ) V [ G ∗ H ] , so V [ G ∗ H ] (cid:15) | B | = 2 κ .(2) Since κ is inaccessible, each W α is collapsed to µ .(3) Suppose z ∈ P κ ( B ). There is some α < κ such that z ∈ V [ G α ]. For β ≥ α , π β [ z ] ∈ W β .(4) The regularity of µ is preserved, and clearly each W α contains all singletons.Let h a ξ : ξ < δ i ⊆ W α with δ < µ . Each a ξ ∈ A is τ G α ξ for some Col ( µ, < α )-name τ ξ . By the µ -closure of Col ( µ, < κ ), h τ ξ : ξ < δ i ∈ V , so h a ξ : ξ <δ i ∈ V [ G α ].(5) By the Mahlo property, S is stationary, and by the κ -c.c. of Col ( µ, < κ )and κ -closure of Add ( κ ), it remains stationary in V [ G ∗ H ]. Suppose α ∈ S .(a) Unions: Let A ∈ P µ ( W α ) have the property that all a in A are capturedbelow α . As above, A ∈ V [ G α ]. Now in V [ G α ], α = µ + and | T β | = µ for β < α . So if π α,β ↾ a is injective, then V [ G α ] (cid:15) | a | < α , and thus V [ G α ] (cid:15) | S A | < α . For distinct x, y ∈ S A , let γ x,y < α be the least γ such that π α,γ ( x ) = π α,γ ( y ). We have γ = sup { γ x,y : x, y ∈ S A } < α .Hence if γ ≤ β < α and all a ∈ A are captured at β , then S A iscaptured at β . 11b) Subsets: Suppose z ∈ W α is captured below α , and z ∈ W α is asubset of z . Then V [ G α ] (cid:15) | z | < α , so by the α -c.c. of Col ( µ, < α ),there is some β < α such that z ∈ V [ G β ]. Thus z is captured below α .(6) First work in V [ G ]. Let ˙ f be an Add ( κ )-name for a function from κ to P κ ( B ) , and let h ˙ b α : α < κ i be as in clause (6). Let ˙ C be a name for aclub, and let p ∈ Add ( κ ) be arbitrary. Build a continuous decreasing chainof conditions below p , h p α : α < κ i ⊆ Add ( κ ), and a continuous increasingchain of ordinals, h ξ α : α < κ i ⊆ κ , with the following properties: For all α , • p α +1 (cid:13) ξ α ∈ ˙ C , • p α +1 decides ˙ f ↾ dom( p α ) and { ˙ b β : β < α } , • dom( p α +1 ) is an ordinal > max { dom( p α ) , ξ α , α } , and • ξ α +1 > dom( p α +1 ).Let g : κ → P κ ( B ) and { b α : α < κ } be the objects defined by whatthe chain h p α : α < κ i decides. For each α < κ , there is a predense set E α ⊆ Col ( µ, < κ ) of size < κ such that g ( α ) and b α are decided by elementsof E α . There is a club D ∈ V such that ∀ α ∈ D , ∀ β < α , E β ⊆ Col ( µ, < α ).For α ∈ D , g ↾ α and { b β : β < α } are in V [ G α ].Back in V [ G ], for α < κ , let γ α be the least γ ≥ α such that π γ α ↾ { b β : β <α } is injective. If α is closed under β γ β , then γ α = α . As S is stationary,there is α ∈ S ∩ D such that γ α = α , ξ α = α , and p α (cid:13) α ∈ ˙ C . We havethat { b β : β < α } is captured at α , and that {h π α [ g ( β )] , π α [ g ( β )] i : β <α } ⊆ W α . Since α is inaccessible in V , if z ⊆ { π α ( b β ) : β < α } is capturedbelow α , then V [ G α ] (cid:15) | z | < α , so { β : π α ( b β ) ∈ z } is bounded below α .Let q ≤ p α be such that for β < µ , q ( α + β ) = 1 if s βα = h π α [ g ( β )] , π α [ g ( β )] i ,and q ( α + β ) = 0 otherwise. Then q (cid:13) α ∈ ˙ C ∩ S , and that items (a), (b),and (c) in clause (6) hold at α . As p was arbitrary, clause (6) is forced. (cid:3) Question
Can W ∗ κ ( λ ) be forced without the use of large cardinals? Can it beforced in a cardinal-preserving way? Does L (cid:15) “For all regular κ , W ∗ κ + ( κ ++ )”? Theorem 3.7 W ∗ κ ( λ ) implies there is a coherent, Suslin ( κ, λ, -forest. Proof
Let κ = µ + , and let T , B , h W α : α < κ i , h A α : α < κ i , and S ⊆ κ witness W ∗ κ ( λ ). We will construct a sequence of functions h f α : α < κ i onthe nodes of T that will generate a coherent family of functions on B with thedesired properties. Each f α will have domain T α and range contained in { , } .Let f be a function from T to 2. Assume we have have constructed asequence of functions h f β : β < α i , with each f β : T β →
2, satisfying thefollowing property:( ∗ ) If r ∈ W β is captured at γ < β , then f β ↾ r disagrees with f γ ◦ π β,γ ↾ r ona set of size < µ . 12et R α = { r ∈ W α : r is caputured below α } . Consider the set F α ofpartial functions on T α of the form f γ ◦ π α,γ ↾ r for r ∈ R α and γ witnessing itsmembership in R α . Assume γ < γ and f γ ◦ π α,γ ↾ r and f γ ◦ π α,γ ↾ r arein F α . By hypothesis ( ∗ ), f γ disagrees with f γ ◦ π γ ,γ at less than µ manypoints in π α,γ [ r ]. Therefore, there are less than µ many points in r ∩ r atwhich f γ ◦ π α,γ and f γ ◦ π α,γ disagree. So F α is a µ -coherent family.Assume first that α / ∈ S . Using Lemma 1.1(a), let f α : T α → { f α } ∪ F α is µ -coherent. Then ( ∗ ) holds for h f β : β ≤ α i .Now assume α ∈ S . Let H α be the closure of F α under < µ modifications.Consider H α as a partial order with f ≤ g iff f ⊇ g . The set A α ⊆ W α codesa set of relations from subsets of T α to 2. If h a , a i ∈ A α , construct a relation h by putting h x, i i ∈ h iff x ∈ a i , and call the set of all such things A ′ α . It maybe the case that every member of A ′ α is a function and a member of H α , andthat A ′ α is a maximal antichain in H α . If not, ignore all these considerations,and let f α be as in the case α / ∈ S , so that ( ∗ ) is preserved.Suppose A ′ α is a maximal antichain in H α . Enumerate R α as h r β : β < µ i .By clauses (4) and (5) of the definition of W ∗ , R α is closed under unions of size < µ . H α is also a µ -closed partial order. If h h i : i < β < µ i is a decreasingsequence, then S i<β dom( h i ) = r ∈ R α , so let γ witness this. By ( ∗ ), each h i disagrees with f γ ◦ π α,γ on a set of size < µ , and so S i<β h i does as well by theregularity of µ .Setting s β = S ξ<β r ξ , we have h s β : β < µ i is an increasing cofinal sequencein R α . For β < µ , let γ β be the least γ < α that witnesses s β ∈ R α . Let h t β : β < µ i enumerate all < µ sized subsets of T α , such that each subset isrepeated µ many times. For a partial function f : T α → β < µ , let f /t β be f with its output values switched at the points in dom( f ) ∩ t β .We will define f α inductively as S β<µ h β . Let h = ∅ . Assume h h i : i < β i has been chosen so that:(1) for i < j < β , h i ⊆ h j ;(2) for i < β , dom( h i ) = s ξ i where ξ i ≥ i , and ξ i > ξ j for j < i ;(3) for i < β , there is a ∈ A ′ α such that h i +1 /t i is a common extension of h i /t i and a .Given h i , there is some a ∈ A ′ α that is compatible with h i /t i . Let ξ i +1 > ξ i besuch that s ξ i +1 ⊇ dom( a ) ∪ s ξ i , and let g ∈ H α be a common extension of a and h i /t i with domain s ξ i +1 . Let h i +1 = g/t i . Clearly (1)–(3) are preserved atsuccessor steps. At limit steps β , we set h β = S i<β h i . This is in H α as well by µ -closure, and the preservation of (1)–(3) is trivial.The point is this: For every t ∈ P µ ( T α ), f α /t extends some a ∈ A ′ α . Forlet i < µ be large enough that s ξ i ⊇ t and t i = t . Then by (3), h i +1 /t extendssome a ∈ A ′ α , and h i +1 /t = ( f α /t ) ↾ s ξ i +1 . We also check that ( ∗ ) is preservedat α : Every r ∈ R α is covered by some s ξ i , and f α ↾ s ξ i = h i , which cohereswith f γ ◦ π α,γ ↾ s ξ i when s ξ i is captured at γ .Now we define the forest. For z ∈ P κ ( B ), let γ z be the least γ < κ suchthat z is captured at γ . Let f z : z → f γ z ◦ π γ z ↾ z . Let F be the closure13f { f z : z ∈ P κ ( B ) } under < µ modifications. Note that by ( ∗ ), if β ≥ γ z ,then f β ◦ π β ↾ z disagrees with f z at < µ many points. Hence F is a coherent( κ, B, κ -c.c. First note that F satisfies the κ + -c.c. by a delta-system argument. So assume towards a contradiction that A = { a α : α < κ } isa maximal antichain. Let z α = dom( a α ), and code each a α as h z α , z α i , where z iα = { b : a α ( b ) = i } . Let h b α : α < κ i enumerate the elements of S α<κ z α .Define: • C = { α < κ : S β<α z β = { b β : β < α }} . • C = { α < κ : { a β : β < α } is a maximal antichain contained in { f ∈ F :( ∃ η < α ) dom( f ) ⊆ { b β : β < η }}} . • C = { α < κ : ( ∀ β < α ) γ z β , γ z β , γ z β < α } .It is easy to see that C , C , and C are club. By clause (6) of the definitionof W ∗ , let α ∈ S ∩ C ∩ C ∩ C be such that { b β : β < α } = S β<α z β is capturedat α , all z ⊆ { π α ( b β ) : β < α } captured below α have sup { β : π α ( b β ) ∈ z } < α ,and A α = {h π α [ z β ] , π α [ z β ] i : β < α } .We claim A ′ α is a maximal antichain in H α . For β < α , z β is captured below α since α ∈ C , so the function coded by h π α [ z β ] , π α [ z β ] i is in H α . 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