aa r X i v : . [ m a t h . L O ] M a y FORCING AND THE HALPERN-L ¨AUCHLI THEOREM
NATASHA DOBRINEN AND DANIEL HATHAWAY
Abstract.
We investigate the effects of various forcings on sev-eral forms of the Halpern-L¨auchli Theorem. For inaccessible κ , weshow they are preserved by forcings of size less than κ . Combiningthis with work of Zhang in [17] yields that the polarized partitionrelations associated with finite products of the κ -rationals are pre-served by all forcings of size less than κ over models satisfying theHalpern-L¨auchli Theorem at κ . We also show that the Halpern-L¨auchli Theorem is preserved by <κ -closed forcings assuming κ ismeasurable, following some observed reflection properties. Introduction
The Halpern-L¨auchli Theorem [7] is a Ramsey theorem for prod-ucts of finitely many trees of height ω which are finitely branching andhave no terminal nodes. It was discovered as a central lemma to theproof in [8] that the Boolean Prime Ideal Theorem (the fact that ev-ery ideal in a Boolean algebra can be extended to a prime ideal) isstrictly weaker than the Axiom of Choice, over Zermelo-Fraenkel settheory. Many variations have been studied, some of which are equiva-lent to the statement that the BPI is strictly weaker than the Axiom ofChoice. Recent compendia of various versions of the Halpern-L¨auchliTheorem appear in [15] and [3]. The Halpern-L¨auchli Theorem hasfound numerous applications in proofs of partition relations for count-able structures, directly to products of rationals (see [11]) and via theclosely related theorem of Milliken for strong trees [12] to finite sets ofrationals (see [1]) and the Rado graph (see [13] and [10]).Some years after the Halpern-L¨auchli Theorem was discovered, Har-rington found a proof which uses the techniques and language of forcing,though without actually passing to a generic extension of the groundmodel. Though this proof was well-known in certain circles, a publishedversion did not appear until [16]. Mathematics Subject Classification.
Fill in.The first author was partially supported from National Science Foundation GrantDMS-1600781.
Shelah applied this proof method of Harrington in [14] to prove apartition theorem (analogue of Milliken’s Theorem) for trees on 2 <κ ,where κ is a cardinal whose measurability is preserved by adding manyCohen subsets of κ . This result was extended and applied by Dˇzamonja,Larson and Mitchell in [4] and [5] to obtain partition relations on the κ -rationals and κ -Rado graph. This work, as well as the expositionin Chapter 3 of Todorcevic’s book [15] informed the authors’ previouswork on variations of the Halpern-L¨auchli Theorem for more than onetree at uncountable cardinals. In [2], we mapped out the implicationsbetween weaker and stronger forms of the Halpern-L¨auchli Theoremon trees of uncountable height, and found a better upper bound for theconsistency strength of the theorem holding for finitely many trees ata measurable cardinal. Building on work in [4] and [2], Zhang proveda stronger tail-cone version at measurable cardinals and applied it toobtain the analogue of Laver’s partition relation for products of finitelymany trees on a measurable cardinal (see [17]).It is intriguing that all theorems for trees of uncountable heightproved so far have required assumptions beyond ZFC. In fact, it isstill unknown whether such partition relation theorems for trees at un-countable heights simply are true in ZFC or whether they entail somelarge cardinal strength. For more discussion of this main problem andother related open problems, see Section 7.In this paper, we are interested in which forcings preserve the Halpern-L¨auchli Theorem for trees of uncountable height, once it is known tohold. Section 2 contains basic definitions, most of which are found in[2] and [17]. It also contains an equivalence between the tail-cone ver-sion of the Halpern-L¨auchli Theorem and a modified version which iseasier to satisfy in practice. Section 3 contains a new method whichconstructs a tree in the ground model using forcing names; this is calledthe derived tree from a name for a tree. The Derived Tree Theorem isproved there.The Derived Tree Theorem is applied in Section 4 to show that smallforcings preserve the Halpern-L¨auchli Theorem and its tail-cone ver-sion. As the partition relation on finite products of κ -rationals holdsin any model where the tail-cone version holds (by work of Zhang in[17]), our work shows that this partition relation is preserved by anyfurther small forcing.Section 5 presents some instances when the somewhere dense version(SDHL) has reflection properties. Thus, if SDHL holds for a stationaryset of cardinals below a strongly inaccessible cardinal κ , then it holdsat κ . Second, we prove that for a measurable cardinal κ , SDHL holdsat κ if and only if the set of cardinals below κ where SDHL holds is a ORCING AND THE HALPERN-L ¨AUCHLI THEOREM 3 member of each normal ultrafilter on κ . We apply this to show that <κ -closed forcings preserve SDHL at measurable cardinals.Finally, in Section 6, we provide a model of ZFC where SDHL holdsat some regular cardinal which is not weakly compact. This producesa different model than the one in [17], one that is obtained by a largecollection of forcings. Section 7 contains several key questions broughtto the fore by this work. Although interesting in their own right, theyare all sub-problems of the main open problem: Is the Halpern-L¨auchliTheorem for trees of any cardinal height simply true in ZFC?2. Basic Definitions
We review here some fundamental definitions from [2]. Given se-quences s and t , the notation s ⊑ t means that s is an initial segmentof t ; the notation s ⊏ t denotes that s is a proper initial segment of t .A set T ⊆ <κ κ is a tree iff it is closed under taking initial segments.For t ∈ <κ κ , let Dom( t ) denote the domain of t . We shall also call thisthe length of t . Given α ≤ Dom( t ), we write t ↾ α for the unique initialsegment of t of length α . Definition 2.1.
A tree T is a regular κ -tree if T ⊆ <κ κ and(1) T is a κ -tree; that is, T has height κ and each level of T hassize < κ ;(2) Every maximal branch of T has length κ ;(3) T is perfect, meaning that for any t ∈ T , there are incomparable s, u ⊒ t in T .Note that if κ is a regular cardinal and there exists a regular κ -tree,then κ must be a strongly inaccessible cardinal. However, there doexist regular κ -trees for singular cardinals κ . Specifically, there existsa regular κ -tree if 2 λ < κ for all λ < cf( κ ).Given a set X ⊆ <κ κ and an ordinal α < κ , let X ( α ) denote the setof sequences in X of length α ; that is,(1) X ( α ) = { t ∈ X : Dom( t ) = α } . Given sets
X, Y ⊆ <κ κ , we say that X dominates Y if to each y ∈ Y there corresponds at least one x ∈ X such that x ⊒ y . Given t ∈ <κ κ ,Cone( t ) is the set of all t ′ ⊒ t in <κ κ . Definition 2.2.
Given 1 ≤ d < ω and a sequence h X i ⊆ <κ κ : i < d i ,define the level product of the X i ’s to be O i Definition 2.3. Let 1 ≤ d < ω . Given κ -trees T , ..., T d − , we calla sequence h X i : i < d i a somewhere dense level matrix if there areordinals α < β < κ and a sequence h t i ∈ T i ( α ) : i < d i such that each X i is a subset of T i ( β ), and further, X i dominates T i ( α + 1) ∩ Cone( t i ).The following is the somewhere dense version of the Halpern-L¨auchliTheorem, which we denote by SDHL( d, σ, κ ). Given a coloring c and aset S , we say c is monochromatic on S if and only if | c “ S | = 1. Definition 2.4. For 1 ≤ d < ω and cardinals 0 < σ < κ with κ infinite,SDHL( d, σ, κ ) is the statement that given any sequence h T i : i < d i ofregular κ -trees and any coloring c : O i Let ≤ d < ω , κ a regular cardinal, and < σ < κ begiven, and assume SDHL ( d, σ, κ ) holds. Let T i ( i < d ) be a sequence ofregular κ -trees (so κ is strongly inaccessible). Let c : N i Definition 2.6. Let T be a regular κ -tree. A tree T ′ ⊆ T is a strongsubtree of T as witnessed by some set A ⊆ κ cofinal in κ if T ′ is regularand for each t ∈ T ′ ( α ) for α < κ ,1) α ∈ A implies every successor of t in T is also in T ′ ;2) α A implies that t has a unique successor in T ′ on level α + 1.We refer to an ordinal α ∈ A as a splitting level of T ′ .The following is the strong tree version of the Halpern-L¨auchli The-orem, which we denote by HL( d, σ, κ ). ORCING AND THE HALPERN-L ¨AUCHLI THEOREM 5 Definition 2.7. For 1 ≤ d < ω and cardinals 0 < σ < κ with κ infinite, HL( d, σ, κ ) is the following statement: Given any sequence h T i : i < d i of regular κ -trees and a coloring c : N i For 1 ≤ d < ω and κ an infinite cardinal, HL tc ( d, <κ, κ ) is the following statement: Given a sequence of regular κ -trees h T i : i < d i , a sequence of nonzero cardinals h σ ζ < κ : ζ < κ i , anda sequence of colorings c ζ : N i Note that SDHL( d, σ, κ ), HL( d, σ, κ ), and HL tc ( d, σ, κ ) are all state-ments about V κ +1 . We will need the following concept later. Definition 2.10. For 1 ≤ d < ω and κ an infinite cardinal, the modified HL tc ( d, < κ, κ ) is just HL tc ( d, < κ, κ ) but with (2) replaced with thefollowing:(2*) There is a function µ : κ → κ such that ( ∀ γ < κ ) µ ( γ ) ≥ γ satisfying the following: For each pair of ordinals ζ ≤ γ < κ ,given any sequence h t i ∈ T ′ i ( α γ ) : i < d i , we have c µ ( ζ ) h t i : i < d i = c µ ( ζ ) h t i ↾ α ζ : i < d i . In other words, the c µ ( ζ ) -color of a tuple is determined by re-stricting to the ζ -th splitting level. Proposition 2.11. Fix ≤ d < ω and κ a strongly inaccessible cardi-nal. Then HL tc ( d, < κ, κ ) and its modified version are equivalent.Proof. It is clear that the unmodified version implies the modified ver-sion holds: Just set µ : κ → κ to be the identity function. For theother direction, assume the modified version holds.Let h T i : i < d i be a sequence of regular κ -trees and h c ζ : ζ < κ i asequence of colorings, where c ζ : N i This section introduces derived trees and proves a theorem which willbe central to the results in Section 4 about small forcings preservingvarious forms of the Halpern-L¨auchli Theorem. Definition 3.1. Let κ be a cardinal, P be a forcing, and without lossof generality, assume that P has a largest member, denoted 1. Assumethat ˙ T is a P -name for which 1 forces that ˙ T is a subtree of < ˇ κ ˇ κ . The derived tree of ˙ T , denoted Der( ˙ T ), is defined as follows. The elementsof Der( ˙ T ) are equivalence classes of pairs ( ˙ τ , α ) satisfying(4) 1 (cid:13) ( ˙ τ ∈ ˙ T and Dom( ˙ τ ) = ˇ α ) , where the equivalence relation ∼ = is defined by(5) ( ˙ τ , α ) ∼ = ( ˙ τ , α ) ⇐⇒ (cid:13) ( ˙ τ = ˙ τ ) . Notice that if ( ˙ τ , α ) ∼ = ( ˙ τ , α ), then α = α . The elements of Der( ˙ T )are ordered as follows:(6) [( ˙ τ , α )] < [( ˙ τ , α )] ⇐⇒ (cid:13) ( ˙ τ ⊑ ˙ τ and ˙ τ = ˙ τ ) . Given S ⊆ Der( ˙ T ), let(7) Names( S ) = { ˙ τ : ( ∃ α ) [( ˙ τ , α )] ∈ S } . We claim that 1 forces that every element of ˙ T is equal to someelement of Names(Der( ˙ T )). To see why, let G be P -generic over V . Let t ∈ ˙ T G and α = Dom( t ). Fix a name ˙ τ such that ˙ τ G = t , and let ˙ b bea name for the leftmost branch of ˙ T . Define ˙ ρ so that(8) 1 (cid:13) [( ˙ ρ = ˙ τ if Dom( ˙ τ ) = ˇ α ) ∧ ( ˙ ρ = ˙ b ↾ α if Dom( ˙ τ ) = ˇ α )] . Then [( ˙ ρ, α )] ∈ Der( ˙ T ) and ˙ ρ G = t .We will now show that Der( ˙ T ) is (isomorphic to) a regular κ -tree inthe ground model whenever 1 forces that ˙ T is a regular κ -tree, and thatgiven an element named by some ( ˙ τ , α ) ∈ Der( ˙ T ), all its successors are NATASHA DOBRINEN AND DANIEL HATHAWAY named by successors of ( ˙ τ , α ) in Der( ˙ T ). This theorem is central to theforcing preservation theorems in following sections.Given a tree T ⊆ <κ κ and a node t ∈ T , by the 0-th successor of t we mean the node t ⌢ α ∈ T with the least possible α ∈ κ . Moregenerally, the γ -th successor of t ∈ T is the node t ⌢ α ∈ T such thatthe set { β < α : t ⌢ β ∈ T } has order type γ . Theorem 3.2 (Derived Tree Theorem) . Let κ be strongly inaccessible, P a forcing of size < κ , and ˙ T a name for a regular κ -tree. ThenDer ( ˙ T ) is isomorphic to a regular κ -tree and ( ∗ ) If [( ˙ τ , α )] ∈ Der ( ˙ T ) and X is the set of all ˙ ρ such that [( ˙ ρ, α +1)] is a successor of [( ˙ τ , α )] in Der ( ˙ T ) , then (cid:13) ( every successorof ˙ τ in ˙ T is named by an element of ˇ X ) .Proof. First note that if [( ˙ τ , α )] is in Der( ˙ T ) and β < α , then there is aname ˙ τ β such that [( ˙ τ β , β )] is in Der( ˙ T ) and [( ˙ τ β , β )] < [( ˙ τ , α )]: simplylet ˙ τ β be a name for ˙ τ ↾ ˇ β . We prove that Der( ˙ T ) is a regular κ -treeby proving it satisfies conditions (1) - (3) of Definition 2.1.To verify (1), we must first show that Der( ˙ T ) is a tree. Suppose[( ˙ τ , α )] , [( ˙ τ , α )] , [( ˙ τ , α )] are members of Der( ˙ T ) satisfying(9) [( ˙ τ , α )] > [( ˙ τ , α )] and [( ˙ τ , α )] > [( ˙ τ , α )] . Assume, without loss of generality, that α ≥ α . Since 1 forces that ˙ T is a tree and that ˙ τ i is an initial segment of ˙ τ of length α i , for i ∈ { , } ,it follows that 1 forces that ˙ τ ↾ α = ˙ τ . Thus, 1 forces that ˙ τ is aninitial segment of ˙ τ , and hence, [( ˙ τ , α )] and [( ˙ τ , α )] are comparablein Der( ˙ T ).For β < κ , let level β denote the set of those [( ˙ τ , α )] ∈ Der( ˙ T ) suchthat α = β . The same argument as above also shows that given [( ˙ τ , α )]in Der( ˙ T ) and β < α , there is a unique [( ˙ ρ, β )] on level β of Der( ˙ T )such that [( ˙ ρ, β )] < [( ˙ τ , α )]. We have now established that Der( ˙ T ) is atree.We must now show that Der( ˙ T ) is a κ -tree. To show that it hasheight κ , given any α < κ , let ˙ τ α be such that 1 (cid:13) ( ˙ τ α = ˙ b ↾ ˇ α ), where ˙ b is a name for the leftmost branch of ˙ T . Then [( ˙ τ α , α )] ∈ Der( ˙ T ). Thus,Der( ˙ T ) has height κ . To show that each level of Der( ˙ T ) has < κ nodes,we will make use of the fact that Der( ˙ T ) consists of elements [( ˙ τ , α )]where 1 (cid:13) (Dom( ˙ τ ) = ˇ α ). (If we drop the α ’s from the definition ofDer( ˙ T ), we can verify ( ∗ ), and (2) and (3) of Definition 2.1, but not(1).) Since 1 (cid:13) ( ˙ T is a ˇ κ -tree), we have that(10) 1 (cid:13) ( ∀ α < ˇ κ )( ∃ γ < ˇ κ )( ∀ t ∈ ˙ T ) α ∈ Dom( t ) ⇒ t ( α ) < γ. ORCING AND THE HALPERN-L ¨AUCHLI THEOREM 9 Since | P | < κ , there is a function g : κ → κ such that(11) 1 (cid:13) ( ∀ α < ˇ κ )( ∀ t ∈ ˙ T ) α ∈ Dom( t ) ⇒ t ( α ) < ˇ g ( α ) . Now, to each pair ( ˙ τ , α ), where [( ˙ τ , α )] ∈ Der( ˙ T ), we may associate asequence h f ξ : ξ < α i , where each f ξ is a function from some maximalantichain of P to g ( ξ ). This sequence represents a nice name for ˙ τ .Since | P | < κ , level α < κ of Der( ˙ T ) is bounded from above by thefollowing:(12) Y ξ<α g ( ξ ) | P | . Since κ is strongly inaccessible, this bound is < κ . We have now shown(1) of Definition 2.1.We will now verify (2), that Det( ˙ T ) has no maximal branches oflength < κ . When we later show that Der( ˙ T ) is perfect, this will implyit has no maximal branches of a successor ordinal length. Thus, itsuffices to show Der( ˙ T ) has no maximal branches of limit length. Let η < κ be a limit ordinal and S = h [( ˙ τ α , α )] : α < η i is an increasingchain in Der( ˙ T ) so that for all ξ < ζ < η ,(13) [( ˙ τ ξ , α ξ )] < [( ˙ τ ζ , α ζ )] . Let ˙ s be a name which P forces to be a function from ˇ η to < ˇ κ ˇ κ suchthat for all α < η ,(14) 1 (cid:13) ˙ s ( ˇ α ) = ˙ τ α . By the definition of the tree relation < in Der( ˙ T ), it follows that(15) 1 (cid:13) ( ∀ α < β < ˇ η ) ˙ s ( α ) ⊑ ˙ s ( β ) . Now let ˙ τ η be a name such that 1 (cid:13) ˙ τ η = S α< ˇ η ˙ s ( α ). It follows from(15) that(16) 1 (cid:13) ˙ τ η ∈ ˇ η ˇ κ and ( ∀ α < ˇ η ) ˙ s ( α ) ⊑ ˙ τ η , and thus, by (14),(17) 1 (cid:13) ( ∀ α < ˇ η ) ˙ τ α ⊑ ˙ τ η . Since P forces that ˙ T has no maximal branches of length < κ , we nowhave that(18) 1 (cid:13) ˙ τ η ∈ ˙ T . So now [( ˙ τ η , η )] ∈ Der( ˙ T ), and this node is above each [( ˙ τ α , α )] for α < η . Thus, we have verified (2) of Definition 2.1.To verify (3), consider any [( ˙ τ , α )] ∈ Der( ˙ T ). Let ˙ b be a name forthe leftmost branch of ˙ T which extends ˙ τ . Let β < κ be the least ordinal greater than or equal to α for which there is some p ∈ P whichforces that there are at least two successors of ˙ b ↾ ˇ β in the tree ˙ T .Such β and p exist since P forces that ˙ T is a perfect tree. Let ˙ τ be aname such that 1 forces ˙ τ = ˙ b ↾ ˇ β . Let ˙ τ be a name which 1 forcesto be the 0-th successor of ˙ τ in ˙ T . Finally, let ˙ τ be a name which1 forces to be the 1-th successor of ˙ τ in ˙ T , if there are at least twosuccessors, and the unique successor if there is only one successor. Onecan see that [( ˙ τ , β + 1)] and [( ˙ τ , β + 1)] are successors of [( ˙ τ , β )] inDer( ˙ T ). Since there is some p which forces ˙ τ = ˙ τ , it follows that[( ˙ τ , β + 1)] = [( ˙ τ , β + 1)]. Thus, [( ˙ τ , β + 1)] and [( ˙ τ , β + 1)] areincomparable extensions of [( ˙ τ , α )] in Der( ˙ T ). Therefore, Der( ˙ T ) is aperfect tree. Hence, Der( ˙ T ) is isomorphic to a regular κ -tree.Finally, the verification of ( ∗ ) follows almost immediately from thedefinition of Der( ˙ T ). Fix [( ˙ τ , α )] ∈ Der( ˙ T ) and let G be P -generic over V . Let s be an arbitrary successor of ˙ τ G in ˙ T G , and let γ be such that s is the γ -th successor of ˙ τ G in ˙ T G . Take ˙ ρ to be a name so that 1forces that ˙ ρ is the ˇ γ -th successor of ˙ τ in ˙ T , if it exists, and the 0-thsuccessor, otherwise. Then ˙ ρ G = s . At the same time 1 forces that ˙ ρ is a successor of ˙ τ in ˙ T , so [( ˙ ρ, α + 1)] is a member of Der( ˙ T ). (cid:3) Remark 3.3. There are two instances in the proof where | P | < κ wasused. The first is non-essential: If P is κ -c.c., or even just ( κ, κ, <κ )-distributive, equation (11) still holds. However, the second use of | P | < κ is essential to the proof. Indeed, given any P which preserves κ and has cardinality at least κ , there is a name ˙ T for a regular κ -treewith the following properties: 1 forces that the first level of ˙ T has sizeat least two (with say elements h i and h i ), and letting { p ζ : ζ < κ } be a set of distinct members of P , there are nice names ˙ τ ζ so that(19) p ζ (cid:13) ˙ τ ζ ∈ ˙ T , Dom( ˙ τ ζ ) = 1 , and ˙ τ ζ (0) = 0 , and all q ∈ P incompatible with p ζ force ˙ τ ζ (0) = 1. Then for all ζ < ξ < κ , ( ˙ τ ζ , = ( ˙ τ ξ , T ) has size at least κ . Thus, the κ -c.c. is not enough to guarantee that the levels of Der( ˙ T )have size less than κ .The Derived Tree Theorem will allow us to use Der( ˙ T ) in theoremsthat require subtrees of <κ κ . Note also that the Derived Tree Theoremprovides the following: Given a name for a regular κ -tree ˙ T , a strongsubtree S of Der( ˙ T ) with splitting levels A ⊆ κ , and a P -generic G , theset W = { ˙ τ G : ˙ τ ∈ Names( S ) } is a strong subtree of ˙ T G , witnessed bythe set of splitting levels A . ORCING AND THE HALPERN-L ¨AUCHLI THEOREM 11 Small Forcings Preserve SDHL, HL, and HL tc In this section we show that if κ is strongly inaccessible and ( ∀ σ <κ ) SDHL( d, σ, κ ) holds, then this still holds after performing any forc-ing of size less than κ . This result then automatically holds for HLreplacing SDHL, since the two are equivalent for κ inaccessible. Fur-ther, we show that HL tc at κ is preserved by forcings of size less than κ . These results make strong use of the Derived Tree Theorem fromthe previous section. Theorem 4.1. Let κ be strongly inaccessible. Let ≤ d < ω and <σ < κ . Let P be a forcing of size < κ . Assume that SDHL ( d, σ · | P | , κ ) holds. Then SDHL ( d, σ, κ ) holds after forcing with P . In particular, thestatement “ ( ∀ σ < κ ) SDHL ( d, σ, κ ) holds” is preserved by all forcingsof size less than κ .Proof. Let h ˙ T i : i < d i be a sequence of names for regular trees in theextension. That is, ( ∀ i < d ) 1 (cid:13) ( ˙ T i ⊆ < ˇ κ ˇ κ is regular). Let ˙ c be suchthat(20) 1 (cid:13) ˙ c : O i< ˇ d ˙ T i → ˇ σ. We must show that P forces that there is a somewhere dense levelmatrix h X i ⊆ ˙ T i : i < ˇ d i such that | ˙ c “ N i< ˇ d X i | = 1. We will dothis by showing that for each p ∈ P , there is some p ′ ≤ p forcing thisstatement. Fix p ∈ P .Consider the trees Der( ˙ T i ) for i < d . Let(21) c ′ : O i Let κ be strongly inaccessible. Let ≤ d < ω . Let P be a forcing of size < κ . Assume that HL tc ( d, < κ, κ ) holds. ThenHL tc ( d, < κ, κ ) holds after forcing with P .Proof. The proof is similar to that of the previous theorem. Fix p ∈ P .By Proposition 2.11, it suffices to find a p ′ ≤ p that forces the modifiedversion of HL tc . Let h ˙ T i : i < d i be a sequence of names for regulartrees, and let h ˙ c ζ : ζ < κ i be a sequence of names for colorings suchthat P forces each ˙ c ζ to take less than ˇ κ colors. Since | P | < κ , thereare ordinals σ ζ < κ for ζ < κ such that(27) 1 (cid:13) ˙ c ζ : O i< ˇ d ˙ T i → ˇ σ ζ . Just as in the previous theorem, the sequence of colorings h ˙ c ζ : ζ < κ i induces a sequence of colorings h c ′ ζ : ζ < κ i where for each ζ < κ ,(28) c ′ ζ : O i Let κ be strongly inaccessible. There is a unique κ -saturated linear order of size κ denoted Q κ , the κ -rationals [17]. Zhangproved in [17] that HL tc ( d, <κ, κ ) implies(34) Q κ ... Q κ → Q κ ... Q κ , ..., | {z } d +1 <κ, ( d +1)! This partition relation means that given any σ < κ and any coloring c : Q i At inaccessible cardinals, the Halpern-L¨auchli Theorem reflects. InProposition 5.1, we show that for κ strongly inaccessible, if SDHL holdson a stationary set below κ , then it holds at κ . In this proposition,SDHL cannot be replaced by HL tc , which we will explain in the nextparagraph. In Proposition 5.2, we prove that SDHL holds at a measur-able cardinal κ if and only if the set of ordinals below κ where SDHLholds is a member of any normal ultrafilter on κ . By Proposition 2.8,the same statement holds for HL. It also holds for HL tc . These twopropositions imply Theorem 5.3, that the Halpern-L¨auchli Theorem ata measurable cardinal κ is preserved by <κ -closed forcings.Let us explain why Proposition 5.1 does not hold for HL tc . Theproblem is we could use the argument of Theorem 6.3 in the nextsection to get HL tc to hold at a cardinal that is not weakly compact,which is impossible by [17]. That is, assume Proposition 5.1 does holdfor HL tc and start with V satisfying HL tc at a measurable κ . Thenperform any nontrivial forcing of size less than κ to obtain some genericextension V [ G ]. In V [ G ], κ is still measurable and HL tc holds at κ . Soin V [ G ], HL tc holds for a stationary set of λ < κ . Now let V [ G ][ H ]be any nontrivial forcing extension of V [ G ] by a <κ -closed forcing.Then in V [ G ][ H ], HL tc holds for a stationary set of λ < κ . Since weare assuming Proposition 5.1 holds for HL tc , then in V [ G ][ H ], HL tc holds at κ . This is impossible, because by a result of Hamkins [9]any nontrivial forcing of size less than κ followed by any nontrivial <κ -closed forcing causes κ to not be weakly compact. Proposition 5.1. Let κ be a cardinal such that either • κ is strongly inaccessible or • cf ( κ ) ≥ ω and κ is the limit of strongly inaccessible cardinals.Let ≤ d < ω and ≤ σ < κ , and assume that SDHL ( d, σ, α ) holdsfor a stationary subset of κ . Then SDHL ( d, σ, κ ) holds.Proof. Let h T i : i < d i be a sequence of regular κ -trees and let c : N i Let κ be a measurable cardinal and U be a normalultrafilter on κ . ThenSDHL ( d, σ, κ ) ⇔ { α < κ : SDHL ( d, σ, α ) } ∈ U . Proof. Let j : V → M be the ultrapower embedding coming from U .Since V κ +1 ⊆ M , SDHL( d, σ, κ ) ⇔ SDHL( d, σ, κ ) M . By Los’s Theorem,SDHL( d, σ, κ ) M ⇔ { α < κ : SDHL( d, σ, α ) } ∈ U . (cid:3) Theorem 5.3. Suppose κ is a measurable cardinal. Let ≤ d < ω and ≤ σ < κ be given, and assume SDHL ( d, σ, κ ) holds. If P preservesstationary subsets of κ and adds no new bounded subsets of κ , thenSDHL ( d, σ, κ ) holds after forcing with P . In particular, if P is <κ -closed, then SDHL ( d, σ, κ ) holds after forcing with P .Proof. As <κ -closed forcings preserve stationary subsets of κ and addno new bounded subsets of κ , we need only prove the first half of thetheorem. ORCING AND THE HALPERN-L ¨AUCHLI THEOREM 17 Fix a normal ultrafilter U on κ . Since SDHL( d, σ, κ ) holds, by Propo-sition 5.2 the set S = { α < κ : SDHL( d, σ, α ) } is in U . Hence, S is stationary. Since P preserves stationary subsetsof κ , 1 (cid:13) ˇ S is stationary. For α < κ , since SDHL( d, σ, α ) is a state-ment about V α +1 , and V α +1 is the same when computed in the forcingextension by P , we have that P does not change the truth value ofSDHL( d, σ, α ) for any α < κ . So,(35) 1 (cid:13) { α < ˇ κ : SDHL( ˇ d, ˇ σ, α ) } is stationary . By Proposition 5.1, P forces that SDHL( ˇ d, ˇ σ, ˇ κ ) holds. (cid:3) SDHL at a Cardinal That is Not Weakly Compact In [2], we proved that SDHL(1 , k, κ ) holds for all finite k and allinfinite cardinals κ . So, by the equivalence of SDHL and HL for allstrongly inaccessible cardinals κ , HL(1 , k, κ ) holds for every stronglyinaccessible κ and every finite k . In [17], Zhang showed that this canbe improved to HL asym (1 , σ, κ ) holding for all σ < κ , but he required κ to be weakly compact. So, it is natural to wonder whether κ > ω needsto be weakly compact in order for HL(2 , σ, κ ) to hold for all σ < κ .While we were writing [2] we discovered the derived tree theorem andthe proof in this section, which answers the question in the negative.In the meantime, Zhang discovered a different proof of the consistencyof ( ∀ σ < κ ) HL(2 , σ, κ ) for a κ that is not weakly compact. Specifically,in Theorem 5.8 of [17] he proved that if for all d < ω , κ is measurablewhenever one adds κ + d many Cohen subsets of κ , then there is a forcingextension in which κ is inaccessible but not weakly compact, and inwhich HL( d, σ, κ ) holds for all d < ω and all σ < κ . The theorem wewill present now implies this, but has a different proof and applies toa broad collection of forcings. Definition 6.1. For 1 ≤ d < ω and an infinite cardinal κ , Ψ d,κ is thestatement ( ∀ σ < κ ) HL( d, σ, κ ) . In [2] we showed the following: Theorem 6.2. Let ≤ d < ω . If κ is measurable whenever one adds κ + d many Cohen subsets of κ , then Ψ d,κ holds (in V). Theorem 6.3. Let ≤ d < ω and κ be measurable. Assume Ψ d,κ holds. Then any non-trivial forcing of size less than κ followed by a non-trivial <κ -closed forcing produces a model in which κ is not weaklycompact and Ψ d,κ holds.Proof. By a theorem of Hamkins [9], any non-trivial forcing of size lessthan κ followed by a non-trivial <κ -closed forcing will force κ to not be weakly compact.Let P be any non-trivial forcing of size < κ . Let G be P -genericover V . Then Ψ d,κ holds in V [ G ] by Theorem 4.1. Let Q be any non-trivial <κ -closed forcing in V [ G ], and let H be Q -generic over V [ G ]. ByHamkins’s result, κ is not weakly compact in V [ G ][ H ]. Since Ψ d,κ holdsin V [ G ] and κ is measurable in this model, it follows from Theorem 5.3that Ψ d,κ also holds in V [ G ][ H ]. (cid:3) Open Problems The main open problem concerning the Halpern-L¨auchli Theorem atuncountable cardinals is the following: Question 7.1. Is it consistent for HL( d, σ, κ ) to fail for some uncount-able cardinal κ ?Because this is unanswered, there are many secondary questions.For example, even though HL( d, σ, κ ) does not imply κ itself must beweakly compact, does it have any large cardinal strength? Does HLhave so much large cardinal strength that it cannot hold in L; or doesHL always hold in L? Does the existence of say 0 imply that within L , HL holds at some, or all, strongly inaccessible cardinals?In all known models in which HL( d, σ, κ ) holds for some stronglyinaccessible κ , GCH fails. Such models appear in [14], [4], [5], [2], [17]and the preceding sections of this article. Is HL( d, σ, κ ) for κ stronglyinaccessible consistent with GCH?In this article, we showed that various forms of HL are preserved bysmall forcings or by <κ -closed forcings. What other types of forcingspreserve HL? An obvious question is the following: Question 7.2. Do κ -c.c. forcings preserve HL( d, σ, κ ), for 0 < d < ω and 0 < σ < κ ?Many variants of these questions can be formulated, and progresson any of them will lead to a better understanding of Halpern-L¨auchliTheorems and associated partition relations on uncountable structures. References 1. Dennis Devlin, Some partition theorems for ultrafilters on ω , Ph.D. thesis, Dart-mouth College, 1979. ORCING AND THE HALPERN-L ¨AUCHLI THEOREM 19 2. Natasha Dobrinen and Daniel Hathaway, The Halpern-L¨auchli Theorem at ameasurable cardinal , Journal of Symbolic Logic (2017), no. 4, 1560–1575.3. Pandelis Dodos and Vassilis Kanellopoulos, Ramsey Theory for Product Spaces ,American Mathematical Society, 2016.4. M. Dˇzamonja, J. Larson, and W. J. Mitchell, A partition theorem for a largedense linear order , Israel Journal of Mathematics (2009), 237–284.5. , Partitions of large Rado graphs , Archive for Mathematical Logic (2009), no. 6, 579–606.6. A. Hajnal and Komj´ath, A strongly non-Ramsey order type , Combinatorica (1997), no. 3.7. J. D. Halpern and H L¨auchli, A partition theorem , Transactions of the AmericanMathematical Society (1966), 360–367.8. J. D. Halpern and A. L´evy, The Boolean prime ideal theorem does not imply theaxiom of choice , Axiomatic Set Theory, Proc. Sympos. Pure Math., Vol. XIII,Part I, Univ. California, Los Angeles, Calif., 1967, American MathematicalSociety, 1971, pp. 83–134.9. Joel David Hamkins, Small forcing makes any cardinal superdestructible , Jour-nal of Symbolic Logic (1998), 51–58.10. Claude Laflamme, Norbert Sauer, and Vojkan Vuksanovic, Canonical partitionsof universal structures , Combinatorica (2006), no. 2, 183–205.11. Richard Laver, Products of infinitely many perfect trees , Journal of the LondonMathematical Society (2) (1984), no. 3, 385–396.12. Keith R. Milliken, A Ramsey theorem for trees , Journal of Combinatorial The-ory, Series A (1979), 215–237.13. Norbert Sauer, Coloring subgraphs of the Rado graph , Combinatorica (2006),no. 2, 231–253.14. Saharon Shelah, Strong partition relations below the power set: consistency –was Sierpinski right? ii , Sets, Graphs and Numbers (Budapest, 1991), vol. 60,Colloq. Math. Soc. J´anos Bolyai, North-Holland, 1991, pp. 637–688.15. Stevo Todorcevic, Introduction to Ramsey Spaces , Princeton University Press,2010.16. Stevo Todorcevic and Ilijas Farah, Some Applications of the Method of Forcing ,Yenisei Series in Pure and Applied Mathematics, 1995.17. Jing Zhang, A tail cone version of the Halpern-L¨auchli Theorem at a largecardinal. The Journal of Symbolic Logic, 1-23. doi:10.1017/jsl.2017.55. Department of Mathematics, University of Denver, C.M. KnudsonHall, Room 300, 2390 S. York St., Denver, CO 80208 U.S.A. E-mail address : [email protected] URL : http://web.cs.du.edu/~ndobrine Department of Mathematics, University of Vermont, 16 ColchesterAve., Burlington, VT 05401 U.S.A. E-mail address : [email protected] URL ::