aa r X i v : . [ m a t h . L O ] J un HIGHER-DIMENSIONAL DELTA-SYSTEMS
CHRIS LAMBIE-HANSON
Abstract.
We investigate higher-dimensional ∆-systems, isolating a particu-lar definition thereof and proving a higher-dimensional version of the classical∆-system lemma. We then present two applications of this lemma to problemsinvolving the interplay between forcing and partition relations involving thereals. Introduction
The starting point for this paper is one of the basic concepts of combinatorialset theory: the ∆ -system . Definition 1.1.
A family U of sets is a ∆ -system if there is a set r , known as the root of the ∆-system, such that u ∩ v = r for all distinct u, v ∈ U .The uniformity provided by ∆-systems can be quite useful, so it is no surprisethat the ∆-system lemma, which isolates conditions under which a family of setscan be thinned out to form a large ∆-system, is one of the foundational results incombinatorial set theory. The most commonly stated form of the lemma, introducedby Shanin [12], is the following. Lemma 1.2.
Suppose that U is an uncountable family of finite sets. Then there isan uncountable subfamily U ∗ ⊆ U such that U ∗ is a ∆ -system. The following is a less pithy but more general formulation. For a proof, we directthe reader to [10, Ch. II, § Lemma 1.3.
Suppose that κ < λ are infinite cardinals such that λ is regular and,for all ν < λ , we have ν <κ < λ . Suppose also that U is a family of sets such that |U| ≥ λ and | u | < κ for all u ∈ U . Then there is U ∗ ⊆ U such that |U ∗ | = λ and U ∗ is a ∆ -system. ∆-systems are inherently one-dimensional objects, in practice often enumeratedas sequences indexed by ordinals. When investigating higher-dimensional combi-natorial objects, however, one frequently encounters families of sets indexed by n -element sets of ordinals for some n > Mathematics Subject Classification.
Key words and phrases.
Delta systems, partition relations, chain conditions.
The higher-dimensional ∆-system lemmas in the aforementioned works havetaken a number of slightly different forms. In this paper, we isolate one partic-ular definition, based most directly on the 2-dimensional ∆-systems of [16] and [2]and on the n -dimensional ∆-systems of [3], with some additional uniformities incor-porated. This definition is presented in Section 2, where we additionally prove an n -dimensional analogue of the classical ∆-system lemma, isolating necessary andsufficient conditions under which an n -dimensional ∆-system of a particular size canbe guaranteed to exist inside of an arbitrary collection of sets indexed by n -elementsets of ordinals. This lemma is naturally seen as an elaboration of the Erd˝os-Radotheorem and is closely connected to the work on canonical partition relations ofErd˝os and Rado [7] and of Baumgartner [1]. Section 3 is a short section presentinga higher-dimensional analogue of the familiar use of ∆-systems to prove that Cohenforcing satisfies the Knaster property. In Section 4, we present an application ofour higher-dimensional ∆-system lemma to a problem involving the interplay offorcing and polarized partition relations. In Section 5, we show that, in certainarguments, the ∆-system lemma presented here can successfully replace a differentlemma (from [13]) that, at least under the currently best known results, requiresstronger assumptions. We apply this to a recent result of Zhang [18] regardingadditive partition relations on the reals, obtaining a slight local improvement to hisresult. Notation and conventions.
For a set X and a cardinal κ , [ X ] κ = { Y ⊆ X || Y | = κ } , and [ X ] <κ = { Y ⊆ X | | Y | < κ } . For a set u of ordinals, otp( u )denotes the order type of u . The class of ordinals is denoted by On. We willoften think of sets of ordinals as increasing sequences of ordinals in the naturalway. So, for instance, if u is a set of ordinals, ρ = otp( u ), and i < ρ , then u ( i )denotes the unique element α ∈ u such that otp( u ∩ α ) = i . If i ⊆ η , then u [ i ]denotes { u ( i ) | i ∈ i } . If X is a set of ordinals and n < ω , then we will usethe notation ( α , . . . , α n − ) ∈ [ X ] n to denote the conjunction of the statements { α , . . . , α n − } ∈ [ X ] n and α < . . . < α n − . If A and B are nonempty sets ofordinals, then we write A < B to assert that α < β for all ( α, β ) ∈ A × B .If κ is an infinite cardinal, then i n ( κ ) is defined by recursion on n < ω bysetting i ( κ ) := κ and i n +1 ( κ ) = 2 i n ( κ ) for all n < ω . Suppose that κ < λ arecardinals. We say that λ is κ -inaccessible if χ κ < λ for all χ < λ . Similarly, λ is <κ -inaccessible if χ <κ < λ for all χ < λ .If P is a forcing notion and p, q ∈ P , then p k q asserts that p and q are compatible,i.e., there is r ∈ P such that r ≤ p and r ≤ q , and p ⊥ q asserts that p and q areincompatible. 2. Uniform n -dimensional ∆ -systems In the context of families of sets of ordinals, the classical ∆-system lemma caneasily be strengthened to require that the root of the ∆-system “sits inside” eachof its elements in the same way. To make this more precise, we introduce thefollowing notation, used to completely describe the order-relations existing amongsets of ordinals.
Definition 2.1.
Suppose that I is a set and, for all i ∈ I , u i is a set of ordinals.Then tp( h u i | i ∈ I i ) is a function from otp( S i ∈ I u i ) to P ( I ) defined as follows. IGHER-DIMENSIONAL DELTA-SYSTEMS 3
First, let S i ∈ I u i be enumerated in increasing order as h α η | η < otp( S i ∈ I u i ) i .Then, for all η < otp( S i ∈ I u i ), let tp( h u i | i ∈ I i )( η ) := { i ∈ I | α η ∈ u i } .We will often slightly abuse notation and write, for instance, tp( u , u , u ) in-stead of tp( h u , u , u i ). Definition 2.2.
A family U of sets of ordinals is a uniform ∆ -system if there isset r such that u ∩ v = r and tp( u, r ) = tp( v, r ) for all distinct u, v ∈ U .Notice that it follows from the statement tp( u, r ) = tp( v, r ) in the above defini-tion that otp( u ) = otp( v ). Proposition 2.3.
Suppose that κ < λ are infinite cardinals such that λ is regularand <κ -inaccessible. Suppose also that U is a family of sets of ordinals such that |U| ≥ λ and | u | < κ for all u ∈ U . Then there is U ∗ ⊆ U such that |U ∗ | = λ and U ∗ is a uniform ∆ -system.Proof. First apply Lemma 1.3 to obtain U ∗ ⊆ U such that |U ∗ | = λ and U ∗ is a∆-system, with root r . Now, for each u ∈ U ∗ , tp( u, r ) is a function from an ordinalof size less than κ to P (2), so, since λ is regular and 2 <κ < λ , we can find a fixedtype t and a set U ∗ ⊆ U ∗ such that |U ∗ | = λ and tp( u, r ) = t for all u ∈ U ∗ . (cid:3) We now turn to our general definition of a uniform n -dimensional ∆-system. Inthe case n = 1, this will coincide with Definition 2.2. For n >
1, our definitiontakes as a starting point Todorcevic’s 2-dimensional double ∆ -system from [16],extending the definition to arbitrary finite dimensions and incorporating uniformityrequirements analogous to those of Definition 2.2. In order to state these uniformityrequirements, the following notion will be useful. Definition 2.4.
Suppose that a and b are sets of ordinals.(1) We say that a and b are aligned if otp( a ) = otp( b ) and, for all γ ∈ a ∩ b , wehave otp( a ∩ γ ) = otp( b ∩ γ ). In other words, if γ is a common element of a and b , then it occupies the same relative position in both a and b .(2) If a and b are aligned then we let r ( a, b ) := { i < otp( a ) | a ( i ) = b ( i ) } .Notice that, in this case, a ∩ b = a [ r ( a, b )] = b [ r ( a, b )]. Definition 2.5.
Suppose that H is a set of ordinals, 0 < n < ω , and, for all b ∈ [ H ] n , u b is a set of ordinals. We call h u b | b ∈ [ H ] n i a uniform n -dimensional ∆ -system if there is an ordinal ρ and, for each m ⊆ n , a set r m ⊆ ρ satisfying thefollowing statements.(1) otp( u b ) = ρ for all b ∈ [ H ] n .(2) For all a, b ∈ [ H ] n and m ⊆ n , if a and b are aligned with r ( a, b ) = m , then u a and u b are aligned with r ( u a , u b ) = r m .(3) For all m , m ⊆ n , we have r m ∩ m = r m ∩ r m .The main result of this section is a higher-dimensional analogue of the ∆-systemlemma, which asserts, roughly speaking, that inside every family of sets of ordinalsindexed by n -tuples from some sufficiently large cardinal µ , we can find a subset H of µ of some specified size such that [ H ] n indexes a uniform n -dimensional ∆-system.In the absence of large cardinals, this H will necessarily be smaller than µ . In thesame way that the ∆-system lemma can fruitfully be seen as as an elaboration of thepigeonhole principle, this n -dimensional ∆-system lemma can fruitfully be seen as CHRIS LAMBIE-HANSON an elaboration of the Erd˝os-Rado theorem, and in fact a version of the Erd˝os-Radotheorem will be folded into our statement to carry along as an inductive hypothesis.The result is also closely related to results on canonical partition relations , intro-duced by Erd˝os and Rado in [7], and in particular to work done by Baumgartneron canonical partition relations [1], which can also be seen as an elaboration ofthe Erd˝os-Rado theorem. Indeed, in the cases in which κ is a successor cardinal,much of our result, Theorem 2.10 below, can be derived from the main result of[1]. When κ is a limit cardinal (and in particular in the important case κ = ℵ , λ = ℵ ), this approach does not seem to work, so we provide a single proof thatworks for all cases. We first introduce the following notation, from [1], that allowsus to indicate precisely the size of the family needed to ensure the existence of alarge uniform n -dimensional ∆-system. Definition 2.6.
Given a regular cardinal λ , recursively define σ ( λ, n ) for 1 ≤ n < ω by letting σ ( λ,
1) = λ and, given 1 ≤ n < ω , letting σ ( λ, n + 1) = (cid:0) <σ ( λ,n ) (cid:1) + .Note that σ ( λ, n ) is regular for each 1 ≤ n < ω .The higher-dimensional ∆-systems that we isolate in our theorem will have anadditional technical uniformity (the “moreover” clause of the forthcoming Theorem2.10) that allows us to control the relationship between u a and u b for certain non-aligned pairs a, b ∈ [ H ] n and is useful in some applications (cf. [11]). In order toproperly state it, we need some further definitions. Readers can safely skip thesetechnical considerations and the “moreover” clause of the theorem on first read, ifdesired; they are not needed in our first application, in Section 4. Definition 2.7.
Suppose that i < ρ are ordinals and a, b ∈ [On] ρ . We say that a and b are aligned above i if a [ ρ \ i ] and b [ ρ \ i ] are aligned.The following notion provides strictly less information than tp( a, b ) but is some-times easier to control. Definition 2.8.
Suppose that a and b are sets of ordinals. Then the intersectiontype of a and b , denoted tp int ( a, b ), is the set { ( i, j ) ∈ otp( a ) × otp( b ) | a ( i ) = b ( j ) } . Definition 2.9.
Suppose that a is a nonempty set of ordinals and i < otp( a ).(1) We say that an ordinal α is i -possible for a if the following two statementshold:(a) if i >
0, then α > a ( i − i + 1 < otp( a ), then α < a ( i + 1).Intuitively, α is i -possible for a if a ( i ) can be replaced by α without changingthe relative positions of the other elements of a .(2) If α is i -possible for a , then a i α is the set ( a \ { a ( i ) } ) ∪ { α } , i.e., the setobtained by replacing the i th element of a with α .We are now ready for the n -dimensional ∆-system lemma. As we will see at theend of this section, unless λ is a weakly compact cardinal, the theorem is optimalin the sense that µ cannot be lowered. Theorem 2.10.
Suppose that • ≤ n < ω ; • κ < λ are infinite cardinals, λ is regular and <κ -inaccessible, and µ = σ ( λ, n ) ; IGHER-DIMENSIONAL DELTA-SYSTEMS 5 • g : [ µ ] n → <κ ; • for all b ∈ [ µ ] n , we are given a set u b ∈ [On] <κ .Then there are H ∈ [ µ ] λ and k < <κ such that (1) g ( b ) = k for all b ∈ [ H ] n ; (2) h u b | b ∈ [ H ] n i is a uniform n -dimensional ∆ -system.Moreover, we can arrange so that, if n ≥ , for all a, b ∈ [ H ] n and all m < n ,if it is the case that a and b are aligned above m and a ( m ) = b ( m ) , then, forany ordinal α ∈ H that is m -possible for both a and b , we have tp int ( u a , u b ) =tp int ( u a m α , u b m α ) .Proof. The proof is by induction on n . When n = 1, the result follows fromProposition 2.3 and the pigeonhole principle. So suppose that 1 < n < ω and wehave established all instances of the theorem for n − θ be a sufficiently large regular cardinal, let ⊳ be a fixed well-ordering of H ( θ ), and let M be an elementary substructure of ( H ( θ ) , ∈ , ⊳ , g, h u b | b ∈ [ µ ] n i )such that M is closed under sequences of length less than σ ( λ, n −
1) and µ M := M ∩ µ ∈ µ . This is possible, since σ ( λ, n −
1) is regular and µ = σ ( λ, n ) = (cid:0) <σ ( λ,n − (cid:1) + .Note that cf( µ M ) ≥ σ ( λ, n − a ∈ [ µ M ] n − , and consider u a ⌢ h µ M i . Let w a := u a ⌢ h µ M i ∩ M and ρ a = otp (cid:0) u a ⌢ h µ M i (cid:1) . Let i a ⊆ ρ a be such that u a ⌢ h µ M i [ i a ] = w a , and let j a := ρ a \ i a . For each j ∈ j a , let γ a,j be the least ordinal γ in M such that u a ⌢ h µ M i ( j ) < γ .We now construct an increasing sequence h α η | η < σ ( λ, n − i of ordinals below µ M as follows. Begin by letting α η = η for all η < n −
1. Now suppose that n − ≤ η < σ ( λ, n −
1) and we have defined h α ξ | ξ < η i . Let A η := { α ξ | ξ < η } .By the closure of M , we know that all of the following are elements of M : • A η ; • h g ( a ⌢ h µ M i ) | a ∈ [ A η ] n − i ; • h ( w a , ρ a , i a , j a ) | a ∈ [ A η ] n − i ; • h γ a,j | a ∈ [ A η ] n − , j ∈ j α i ;Moreover, tp( h u a ⌢ h µ M i | a ∈ [ A η ] n − i ) is a function from an ordinal less than σ ( λ, n −
1) to P ([ A η ] n − ). Since [ A η ] n − also has size less than σ ( λ, n − M implies that tp( h u a ⌢ h µ M i | a ∈ [ A η ] n − i ) ∈ M .For each a ∈ [ A η ] n − and j ∈ j a , let ǫ a,j := sup { sup( u b ∩ γ a,j ) | b ∈ [ A η ] n } . Note that cf( γ a,j ) ≥ σ ( λ, n − x ⊆ γ a,j such that x ⊆ M . Therefore, we have ǫ a,j ∈ M ∩ γ a,j and, again by closure, h ǫ a,j | a ∈ [ A η ] n − , j ∈ j a i ∈ M .In H ( θ ), the ordinal µ M witnesses the truth of the statement asserting the exis-tence of an ordinal β such that: • sup( A η ) < β < µ ; • g ( a ⌢ h β i ) = g ( a ⌢ h µ M i ) for all a ∈ [ A η ] n − ; • tp( h u a ⌢ h β i | a ∈ [ A η ] n − i ) = tp( h u a ⌢ h µ M i | a ∈ [ A η ] n − i ); • u a ⌢ h β i [ i a ] = w a for all a ∈ [ A η ] n − ; • u a ⌢ h β i ( j ) is in the interval ( ǫ a,j , γ a,j ) for all a ∈ [ A η ] n − and all j ∈ j a .All of the parameters in the above statement are in M (note, for instance, that,in the second item, a ⌢ h µ M i is not in M , but h g ( a ⌢ h µ M i ) | a ∈ [ A η ] n − i is). CHRIS LAMBIE-HANSON
Therefore, by elementarity, we can choose α η ∈ M satisfying the statement, andwe continue to the next step of the construction.After completing the construction, let A = { α η | η < σ ( λ, n − } , and define afunction g ∗ on [ A ] n − by letting g ∗ ( a ) = h g ( a ⌢ h µ M i ) , ρ a , i a , j a i for all a ∈ [ A ] n − .Since we know that • g : [ µ ] n → <κ ; • ρ a < κ ; and • i a , j a ⊆ ρ a ;it follows that g ∗ can be coded as a function from [ A ] n − to 2 <κ . By the inductionhypothesis applied to g ∗ and h u a ⌢ h µ M i | a ∈ [ A ] n − i , we can therefore find H ⊆ A , k < <κ , ρ < κ , and sets i , j ⊆ ρ such that the following statements all hold: • otp( H ) = λ ; • g ( a ⌢ h µ M i ) = k for all a ∈ [ H ] n − ; • h ρ a , i a , j a i = h ρ, i , j i for all a ∈ [ H ] n − ; • h u a ⌢ h µ M i | a ∈ [ H ] n − i is a uniform ( n − s m ⊆ ρ for each m ⊆ n − • If n ≥
3, then h u a ⌢ h µ M i | a ∈ [ H ] n − i satisfies the “moreover” clause inthe statement of the theorem.Let E := { η < σ ( λ, n − | α η ∈ H } . We will thin out H to a furtherunbounded subset H ⊆ H before the end of the proof. For now, let us beginverifying clauses (1) and (2) in the statement of the theorem, noting that what weverify for H will remain true after further thinning out.We first take care of clause (1). Fix b ∈ [ H ] n . Then b is of the form a ⌢ h α η i for some η ∈ E and a ∈ [ A η ] n − . By our choice of α η , we have g ( a ⌢ h α η i ) = g ( a ⌢ h µ m i ), and, by our choice of H and k , we have g ( a ⌢ h µ m i ) = k . Therefore, g ( b ) = k , as desired.We now turn our attention to clause (2). We first specify the values for h r m | m ⊆ n i that will eventually witness that h u b | b ∈ [ H ] n i is a uniform n -dimensional∆-system. For each m ⊆ n , let m − = m ∩ ( n − n − ∈ m , then set r m = s m − .If n − / ∈ m , then set r m = s m − ∩ i . Claim 2.11.
For all m , m ⊆ n , we have r m ∩ m = r m ∩ r m .Proof. This follows immediately from the fact that s m − ∩ m − = s m − ∩ s m − for all m , m ⊆ n . (cid:3) It remains to verify clause (2) of Definition 2.5, i.e., if a, b ∈ [ H ] n are alignedand r ( a, b ) = m , then u a and u b are aligned, and r ( u a , u b ) = r m . We split thisverification into two cases, depending on whether or not n − m . Claim 2.12.
Suppose that b , b ∈ [ H ] n are aligned and n − ∈ m = r ( b , b ) .Then u b and u b are aligned and r ( u b , u b ) = r m .Proof. Since n − ∈ m , we have r m = s m − . It also follows from the fact that n − ∈ m that there is η ∈ E such that b and b are of the form a ⌢ h α η i and a ⌢ h α η i respectively, where a , a ∈ [ A η ] n − are aligned and r ( a , a ) = m − .By our choice of H and s m − , it follows that u a ⌢ h µ M i and u a ⌢ h µ M i are alignedand r ( u a ⌢ h µ M i , u a ⌢ h µ M i ) = s m − . By our choice of α η , we have tp( u b , u b ) =tp( u a ⌢ h µ M i , u a ⌢ h µ M i ), and therefore u b and u b are aligned, with r ( u b , u b ) = s m − = r m , as desired. (cid:3) IGHER-DIMENSIONAL DELTA-SYSTEMS 7
We next deal with the case in which n − / ∈ m . This will take a bit more work.We first establish the following claim. Claim 2.13.
Suppose that b , b ∈ [ H ] n are aligned and n − / ∈ m = r ( b , b ) .Then u b [ r m ] = u b [ r m ] .Proof. Since n − / ∈ m , we have m − = m and r m = s m ∩ i . We also know that b and b are of the form a ⌢ h α η i and a ⌢ h α ε i , respectively, where η = ε and a and a are aligned, with r ( a , a ) = m . Fix i ∈ r m . Since i ∈ i , our choices of α η and α ε imply that u b ( i ) = u a ⌢ h µ M i ( i ) and u b ( i ) = u a ⌢ h µ M i ( i ). Since i ∈ s m and r ( a , a ) = m , we know that u a ⌢ h µ M i ( i ) = u a ⌢ h µ M i ( i ). Together, this impliesthat u b ( i ) = u b ( i ). (cid:3) For b and b as in the previous claim, showing that u b and u b are disjointoutside of u b [ r m ] will take some more work and possibly a thinning out of H .For m < n and a ∈ [ H ] m , choose any b ∈ [ H ] n with a = b [ m ] (i.e., b is anend-extension of a ), and define u a := u b [ r m ]. We claim that this definition isindependent of our choice of b . To this end, suppose that b and b are distinctelements of [ H ] n with b [ m ] = b [ m ] = a . If b and b are aligned, then Claims2.12 and 2.13 imply that u b [ r m ] = u b [ r m ]. Otherwise, find b ∈ [ H ] n such that b [ m ] = a and b ( m ) > max( b ∪ b ). Then, for each ℓ <
2, we know that b ℓ and b are aligned, so, again by Claim 2.13, we have u b [ r m ] = u b [ r m ] = u b [ r m ].For the rest of the proof, we adopt the convention that max( ∅ ) = − Claim 2.14.
Suppose that m < n and a ∈ [ H ] m . Then h u a ⌢ h β i | β ∈ H \ (max( a ) + 1) i is a ∆ -system with root u a .Proof. Suppose first that m = n −
1, in which case r m = i . Fix ( η, ε ) ∈ [ E ] with α η > max( a ), and consider u a ⌢ h α η i ∩ u a ⌢ h α ε i . If i ∈ i , then Claim 2.13 impliesthat u a ⌢ h α η i ( i ) = u a ⌢ h α ε i ( i ). If j ∈ j , then, by our choice of α ε , we know thatsup( u a ⌢ h α η i ∩ γ a,j ) < u a ⌢ h α ε i ( j ) < γ a,j , and hence u a ⌢ h α ε i ( j ) / ∈ u a ⌢ h α η i . It follows that u a ⌢ h α η i ∩ u a ⌢ h α ε i = u a ⌢ h α ε i [ i ] = u a , as desired.Next, suppose that m < n −
1. Fix ( β , β ) ∈ [ H ] with β > max( a ), andconsider u a ⌢ h β i ∩ u a ⌢ h β i . Fix c ∈ [ H ] n − m − with min( c ) > β and set b ℓ := a ⌢ h β ℓ i ⌢ c for ℓ <
2. Note that b ℓ ∈ [ H ] n , that u a ⌢ h β ℓ i = u b ℓ [ r m +1 ], and that u a = u b ℓ [ r m ]. Observe also that b and b are aligned and that r ( b , b ) = n \ { m } ,so, by Claim 2.12, we have u b ∩ u b = u b [ r n \{ m } ] = u b [ r n \{ m } ]. Putting thistogether, we obtain u a ⌢ h β i ∩ u a ⌢ h β i = u b [ r m +1 ] ∩ u b [ r m +1 ]= u b [ r m +1 ] ∩ u b [ r m +1 ] ∩ u b [ r n \{ m } ] ∩ u b [ r n \{ m } ]= u b [ r m ] ∩ u b [ r m ]= u a . (cid:3) CHRIS LAMBIE-HANSON
We are now ready to thin out H to our final set H witnessing the conclusionof the theorem. We will recursively define an increasing sequence h β ξ | ξ < λ i ofordinals from H and then define H := { β ξ | ξ < λ } .Begin by letting β = min( H ). Next, suppose that 0 < ζ < λ and h β ξ | ξ < ζ i has been defined. Let B ζ := { β ξ | ξ < ζ } . Suppose that a ∈ [ B ζ ]
0, thenmax( b ) > b ( m ∗ − H to H , we know that u b \ u b [ n − is disjoint from u b [ m ∗ ] . Since γ ∈ u b [ m ∗ ] by the previous claim, itfollows that γ ∈ u b [ n − . (cid:3) For ℓ <
2, let a ℓ = b ℓ [ n −
1] and β ℓ = b ℓ ( n − γ ∈ u a ∩ u b = u b [ i ] ∩ u b [ i ] = w a ∩ w a = u a ⌢ h µ M i [ i ] ∩ u a ⌢ h µ M i [ i ] . In particular, we have i ∈ i and, since u b [ i ] = u a ⌢ h µ M i [ i ], we also know that u a ⌢ h µ M i ( i ) = γ .Since b and b are aligned, we know that a and a are aligned, and, since n − / ∈ m , we also have r ( a , a ) = m . Therefore, by our choice of H , it followsthat u a ⌢ h µ M i and u a ⌢ h µ M i are aligned and r ( u a ⌢ h µ M i , u a ⌢ h µ M i ) = s m . Since γ ∈ u a ⌢ h µ M i ∩ u a ⌢ h µ M i , it follows that i ∈ s m . But since i ∈ i and r m = s m ∩ i ,it follows that i ∈ r m , which finishes the proof of clause (2). IGHER-DIMENSIONAL DELTA-SYSTEMS 9
We finally turn our attention to the “moreover” clause. To this end, fix m < n and a, b ∈ [ H ] n such that a and b are aligned above m and a ( m ) = b ( m ). Fix α ∈ H such that α is m -possible for both a and b . We must show that tp int ( u a , u b ) =tp int ( u a m α , u b m α ). There are a few cases to consider. First note that, if n = 2,then it must in fact be the case that a and b are aligned, a m α , b m α , are aligned,and r ( a, b ) = r ( a m α , b m α ). Now tp int ( u a , u b ) = tp int ( u a m α , u b m α ) easilyfollows from the fact that h u a | a ∈ [ H ] n i is a uniform n -dimensional ∆-system.We may thus assume that n ≥
3. Let a − = a [ n −
1] and b − = b [ n − a + = a − ⌢ h µ M i and b + = b − ⌢ h µ M i .Suppose first that m = n −
1, so a ( n −
1) = b ( n − h α η | η < σ ( λ, n − i , we know that tp( u a , u b ) = tp( u a + , u b + ) = tp( u a m α , u b m α ),and hence tp int ( u a , u b ) = tp int ( u a m α , u b m α ).Suppose next that m < n −
1. By the fact that h u a ⌢ h µ M i | a ∈ [ H ] n − i satisfiesthe “moreover” clause in the statement of the theorem, we know thattp int ( u a + , u b + ) = tp int (cid:16) u a + m α , u b + m α (cid:17) . ( ∗ )Suppose in addition that a ( n −
1) = b ( n − h α η | η < σ ( λ, n − i , we know thattp( u a , u b ) = tp( u a + , u b + ) and tp ( u a m α , u b m α ) = tp (cid:16) u a + m α , u b + m α (cid:17) . Putting this together yields tp int ( u a , u b ) = tp int ( u a m α , u b m α ), as desired.The remaining case is that in which a ( n − = b ( n − int ( u a , u b ) ⊆ tp int ( u a m α , u b m α ). A symmetric argument will yield the reverseinclusion. To this end, fix ( i, j ) ∈ tp int ( u a , u b ). Thus, we have u a ( i ) = u b ( j ) = γ for some ordinal γ . Since a and b are aligned above m , a ( m ) = b ( m ), and a ( n − = b ( n − b ( n − / ∈ a and a ( n − / ∈ b . An argument exactlyas in the proofs of Claims 2.15 and 2.16 then shows that γ ∈ u a − ∩ u b − = u a [ i ] ∩ u b [ i ],and hence we have i, j ∈ i .Since i, j ∈ i , our construction of h α η | η < σ ( λ, n − i implies that u a ( i ) = u a + ( i ) and u b ( j ) = u b + ( j ), and hence ( i, j ) ∈ tp int ( u a + , u b + ). By equation ( ∗ )above, we have ( i, j ) ∈ tp int (cid:16) u a + m α , u b + m α (cid:17) . Again by our construction of h α η | η < σ ( λ, n − i and the facts that i, j ∈ i , we have u a m α ( i ) = u a + m α ( i ) and u b m α ( j ) = u b + m α , so ( i, j ) ∈ tp int ( u a j α , u b j α ), thus finishing the proof. (cid:3) The following corollary gives an important special case, obtained from setting κ = ℵ and λ = ℵ ℓ in Theorem 2.10 for some 1 ≤ ℓ < ω . Corollary 2.17.
Suppose that ≤ ℓ, n < ω , and let µ = i n − ( ℵ ℓ − ) + . If h u b | b ∈ [ µ ] n i is a sequence of finite sets of ordinals, then there is H ∈ [ µ ] ℵ ℓ suchthat h u b | b ∈ [ H ] n i is a uniform n -dimensional ∆ -system. We end this section with a discussion of the optimality of Theorem 2.10. It canbe argued that, if κ < λ ≤ µ are infinite cardinals, 1 ≤ n < ω , and µ → ( λ ) n <κ ,then any sequence h u a | a ∈ [ µ ] n i consisting of elements of [On] <κ can be thinnedout to a uniform n -dimensional ∆-system of size λ (see [3] for such an argument).In general, µ → ( λ ) n <κ is a stronger assertion than µ ≥ σ ( λ, n ), which is ourassumption in Theorem 2.10, so this argument yields weaker results than those ofTheorem 2.10. However, if λ is weakly compact, then we have λ → ( λ ) n <κ for all1 ≤ n < ω and all κ < λ , so we obtain the following corollary. Corollary 2.18.
Suppose that ≤ n < ω and that κ < λ are infinite cardinals, with λ being weakly compact. Suppose also that h u a | a ∈ [ λ ] n i is a sequence consisting ofelements of [On] <κ . Then there is H ∈ [ λ ] λ such that h u a | a ∈ [ H ] n i is a uniform n -dimensional ∆ -system. If λ is not weakly compact, however, then our result is optimal, even focusing onlyon the higher-dimensional ∆-systems and disregarding clause (1) or the “moreoverclause”, for essentially the same reason that the Erd˝os-Rado theorem is optimal. Proposition 2.19.
Suppose that ≤ n < ω and λ is a regular uncountable cardinalthat is not weakly compact, and suppose that µ < σ ( λ, n ) . Then there is a sequence h u a | a ∈ [ µ ] n i consisting of finite sets of ordinals such that there is no H ∈ [ µ ] λ for which h u a | a ∈ [ H ] n i is a uniform n -dimensional ∆ -system.Proof. If n = 1, then we have µ < λ , so the result is trivial. So assume that n >
1. First observe that, since λ is uncountable and not weakly compact, aneasy argument shows that 2 <λ ( λ ) . Therefore, by successive applications of[6, Lemma 5A], which is the lemma establishing the optimality of the Erd˝os-Radotheorem, we have, for all m < ω , i m (2 <λ ) ( λ ) m . Notice that σ ( λ, n ) = (cid:0) i n − (2 <λ ) (cid:1) + for all 2 ≤ n < ω . Therefore, we have µ ≤ i n − (2 <λ ), so there isa function c : [ µ ] n → H ∈ [ µ ] λ . For each a ∈ [ µ ] n ,simply let u a be any set of ordinals of cardinality c ( a ) + 1. Then, for any H ∈ [ µ ] λ ,we have |{ otp( u a ) | a ∈ [ H ] n }| >
1, so, a fortiori , h u a | a ∈ [ H ] n i is not a uniform n -dimensional ∆-system. (cid:3) Chain conditions
One of the primary uses of the classical ∆-system lemma is in proving that certainforcing notions satisfy chain conditions. For example, one of the first applicationsthat many people learn is in the proof that the forcing notion to add any numberof Cohen reals is κ -Knaster for every regular uncountable κ : Lemma 3.1.
Let χ be any infinite cardinal, and let P = Add( ω, χ ) be the forcingto add χ -many Cohen reals. Suppose that κ is a regular uncountable cardinal and h p α | α < κ i is a sequence of conditions from P . Then there is an unbounded A ⊆ κ such that h p α | α ∈ A i consists of pairwise compatible conditions. During forcing constructions involving higher-dimensional combinatorial state-ments, one frequently encounters sequences of conditions indexed not by singleordinals but by n -element sets of ordinals for some n >
1. One would then like tofind a large set such that the restriction of the sequence to that set satisfies certainuniformities analogous to the uniformities exhibited by h p α | α ∈ A i in Lemma 3.1.A first, na¨ıve attempt at formulating a statement to this effect might look vaguelyas follows:Let χ be an infinite cardinal and n < ω , and let P be the forcing toadd χ -many Cohen reals. Then there is a sufficiently large regularcardinal µ ≤ χ such that, for every sequence h p a | a ∈ [ µ ] n i ofconditions in P , there is a “large” set H ⊆ µ such that h p a | a ∈ [ H ] n i consists of pairwise compatible conditions.It is easily seen that such a statement cannot possibly hold if n >
1, however.Indeed suppose that n = 2 and, for all ( α, β ) ∈ [ µ ] , define a condition p α,β ∈ P by IGHER-DIMENSIONAL DELTA-SYSTEMS 11 letting dom( p α,β ) = { α, β } , p ( α ) = 0, and p ( β ) = 1 (we are thinking of conditionsin P as being finite partial functions from θ to 2). Then p α,β ⊥ p β,γ for all ( α, β, γ ) ∈ [ µ ] , so we could not even find a set H of size 3 as in the above statement. Theobvious problem here is that the sets { α, β } and { β, γ } are not aligned, and it turnsout that this is the only obstacle. By only requiring the compatibility of p a and p b when a and b are aligned, we obtain a consistent statement. For example: Lemma 3.2.
Suppose that λ is a regular uncountable cardinal, n < ω , and µ = σ ( λ, n ) , and suppose that P is the forcing notion to add χ -many Cohen reals forsome infinite cardinal χ . Then, for every sequence h p a | a ∈ [ µ ] n i of conditions in P , there is a set H ∈ [ µ ] λ such that, for all a, b ∈ [ H ] n , if a and b are aligned, then p a k p b .Proof. Fix a sequence h p a | a ∈ [ µ ] n i consisting of conditions in P . For each a ∈ [ µ ] n , let u a = dom( p a ) and k a = otp( u a ), and let ¯ p a : k a → p a , i.e., ¯ p a ( i ) = p ( u a ( i )) for all i < k a . Now apply Theorem2.10 to h u a | a ∈ [ µ ] n i and the function a ¯ p a to find an H ∈ [ µ ] λ , a k < ω , and afunction ¯ p : k → h u a | a ∈ [ H ] n i is a uniform n -dimensional ∆-systemand ¯ p a = ¯ p for all a ∈ [ H ] n .We claim that p a k p b for all aligned a, b ∈ [ H ] n . To this end, fix a, b ∈ [ H ] n such that a and b are aligned. The only way we could have p a ⊥ p b is if thereis α ∈ u a ∩ u b such that p a ( α ) = p b ( α ). Since h u a | a ∈ [ H ] n i is a uniform n -dimensional ∆-system, we know that u a and u b are aligned. Moreover, we knowthat ¯ p a = ¯ p b = ¯ p . Therefore, if α ∈ u a ∩ u b , then there is i < k such that α = u a ( i ) = u b ( i ). But then p a ( α ) = ¯ p ( i ) = p b ( α ). Therefore, we have p a k p b . (cid:3) Remark 3.3. If λ is weakly compact, then, by Corollary 2.18, Lemma 3.2 stillholds with µ = λ rather than µ = σ ( λ, n ).4. An application to polarized partition relations
In this section, we give a relatively simple application illustrating a typical useof Theorem 2.10 in a forcing argument.
Definition 4.1.
Let 1 ≤ n < ω . Then Θ n is the least cardinal θ such that, forevery function f : θ n → ω , there is a sequence h A i | i < n i of infinite subsets of θ such that f ↾ Q i
Suppose that ≤ n < ω , κ is a cardinal, and Θ n > κ . Then Θ n +1 > κ + .Proof. Since Θ n > κ , we can fix a function g : κ n → ω such that g is not constanton any product of n infinite subsets of κ . For each β < κ + , fix an injective function e β : β → κ . Then the function g β : β n → ω defined by letting g β ( h α , . . . , α n − i ) = g ( h e β ( α ) , . . . , e β ( α n − ) i )for all h α , . . . , α n − i ∈ β n has the property that g β is not constant on any productof n infinite subsets of β .We now define a function f : ( κ + ) n +1 → ( n + 2) × ω that will not be constant onany product of ( n + 1) infinite subsets of κ + . This can easily be coded as a functioninto ω , so this suffices to prove the proposition. Given ~α = ( α , . . . , α n ) ∈ ( κ + ) n +1 and i ≤ n , let ~α i denote the sequence formedby removing α i from ~α , i.e., ~α i = h α , . . . , α i − , α i +1 , . . . , α n ). Let us now define f ( ~α ). If there are i < j ≤ n such that α i = α j , then let f ( ~α ) = ( n +1 , i ≤ n be such that α j < α i for all j ∈ ( n + 1) \ { i } , and let f ( ~α ) = ( i, g α i ( ~α i )).Suppose for sake of contradiction that h A i | i ≤ n i is a sequence of infinitesubsets of κ + such that f ↾ Q i ≤ n A i is constant, taking value ( i, k ). First note thatwe can always find a sequence ~α ∈ Q i ≤ n A i whose coordinates are all distinct, so itcannot be the case that i = n + 1. Thus, i ≤ n , so, by our definition of f , it followsthat A j < A i for all j ∈ ( n + 1) \ { i } . Fix β ∈ A i , and define h A ∗ j | j < n i by letting A ∗ j = A j for j < i and A ∗ j = A j +1 for i ≤ j < n . Then each A ∗ j is an infinite subsetof β and, by our definition of f , it follows that g β ↾ Q j Corollary 4.3. Θ n ≥ ℵ n for all ≤ n < ω . It follows easily from the Erd˝os-Rado theorem that Θ n ≤ i + n − for all 1 ≤ n < ω . In particular, if GCH holds, then Θ n = ℵ n for all 1 ≤ n < ω . We nowapply Theorem 2.10 to prove that adding any number of Cohen reals preserves theinequality Θ n ≤ ( i + n − ) V . (In fact, we will prove that a slightly stronger partitionrelation, which easily implies Θ n ≤ ( i + n − ) V , holds after forcing to add the Cohenreals.) Thus, if we start with a model of GCH, then we can force to make thecontinuum arbitrarily large while keeping Θ n = ω n for all 1 ≤ n < ω . Theorem 4.4. Fix < n < ω and an infinite cardinal χ . Let µ = i + n − , and let P be the forcing to add χ -many Cohen reals. Then the following statement holds in V P : For every function c : [ µ ] n → ω , there is a sequence h A m | m < n i such that • for all m < n , A m is a subset of µ of order type ω + 1 ; • for all m < m ′ < n , we have A m < A m ′ ; • c ↾ Q m 1. Fix a condition p ∈ P and a P -name ˙ c forced by p to be a function from[ µ ] n to ω . For each b ∈ [ µ ] n , find a condition q b ≤ p and a color k b < ω such that q b (cid:13) “ ˙ c ( b ) = k b ”. Let u b = dom( q p ), and define a function g : [ µ ] n → <ω × ω byletting g ( b ) = h ¯ q b , k b i for all b ∈ [ µ ] n . Apply Theorem 2.10 to find H ∈ [ µ ] ℵ suchthat h u b | b ∈ [ H ] n i is a uniform n -dimensional ∆-system and g ↾ [ H ] n is constant,taking value h ¯ q, k i . By taking an initial segment if necessary, assume that we infact have otp( H ) = ω . Note that, if b and b ′ are aligned elements of [ H ] n , then q b and q b ′ are compatible in P .Let ρ = | ¯ q | , and let h r m ⊆ ρ | m ⊆ n i witness the fact that h u b | b ∈ [ H ] n i is auniform n -dimensional ∆-system. For each m < n and each a ∈ [ H ] m , define u a byletting b be any element of [ H ] n such that b [ m ] = a and then letting u a := u b [ r m ](we are thinking of m as an initial subset of n here). Then set q a := q b ↾ u a . By IGHER-DIMENSIONAL DELTA-SYSTEMS 13 the facts that h u b | b ∈ [ H ] n i forms a uniform n -dimensional ∆-system and that¯ q b = ¯ q for all b ∈ [ H ] n , it follows that our definition of u a and q a is independent ofour choice of b .By the arguments of Claim 2.14, we know that, for every m < n and every a ∈ [ H ] m , the sequence h u a ⌢ h β i | β ∈ H \ (max( a ) + 1) i is a 1-dimensional ∆-system, with root u a . Since q b ≤ p for all b ∈ [ H ] n , it follows that dom( p ) ⊆ u ∅ and q ∅ ≤ p . We will show that q ∅ forces the existence of a sequence h A m | m < n i in V P such that • each A m is a subset of µ of order type ω + 1; • A m < A m ′ for all m < m ′ < n ; • the realization of ˙ c is constant when restricted to Q m Suppose that m < n , a ∈ [ H ] m , and γ ∈ H . Then the set D a,γ = { q a ⌢ h β i | β ∈ H \ γ } is predense below q a in P .Proof. By increasing γ if necessary, we may assume that γ > max( a ). Fix acondition r ≤ q a . We will find an element of D a,γ compatible with r . Since h u a ⌢ h β i | β ∈ H \ γ } is an infinite 1-dimensional ∆-system with root u a , and sincedom( r ) is finite, we can find β ∈ H \ γ such that u a ⌢ h β i \ u a is disjoint fromdom( r ). But then q a ⌢ h β i ↾ dom( r ) = q a , so, since r ≤ q a , it follows that r ∪ q a ⌢ h β i is a condition in P , so q a ⌢ h β i is an element of D a,γ compatible with r . (cid:3) Now suppose that G is P -generic over V with q ∅ ∈ G , and let c be the realizationof ˙ c in V [ G ]. By applying Claim 4.5 n times, working in V [ G ], we can recursivelychoose an increasing sequence h δ m | m < n i of elements of H such that, letting d = { δ m | m < n } , we have • q d ∈ G ; • H ∩ δ is infinite; • for all m < n − H ∩ ( δ m +1 \ ( δ m + 1)) is infinite.Let A ′ denote the set of the first ω -many elements of H ∩ δ and, for all m < n − A ′ m +1 denote the set of the first ω -many elements of H ∩ ( δ m +1 \ ( δ m + 1)).We now construct an n × ω matrix h α m,ℓ | m < n, ℓ < ω i such that • for all m < n , h α m,ℓ | ℓ < ω i is an increasing sequence of elements of A ′ m ; • letting A m = { α m,ℓ | ℓ < ω } ∪ { δ m } for each m < n , we have q b ∈ G for all b ∈ Q m 0) = { δ m ′ } , so q ∗ , = q d ∈ G . Thus, our recursion hypothesis is initially satisfied. Nowsuppose that ( m, ℓ ) ∈ n × ω and we have defined h α m ′ ,ℓ ′ | ( m ′ , ℓ ′ ) < ( m, ℓ ) i so thatthe resulting condition q ∗ m,ℓ is in G . Temporarily move back to V , noting that each A m ′ ↾ ( m, ℓ ) is finite and hence in V , and A ′ m is also in V , as it is definable from H (and δ m − , if m > B = Q m ′ The set E = { q ∗ α | α ∈ A ′ m \ γ } is predense below q ∗ m,ℓ in P .Proof. Fix r ≤ q ∗ m,ℓ . We will find an element of E compatible with r . Let m = n \ { m } . For each ( b , b ) ∈ B × B , the sequence h u b ∪{ α }∪ b | α ∈ A ′ m \ γ i formsa 1-dimensional ∆-system whose root is equal to u b ∪{ α }∪ b [ r m ] for all α ∈ A ′ m \ γ .Since A ′ m \ γ is infinite and dom( r ), B , and B are all finite, we can find α ∈ A ′ m \ γ such that, for all ( b , b ) ∈ B × B , the set u b ∪{ α }∪ b \ ( u b ∪{ α }∪ b [ r m ]) is disjointfrom dom( r ).We claim that q ∗ a and r are compatible. To see this, it suffices to show that q b ∪{ α }∪ b and r are compatible for every ( b , b ) ∈ B × B . Thus, fix ( b , b ) ∈ B × B . We know that u b ∪{ α }∪ b ∩ dom( r ) ⊆ u b ∪{ α }∪ b [ r m ]. But we also knowthat u b ∪{ α }∪ b [ r m ] = u b ∪{ δ m }∪ b [ r m ], since b ∪ { α } ∪ b and b ∪ { δ m } ∪ b arealigned, with r ( b ∪ { α } ∪ b , b ∪ { δ m } ∪ b ) = m . Then q b ∪{ α }∪ b ↾ ( u b ∪{ α }∪ b [ r m ]) = q b ∪{ δ m }∪ b ↾ ( u b ∪{ δ m }∪ b [ r m ]) . But we know that q ∗ m,ℓ ≤ q b ∪{ δ m }∪ b , since b ∪ { δ m } ∪ b ∈ Q m ′ With some appropriate bookkeeping, the order type ω + 1 in thestatement of Theorem 4.4 can be replaced by any countable ordinal α .5. A variation, and monochromatic sumsets of reals In this section, we discuss an alternative form of higher-dimensional ∆-systemthat has appeared in the literature. The following theorem is due to Shelah andfollows from results in [13]. Theorem 5.1. Suppose that κ ≤ λ ≤ µ are infinite cardinals, ≤ n < ω , and µ → ( λ ) n κ . Suppose moreover that h u a | a ∈ [ µ ] n i is a sequence of elements from [On] ≤ κ . Then there is H ∈ [ µ ] λ and a sequence h u ∗ a | a ∈ [ H ] ≤ n i of elements from [On] ≤ κ such that IGHER-DIMENSIONAL DELTA-SYSTEMS 15 (1) u ∗ a ⊇ u a for all a ∈ [ H ] n ; (2) for all a, b ∈ [ H ] n , we have tp( u ∗ a , u a ) = tp( u ∗ b , u b ) ; (3) for all a, b ∈ [ H ] ≤ n , we have u ∗ a ∩ u ∗ b = u ∗ a ∩ b ; (4) for all a ⊆ a and b ⊆ b , where a , b ∈ [ H ] ≤ n , if tp( a , a ) = tp( b , b ) ,then tp( u ∗ a , u ∗ a ) = tp( u ∗ b , u ∗ b ) . It is currently unclear whether arguments similar to those in the proof of Theorem2.10 can be used to obtain the conclusion of Theorem 5.1 from a weaker assumptionon µ , such as µ ≥ σ ( λ, n ). It is the case, however, that certain results that havebeen proven using Theorem 5.1 can be proven by instead using Theorem 2.10. Thiscan yield some improvements, since Theorem 2.10 places weaker assumptions onthe cardinal µ . We give one example of such a result here.In [18], Zhang uses Theorem 5.1 to prove that, in the forcing extension obtainedby adding i ω -many Cohen reals, we have R → + ( ℵ ) r for every r < ω , i.e., forevery r < ω and every function f : R → r , there is an infinite set X ⊆ R suchthat f ↾ ( X + X ) is constant. We remark that, by a result of Hindman, Leader,and Strauss [8], if 2 ℵ < ℵ ω , then there is r < ω such that R + ( ℵ ) r , so, over amodel of GCH, it is necessary to add at least i ω -many reals to obtain R → + ( ℵ ) r for every r < ω .Let us examine, though, the number of reals that must be added to obtain R → + ( ℵ ) r for some fixed r < ω . Zhang in fact proves that R → + ( ℵ ) holdsin ZFC and, for a fixed r > 2, in proving that R → + ( ℵ ) r holds in the forcingextension, Theorem 5.1 is employed with κ = ℵ , λ = ℵ , and n = 2 r . Hence, µ canbe taken to be least such that µ → ( ℵ ) r ℵ . By the Erd˝os-Rado theorem, then, wecan take µ = i +4 r . Zhang’s proof uses the fact that we have added at least µ -manyCohen reals and therefore shows that, for this fixed value of r > 2, the statement R → + ( ℵ ) r holds in the forcing extension obtained by adding i +4 r -many Cohenreals.Inspection of Zhang’s proof reveals that, Theorem 2.10, with κ = ℵ , λ = i +1 ,and n = 2 r , can be used in place of Theorem 5.1. We can therefore take µ = σ ( i +1 , r ) = i +2 r , obtaining the following corollary: Corollary 5.2. Suppose that < r < ω and P is the forcing to add at least i +2 r -many Cohen reals. Then, in V P , we have R → + ( ℵ ) r . This is an improvement on the bound of i +4 r given by Zhang’s proof, though ofcourse it does not improve on Zhang’s bound for obtaining R → + ( ℵ ) r simultane-ously for all r < ω . We omit the adaptation of Zhang’s proof using Theorem 2.10instead of Theorem 5.1 here, as it would entail introducing a considerable numberof definitions and only involves very minor changes to Zhang’s proof. Instead, wedirect the reader to [18] and [11], in which Zhang’s original proof and the adaptationusing Theorem 2.10 are spelled out in detail. References 1. James E. Baumgartner, Canonical partition relations , J. Symbolic Logic (1975), no. 4,541–554. MR 3988372. Jeffrey Bergfalk, Michael Hruˇs´ak, and Saharon Shelah, Ramsey theory for highly connectedmonochromatic subsets , (2019), Preprint. https://arxiv.org/abs/1812.06386.3. Jeffrey Bergfalk and Chris Lambie-Hanson, Simultaneously vanishing higher derived limits ,(2019), Preprint. https://arxiv.org/abs/1907.11744. 4. Natasha Dobrinen and Dan Hathaway, The Halpern-L¨auchli theorem at a measurable cardinal ,J. Symb. Log. (2017), no. 4, 1560–1575. MR 37436235. M. Dˇzamonja, J. A. Larson, and W. J. Mitchell, A partition theorem for a large dense linearorder , Israel J. Math. (2009), 237–284. MR 25201106. P. Erd˝os, A. Hajnal, and R. Rado, Partition relations for cardinal numbers , Acta Math. Acad.Sci. Hungar. (1965), 93–196. MR 2026137. P. Erd¨os and R. Rado, A combinatorial theorem , J. London Math. Soc. (1950), 249–255.MR 378868. Neil Hindman, Imre Leader, and Dona Strauss, Pairwise sums in colourings of the reals , Abh.Math. Semin. Univ. Hambg. (2017), no. 2, 275–287. MR 36961519. Ashutosh Kumar and Dilip Raghavan, Separating families and order dimension of Turingdegrees , Preprint.10. Kenneth Kunen, Set theory , Studies in Logic and the Foundations of Mathematics, vol. 102,North-Holland Publishing Co., Amsterdam-New York, 1980, An introduction to independenceproofs. MR 59734211. Chris Lambie-Hanson, A note on a result of Zhang about monochromatic sumsets of reals ,(2020), Unpublished note.12. N. A. Shanin, A theorem from the general theory of sets , C. R. (Doklady) Acad. Sci. URSS(N.S.) (1946), 399–400. MR 001881413. S. Shelah, Strong partition relations below the power set: consistency; was Sierpi´nski right?II , Sets, graphs and numbers (Budapest, 1991), Colloq. Math. Soc. J´anos Bolyai, vol. 60,North-Holland, Amsterdam, 1992, pp. 637–668. MR 121822414. Saharon Shelah, Consistency of positive partition theorems for graphs and models , Set theoryand its applications (Toronto, ON, 1987), Lecture Notes in Math., vol. 1401, Springer, Berlin,1989, pp. 167–193. MR 103177315. Stevo Todorcevic, Walks on ordinals and their characteristics , Progress in Mathematics, vol.263, Birkh¨auser Verlag, Basel, 2007. MR 235567016. Stevo Todorˇcevi´c, Reals and positive partition relations , Logic, methodology and philosophy ofscience, VII (Salzburg, 1983), Stud. Logic Found. Math., vol. 114, North-Holland, Amsterdam,1986, pp. 159–169. MR 87478617. Jing Zhang, A tail cone version of the Halpern-L¨auchli theorem at a large cardinal , J. Symb.Log. (2019), no. 2, 473–496. MR 396160918. , Monochromatic sumset without large cardinals , Fund. Math. (2020), To appear. Department of Mathematics and Applied Mathematics, Virginia Commonwealth Uni-versity, Richmond, VA 23284, United States E-mail address : [email protected] URL ::