aa r X i v : . [ m a t h . L O ] M a y A NOTE ON HIGHLY CONNECTED AND WELL-CONNECTEDRAMSEY THEORY
CHRIS LAMBIE-HANSON
Abstract.
We study a pair of weakenings of the classical partition relation ν → ( µ ) λ recently introduced by Bergfalk-Hruˇs´ak-Shelah and Bergfalk, respec-tively. Given an edge-coloring of the complete graph on ν -many vertices, theseweakenings assert the existence of monochromatic subgraphs exhibiting highdegrees of connectedness rather than the existence of complete monochromaticsubgraphs asserted by the classical relations. As a result, versions of theseweakenings can consistently hold at accessible cardinals where their classicalanalogues would necessarily fail. We prove some complementary positive andnegative results indicating the effect of large cardinals, forcing axioms, andsquare principles on these partition relations. We also prove a consistency re-sult indicating that a non-trivial instance of the stronger of these two partitionrelations can hold at the continuum. Introduction
In this paper, we study some natural variations of the classical partition relationfor pairs. Recalling the arrow notation of Erd˝os and Rado, given cardinals µ , ν ,and λ , the expression ν → ( µ ) λ denotes the assertion that, for every coloring c : [ ν ] → λ , there is a set X ⊆ ν ofcardinality µ such that c ↾ [ X ] is constant. This can be usefully interpreted in thelanguage of graph theory as asserting that for every edge-coloring of the completegraph K ν on ν -many vertices using λ -many colors, there is a monochromatic sub-graph of K ν isomorphic to K µ . With this notation, the infinite Ramsey theoremfor pairs can be succintly expressed as ℵ → ( ℵ ) k for all k < ω .When one tries to generalize the infinite Ramsey theorem to uncountable car-dinals in the most straightforward way, by replacing each instance of ‘ ℵ ’ in theabove expression by some fixed uncountable cardinal ‘ ν ’, one immediately runsinto a statement that can only hold at large cardinals, as the assertion that ν isuncountable and ν → ( ν ) is equivalent to the assertion that ν is weakly compact.To achieve a consistent statement at accessible uncountable cardinals, then, onemust weaken the statement ν → ( ν ) . One natural approach is to increase the value Date : May 22, 2020.2010
Mathematics Subject Classification.
Key words and phrases. partition relations, highly connected, well-connected, guessing models,square principles.We thank Jeffrey Bergfalk and Michael Hruˇs´ak for reading an early draft and providing helpfulcomments and corrections. of the cardinal on the left side of the expression. This is precisely what was doneby Erd˝os and Rado in [5]. A special case of what became known as the Erd˝os-Radotheorem states that, for every infinite cardinal κ , we have(2 κ ) + → ( κ + ) κ . This result is sharp, in the sense that there are colorings witnessing the negativerelations 2 κ (3) κ and 2 κ ( κ + ) . (We refer the reader to [6, §
7] for proofs andfurther discussion of these facts.)Another approach is to weaken the meaning of the arrow notation, in particu-lar by replacing the requirement that the monochromatic subgraph witnessing thepartition relation be complete by a weaker but still nontrivial notion of largeness.This is the approach taken by Bergfalk-Hruˇs´ak-Shelah and Bergfalk in [2] and [1],where they introduce the notions of partition relations for highly connected and well-connected subsets, respectively. It is these notions that provide the primarysubject of this paper; let us turn to their definitions, beginning with the partitionrelation for highly connected subsets.Throughout the paper, if G = ( X, E ) is a graph and Y ⊆ X , then we let G \ Y denote the graph ( X \ Y, E ∩ [ X \ Y ] ). We will also sometime write | G | to mean | X | . Definition 1.1.
Given a graph G = ( X, E ) and a cardinal κ , we say that G is κ -connected if G \ Y is connected for every Y ∈ [ X ] <κ . We say that G is highlyconnected if it is | G | -connected. Definition 1.2. (Bergfalk-Hruˇs´ak-Shelah [2]) Suppose that µ , ν , and λ are car-dinals. The partition relation ν → hc ( µ ) λ is the assertion that, for every coloring c : [ ν ] → λ , there is an X ∈ [ ν ] µ and a highly connected subgraph ( X, E ) of( ν, [ ν ] ) such that c ↾ E is constant.As was noted in [2], a finite graph is highly connected if and only if it is complete.As a result, if µ is finite, then the relation ν → hc ( µ ) λ is simply the classical relation ν → ( µ ) λ . In the context of infinite µ , however, the → hc version of a relation canconsistently hold in situations where the classical version necessarily fails. Forexample, the following is proven in [2]. Theorem 1.3 (Bergfalk-Hruˇs´ak-Shelah [2]) . It is consistent, relative to the con-sistency of a weakly compact cardinal, that ℵ → hc (cid:0) ℵ (cid:1) ℵ . We now recall the partition relation for well-connected subsets.
Definition 1.4. (Bergfalk [1]) Suppose that µ , ν , and λ are cardinals. Given acoloring c : [ ν ] → λ and a fixed color i < λ , we say that a subset X ⊆ ν is well-connected in color i (with respect to c ) if, for every α < β in X , there is a finitepath h α , . . . , α n i (not necessarily contained in X ) such that • α = α and α n = β ; • α k ≥ α for all k ≤ n ; and • c ( α k , α k +1 ) = i for all k < n .The partition relation ν → wc ( µ ) λ is the assertion that, for every coloring c : [ ν ] → λ , there are X ∈ [ ν ] µ and i < λ such that X is well-connected in color i . NOTE ON HIGHLY CONNECTED AND WELL-CONNECTED RAMSEY THEORY 3
As usual, ν hc ( µ ) λ and ν wc ( µ ) λ denote the negations of the respectivepartition relations.The → wc relation is a clear weakening of the classical → relation and is in facta weakening of → hc . Indeed, by [1, Lemma 6], given cardinals µ , ν , and λ , we have (cid:0) ν → ( µ ) λ (cid:1) ⇒ (cid:0) ν → hc ( µ ) λ (cid:1) ⇒ (cid:0) ν → wc ( µ ) λ (cid:1) . We have already seen that the left implication in the above statement can consis-tently fail. The right implication can consistently fail as well. For example, thefollowing is proven in [1].
Theorem 1.5 (Bergfalk [1]) . It is consistent, relative to the consistency of a weaklycompact cardinal, that ℵ → wc ( ℵ ) ℵ holds but ℵ → hc ( ℵ ) ℵ fails. The model for the above theorem is Mitchell’s model for the tree property at ℵ from [10]. The question of the consistency of ℵ → hc ( ℵ ) ℵ was asked in [2] andremains open.In this paper, we prove some further results about the relations → hc and → wc ,some of them addressing questions from [2] and [1]. We provide here a brief outlineof the remainder of the paper. In Section 2, we isolate some instances in which therelations ν → hc ( ν ) λ and ν → wc ( ν ) λ necessarily hold due either to the existence ofcertain large cardinals or to certain forcing axioms holding. In particular, we showthat, if PFA holds, then ν → wc ( ν ) ℵ holds for every regular cardinal ν ≥ ℵ . Thisis optimal in the sense that it was proven in [1] that ℵ wc (3) ℵ In Section 3, weprove some complementary negative results, in particular indicating that certainsquare principles imply the failure of instances of ν → wc ( ν ) λ . In the process,we introduce natural square-bracket versions of the partition relations for highlyconnected and well-connected subsets. In Section 4, we prove the consistency,relative to the consistency of a weakly compact cardinal, of the partition relation2 ℵ → hc [2 ℵ ] ℵ , . This is sharp due to the fact that, as shown in [2], the negativerelation 2 ℵ hc (2 ℵ ) ℵ is provable in ZFC. Finally, in Section 5, we state somequestions and problems that remain open. Notation and conventions. If X is a set and µ is a cardinal, then [ X ] µ = { Y ⊆ X | | Y | = µ } . A graph for us is a pair G = ( X, E ), where X is a set and E ⊆ [ X ] .Given a function c with domain [ X ] , we often slightly abuse notation and write,for instance, c ( a, b ) instead of c ( { a, b } ). While elements of [ X ] are unordered pairs,we will sometimes care about the relative order of the elements of such a pair. Inparticular, if X is a set of ordinals, then we will use the notation ( α, β ) ∈ [ X ] toindicate the conjunction of the two statements { α, β } ∈ [ X ] and α < β .A path in a graph G = ( X, E ) is a finite sequence h x , . . . , x n i of pairwise distinctelements of X such that { x i , x i +1 } ∈ E for all i < n . We say that G is connected if, for all distinct x, y ∈ X , there is a path h x , . . . , x n i in G such that x = x and x n = y .If x is a set of ordinals, then the set of accumulation points of x , denoted byacc( x ), is defined to be { α ∈ x | sup( x ∩ α ) = α } . In particular, if β is an ordinal,then acc( β ) is the set of limit ordinals below β .2. Positive results
In [1], Bergfalk asked about conditions under which the relations µ + → hc ( µ ) µ ) and µ + → wc ( µ ) µ ) hold, where µ is a singular cardinal. In this section, we provide CHRIS LAMBIE-HANSON two scenarios in which such relations (and more) hold. The first scenario simplyinvolves the presence of large cardinals and produces instances of the the highlyconnected partition relation. The second involves the existence of guessing modelsand produces instances of the well-connected partition relation. Complementarynegative results appear in the next section.
Definition 2.1.
Suppose that θ ≤ κ are regular, uncountable cardinals. κ is θ -strongly compact if, for every set X and every κ -complete filter F over X , F can beextended to a θ -complete ultrafilter over X . κ is strongly compact if it is κ -stronglycompact. Theorem 2.2.
Suppose that θ ≤ κ are regular uncountable cardinals and κ is θ -strongly compact. Suppose moreover that λ and ν are cardinals with λ < θ and cf( ν ) ≥ κ . Then ν → hc ( ν ) λ . Proof.
Fix a coloring c : [ ν ] → λ . We must find a color i < λ and a set X ∈ [ ν ] ν such that the graph ( X, c − ( i ) ∩ [ X ] ) is highly connected.Using the fact that κ is θ -strongly compact, let U be a θ -complete ultrafilterover ν extending the κ -complete filter F = { X ⊆ ν | | ν \ X | < ν } . In particular, every element of U has cardinality ν . Now, for each α < ν , use the θ -completeness of U to find a color i α < λ and a set X α ⊆ ν \ ( α + 1) such that X α ∈ U and c ( α, β ) = i α for all β ∈ X α . Now use the θ -completeness of U againto find a fixed color i < λ and a set X ∈ U such that i α = i for all α ∈ X .Let G = ( X, c − ( i ) ∩ [ X ] ). We claim that X is highly connected. To this end,let Y ∈ [ X ] <ν , and fix α < β in X \ Y . To show that α and β are connected in G \ Y , find γ ∈ ( X ∩ X α ∩ X β ) \ Y, and note that c ( α, γ ) = c ( β, γ ) = i . Then h α, γ, β i is a path from α to β in G \ Y . (cid:3) Corollary 2.3.
Suppose that µ is a singular limit of strongly compact cardinals.Then µ + → hc ( µ + ) λ for all λ < µ . Recall the following definition, which comes from [13] and is a generalization ofa notion from [14].
Definition 2.4.
Suppose that κ < χ are regular uncountable cardinals, M ≺ ( H ( χ ) , ∈ ), and κ ≤ | M | < χ .(1) Suppose that x ∈ M and d ⊆ x .(a) We say that d is ( κ, M ) -approximated if d ∩ z ∈ M for every z ∈ M ∩ P κ ( H ( χ )).(b) We say that d is M -guessed if there is e ∈ M such that e ∩ M = d ∩ M .(2) M is κ -guessing if every ( κ, M )-approximated set is M -guessed.Given an infinite regular cardinal θ , let ( T θ ) be the statement asserting thatthere are arbitrarily large regular cardinals χ such that the set { M ≺ ( H ( χ ) , ∈ ) | | M | = θ + , <θ M ⊆ M, and M is θ + -guessing } NOTE ON HIGHLY CONNECTED AND WELL-CONNECTED RAMSEY THEORY 5 is stationary in P θ ++ ( H ( χ )).It is proven by Viale and Weiss in [14] that the Proper Forcing Axiom ( PFA ) im-plies ( T ℵ ). Trang, in [12], proves the consistency of ( T ℵ ), assuming the consistencyof a supercompact cardinal. The proof uses a Mitchell-type forcing constructionand is easily modified to show that, if θ is a regular cardinal and there is a super-compact cardinal above θ , then there is a θ -closed forcing extension in which ( T θ + )holds. Theorem 2.5.
Suppose that θ is a regular uncountable cardinal and ( T θ ) holds.Then, for every regular cardinal ν > θ + and every λ ≤ θ , we have ν → wc ( ν ) λ . Proof.
Fix a regular cardinal ν > θ + , a cardinal λ ≤ θ , and a coloring c : [ ν ] → λ .We must find a color i < λ and a set X ∈ [ ν ] ν such that X is well-connected in color i . Given a color i < λ and α < β < ν , we say that α < i β if { α, β } is well-connectedin color i . It is easily verified that, for every i < λ , ( ν, ≤ i ) is a tree, i.e., it is apartial order and, for all β ∈ ν , the set of < i -predecessors of β is well-ordered by < i (cf. [1, Lemma 12]).Using the fact that ( T θ ) holds, we may fix a regular cardinal χ >> ν , a well-ordering ⊳ of H ( χ ), and an elementary submodel M ≺ ( H ( χ ) , ∈ , ⊳ , λ, θ, ν, c ) suchthat • | M | = θ + ; • <θ M ⊆ M ; and • M is θ + -guessing.Let ν M = sup( M ∩ ν ). Claim 2.6. cf( ν M ) = θ + .Proof. Since | M | = θ + , we clearly have cf( ν M ) ≤ θ + . Suppose for sake of contra-diction that cf( ν M ) = µ < θ + . Let A be a cofinal subset of M ∩ ν M of order type µ , and let h α η | η < µ i be the increasing enumeration of A .We first show that A is ( θ + , M )-approximated. To this end, fix z ∈ M ∩P θ + ( H ( χ )). Since A ⊆ ν , we may assume that z ⊆ ν . Since | z | < θ + and ν > θ + , we know that z is bounded below ν . Since z ∈ M , it follows by ele-mentarity that z is bounded below ν M , so there is ξ < µ such that z ⊆ α ξ . Butthen A ∩ z ⊆ { α η | η < ξ } , and hence A ∩ z is a subset of M and | A ∩ z | < µ ≤ θ .Since <θ M ⊆ M , it follows that A ∩ z ∈ M . Since z was arbitrary, we have shownthat A is ( θ + , M )-approximated.Since M is θ + -guessing, there is B ∈ M such that B ∩ M = A ∩ M = A . Since A is unbounded in M ∩ ν , it follows by elementarity that B is unbounded in ν ,and hence otp( B ) = ν . Let π : ν → B be the order-preserving bijection, and notethat π ∈ M . Since <θ M ⊆ M and θ ∈ M , we have θ + 1 ⊆ M , and in particular µ + 1 ⊆ M . But then π “( µ + 1) ⊆ B ∩ M = A ∩ M , contradicting the fact thatotp( A ) = µ . (cid:3) Since cf( ν M ) = θ + > λ , there is fixed color i < θ and an unbounded set d ⊆ M ∩ ν M such that c ( β, ν M ) = i for all β ∈ d . Let d be the < i -downward closureof d , i.e., d = { α < ν | there is β ∈ d such that α < i β } . CHRIS LAMBIE-HANSON (Note that it may not be the case that d ⊆ M .) We claim that d is ( θ + , M )-approximated. To see this, fix z ∈ M ∩ P θ + ( H ( χ )). As in the proof of the claim,we may assume that z ⊆ ν , and again as in the proof of the claim it follows thatthere is β ∈ d such that z ⊆ β . Then, using the fact that < i is a tree ordering, wehave d ∩ z = { α ∈ z | α < i β } . Since everything needed to define this latter set is in M , we have d ∩ z ∈ M . Since z was arbitrary, we have shown that d is ( θ + , M )-approximated.As M is θ + -guessing, there is e ∈ M such that e ∩ M = d ∩ M . By elementarity,it follows that e is a cofinal subset of ν that is linearly ordered by < i . In otherwords, e is well-connected in color i . Since ν is regular, we have | e | = ν , so we haveproven our theorem. (cid:3) Corollary 2.7.
Suppose that
PFA holds. Then, for every regular cardinal ν ≥ ℵ ,we have ν → wc ( ν ) ℵ . Indexed squares and negative results
In this section, we use square principles to isolate situations in which instancesof → wc necessarily fail. The results in this section are refinements of [1, Lemma9]. In order to fully state our results, we introduce square bracket versions of thepartition relations being studied. Definition 3.1.
Suppose that ν and λ are cardinals. Given a coloring c : [ ν ] k → λ and a collection of colors Λ ⊆ λ , we say that a subset X ⊆ ν is well-connected in Λ(with respect to c ) if, for every α < β in X , there is a finite path h α , . . . , α n i (notnecessarily contained in X ) such that • α = α and α n = β ; • α k ≥ α for all k ≤ n ; and c ( α k , α k +1 ) ∈ Λ for all k < n . Definition 3.2.
Suppose that µ , ν , λ , and κ are cardinals.(1) The partition relation ν → hc [ µ ] λ,κ ( resp. ν → hc [ µ ] λ,<κ ) is the assertionthat, for every coloring c : [ ν ] → λ , there is an X ∈ [ ν ] µ , a highly connectedsubgraph ( X, E ) of ( ν, [ ν ] ), and a set Λ ∈ [ λ ] ≤ κ ( resp. Λ ∈ [ λ ] <κ ) such that c “ E ⊆ Λ.(2) The partition relation ν → wc [ µ ] λ,κ ( resp. ν → wc [ µ ] λ,<κ ) is the assertionthat, for every coloring c : [ ν ] → λ , there are X ∈ [ ν ] µ and Λ ∈ [ λ ] ≤ κ ( resp. Λ ∈ [ λ ] <κ ) such that X is well-connected in Λ.As usual, the negations of these partition relations will be denoted by, e.g., ν hc [ µ ] λ,κ . We now recall certain square principles known as indexed squareprinciples . Definition 3.3 (Cummings-Foreman-Magidor [4]) . Suppose that µ is a singularcardinal. A (cid:3) ind µ, cf( µ ) -sequence is a sequence h C α,i | α ∈ acc( µ + ) , i ( α ) ≤ i < cf( µ ) i such that(1) for all α ∈ acc( µ + ), we have i ( α ) < cf( µ );(2) for all α ∈ acc( µ + ) and all i ( α ) ≤ i < cf( µ ), C α,i is a club in α ;(3) for all α ∈ acc( µ + ) and all i ( α ) ≤ i < j < cf( µ ), we have C α,i ⊆ C α,j ; NOTE ON HIGHLY CONNECTED AND WELL-CONNECTED RAMSEY THEORY 7 (4) for all α < β in acc( µ + ) and all i ( β ) ≤ i < cf( µ ), if α ∈ acc( C β,i ), then i ( α ) ≤ i and C α,i = C β,i ∩ α ;(5) for all α < β in acc( µ + ), there is i such that i ( β ) ≤ i < cf( µ ) and α ∈ acc( C β,i );(6) there is an increasing sequence h µ i | i < cf( µ ) i of regular cardinals suchthat(a) sup( { µ i | i < cf( µ ) } ) = µ ;(b) for all α ∈ acc( µ + ) and all i ( α ) ≤ i < cf( µ ), we have | C α,i | < µ i . (cid:3) ind µ, cf( µ ) is the assertion that there is a (cid:3) ind µ, cf( µ ) -sequence. Definition 3.4 ([8]) . Suppose that λ < µ are infinite regular cardinals. Then a (cid:3) ind ( µ, λ )-sequence is a sequence h C α,i | α ∈ acc( µ ) , i ( α ) ≤ i < λ i such that(1) for all α ∈ acc( µ ), we have i ( α ) < λ ;(2) for all α ∈ acc( µ ) and all i ( α ) ≤ i < λ , C α,i is a club in α ;(3) for all α ∈ acc( µ ) and all i ( α ) ≤ i < j < λ , we have C α,i ⊆ C α,j ;(4) for all α < β in acc( µ ) and all i ( β ) ≤ i < λ , if α ∈ acc( C β,i ), then i ( α ) ≤ i and C α,i = C β,i ∩ α ;(5) for all α < β in acc( µ ), there is i such that i ( β ) ≤ i < λ and α ∈ acc( C β,i );(6) for every club D in µ and every i < λ , there is α ∈ acc( D ) such that D ∩ α = C α,i . (cid:3) ind ( µ, λ ) is the assertion that there is a (cid:3) ind ( µ, λ )-sequence. Remark 3.5.
These indexed square principles follow from more familiar non-indexed square principles. For example, if µ is singular, then (cid:3) µ implies (cid:3) ind µ, cf( µ ) (cf. [7]), while, if λ < µ are regular infinite cardinals, then (cid:3) ( µ ) implies (cid:3) ind ( µ, λ )(cf. [9]). Theorem 3.6.
Suppose that µ is a singular cardinal and (cid:3) µ holds. Then µ + wc [ µ ] µ ) ,< cf( µ ) . Proof.
By Remark 3.5, (cid:3) µ implies (cid:3) ind µ, cf( µ ) . Therefore, we may fix a (cid:3) ind µ, cf( µ ) -sequence h C α,i | α ∈ acc( µ + ) , i ( α ) ≤ i < cf( µ ) i and a sequence h µ i | i < cf( µ ) i ofregular cardinals such that • sup( { µ i | i < cf( µ ) } ) = µ ; • for all α ∈ acc( µ + ) and all i ( α ) ≤ i < cf( µ ), we have | C α,i | < µ i .Define a coloring c : [acc( µ + )] → cf( µ ) by letting c ( α, β ) be the least ordinal i < cf( µ ) such that i ( β ) ≤ i and α ∈ acc( C β,i ) for all α < β in acc( µ + ).We claim that c witnesses the negative relation µ + wc [ µ ] µ ) ,< cf( µ ) . (Moreformally, the composition of c with the unique order-preserving bijection from µ + to acc( µ + ) will witness the negative relation.) To show this, we will prove thatif i < cf( µ ) is a color, Λ ⊆ i , and X ⊆ acc( µ + ) is well-connected in Λ, thenotp( X ) ≤ µ i .To this end, fix a color i < cf( µ ), a set Λ ⊆ i , and a set X ⊆ acc( µ + ) that iswell-connected in Λ. Claim 3.7.
For all α < β in X , we have α ∈ acc( C β,i ) .Proof. The proof will be by induction on the minimal length of a path connecting α and β as in Definition 3.1. CHRIS LAMBIE-HANSON
Fix α < β in X . Since X is well-connected in Λ, we can fix a path ~α = h α , . . . , α n i such that • α = α and α n = β ; • α k ≥ α for all k ≤ n ; and • c ( α k , α k +1 ) ∈ Λ for all k < n .Assume moreover that ~α has minimal length among all such paths. If n = 1, then wewill have c ( α, β ) ∈ Λ. Therefore, there is j < k such that α ∈ acc( C β,j ) ⊆ acc( C β,i ),so α ∈ acc( C β,i ), as desired.Suppose therefore that n > n . In particular, it follows that α ∈ acc( C α n − ,i ). Note also that c ( α n − , β ) ∈ Λ, so either α n − ∈ acc( C β,i ) or viceversa. Let γ = min( { α n − , β } ). Then C α n − ,i ∩ γ = C β,i ∩ γ , so, since α ∈ acc( C α n − ,i ), it follows that α ∈ acc( C β,i ), as desired. (cid:3) Now suppose for sake of contradiction that otp( X ) > µ i . It follows that there is β ∈ X such that otp( X ∩ β ) = µ i . But then, by the claim, we have X ∩ β ⊆ acc( C β,i ),contradicting the fact that | C β,i | < µ i . Therefore, for every color i < cf( µ ), everyΛ ⊆ i , and every set X ⊆ acc( µ + ) that is well-connected in Λ, we have | X | < µ ,and hence c witnesses µ + wc [ µ ] µ ) ,< cf( µ ) . (cid:3) Theorem 3.8.
Suppose that λ < µ are infinite regular cardinals and (cid:3) ( µ ) holds.Then µ wc [ µ ] λ,<λ . Proof.
The proof is quite similar to that of Theorem 3.6, so we only indicate its dif-ferences. By Remark 3.5, we can fix a (cid:3) ind ( µ, λ )-sequence h C α,i | α ∈ acc( µ ) , i ( α ) ≤ i < λ i . Define a coloring c : [acc( µ )] → λ by letting c ( α, β ) be the least ordinal i < λ such that i ( β ) ≤ i and α ∈ acc( C β,i ).We claim that c witnesses the negative partition relation in the statement ofthe theorem. Suppose that i < λ , Λ ⊆ i , and X ⊆ acc( µ ) is well-connected in Λ.Exactly as in the proof of Theorem 3.6, we can prove that, for all α < β in X , wehave α ∈ acc( C β,i ). Now suppose for sake of contradiction that | X | = µ , and let D = S α ∈ X C α,i . Since C β,i end-extends C α,i for all α < β in X , it follows that D is a club in µ .We claim that D ∩ α = C α,i for all α ∈ acc( D ). Indeed, if α ∈ acc( D ), then,letting β = min( X \ ( α + 1)), we have D ∩ β = C β,i , so D ∩ α = C β,i ∩ α .Since α ∈ acc( D ), it follows that α ∈ acc( C β,i ), so, by Definition 3.4, we have C α,i = C β,i ∩ α = D ∩ α . D is then a counterexample to clause (6) of Definition 3.4, so it follows that | X | < µ . Therefore, c witnesses µ wc [ µ ] λ,<λ . (cid:3) Recall that, if µ is a regular uncountable cardinal and (cid:3) ( µ ) fails, then µ is weaklycompact in L. As a result, we immediately obtain the following equiconsistencyresult. Corollary 3.9.
The following statements are equiconsistent over
ZFC . (1) There exists a weakly compact cardinal. (2)
There exist infinite regular cardinals λ < µ such that µ → wc [ µ ] λ,<λ holds. NOTE ON HIGHLY CONNECTED AND WELL-CONNECTED RAMSEY THEORY 9 A sharp positive result at the continuum
In [2, Proposition 8], it is shown that, for all infinite cardinals µ and λ with µ ≤ λ , we have the negative relation µ hc ( µ ) λ . In particular, 2 λ hc (2 λ ) λ . In [2], this was seen to be sharp in the sense that reducing the number of colorsresults in a consistent statement. In particular, it was shown that assuming theconsistency of a weakly compact cardinal, it is consistent that, for example, thepositive relation 2 ℵ → hc (2 ℵ ) ℵ holds.In this section, we show it is sharp in a different sense, namely that increasingthe number of colors allowed to appear in the desired homogeneous set also resultsin a consistent statement. More precisely, we show that assuming the consistency ofa weakly compact cardinal, the positive relation 2 λ → hc [2 λ ] λ, consistently holds.Note that, by Corollary 3.9, the weakly compact cardinal is necessary in both ofthese results.Before stating our theorem, we introduce the following useful definition. Definition 4.1.
Suppose that u and v are sets of ordinals. We say that u and v are aligned if otp( u ) = otp( v ) and, for all α ∈ u ∩ v , we have otp( u ∩ α ) = otp( v ∩ α ).In other words, if an ordinal appears in both u and v , then it appears at the samerelative position in each. Theorem 4.2.
Suppose that λ < θ are infinite regular cardinals and θ is weaklycompact. Let P be the forcing to add θ -many Cohen subsets to λ . Then, afterforcing with P , we have λ → hc [2 λ ] λ ′ , for all λ ′ < λ .Proof. Elements of P are partial functions p : θ → | dom( p ) | < λ ,ordered by reverse inclusion. Note that, in V P , we have 2 λ = θ , so we must showthat θ → hc [ θ ] λ ′ , holds after forcing with P for all λ ′ < θ . Fix a cardinal λ ′ < θ ,a condition p ∈ P and a P -name ˙ c that is forced by p to be a name for a functionfrom [ θ ] to λ ′ . We will find a condition r ≤ p and colors i , i < λ ′ such that r forces the existence of a highly connected subgraph ( ˙ X, ˙ E ) of ( θ, [ θ ] ) such that˙ c “ ˙ E = { i , i } .For all α < β < θ , fix a condition q α,β ≤ p and a color i α,β < λ ′ such that q α,β (cid:13) “ ˙ c ( α, β ) = i α,β ” . Without loss of generality, let us assume that { α, β } ⊆ dom( q α,β ). We will takeadvantage of the weak compactness of θ to find an unbounded A ⊆ θ such thatthe conditions { q α,β | ( α, β ) ∈ [ A ] } enjoy a certain uniformity. For notationalconvenience, for all ( α, β ) ∈ [ A ] , let u α,β = dom( q α,β ).First, we can appeal to the weak compactness of θ to find ordinals i < λ ′ and ξ < λ , a function d : ξ →
2, and an unbounded A ⊆ θ such that, for all( α, β ) ∈ [ A ] , we have(1) i α,β = i ;(2) otp( u α,β ) = ξ ; (3) letting u α,β be enumerated in increasing order as { γ ζ | ζ < ξ } , we havethat q α,β ( γ ζ ) = d ( ζ ) for all ζ < ξ .We now appeal to [2, Lemma 18] (see also [11]), which can be seen as a two-dimensional ∆-system lemma, to find an unbounded A ⊆ A such that(4) for all α ∈ A , the set { u α,β | β ∈ A \ ( α + 1) } is a ∆-system, with root u + α ;(5) letting A ∗ = A \ { min( A ) } , for all β ∈ A ∗ , the set { u α,β | α ∈ A ∩ β } isa ∆-system, with root u − β ;(6) the sets { u + α | α ∈ A } and { u − α | α ∈ A ∗ } are both ∆-systems, with roots u + ∅ and u −∅ , respectively.It is in fact easy to see, given the above discussion, that the roots u + ∅ and u −∅ ofitem (6) must both be equal to the set u ∅ := \ ( α,β ) ∈ [ A ] u α,β Moreover, the proof of [2, Lemma 18] makes it clear that the following can bearranged:(7) for all ( α, β ) , ( γ, δ ) ∈ [ A ] , if { α, β } and { γ, δ } are aligned, then u α,β and u γ,δ are aligned.(See the proof of [3, Lemma 3.3] for similar arguments.)By items (3) and (7) above, it follows that, if ( α, β ) , ( γ, δ ) ∈ [ A ] are aligned,then q α,β ∪ q γ,δ is a condition in P , extending both q α,β and q γ,δ . It also followsthat, for all α ∈ A , we can define a condition q + α by letting q + α = q α,β ↾ u + α forsome β ∈ A \ ( α + 1), and that this definition is independent of our choice of β .Similarly, for all β ∈ A ∗ , we can define q − β by letting q − β = q α,β ↾ u − β for some α ∈ A ∩ β , and we can define q ∅ by letting q ∅ = q α,β ↾ u ∅ for some ( α, β ) ∈ [ A ] .Again, these definitions are independent of our choices.By the previous paragraph, for all ( α, β ) ∈ [ A ∗ ] , we have that q − α ∪ q + β is acondition in P extending both q − α and q + β . For ( α, β ) ∈ [ A ∗ ] , fix a condition r α,β ≤ q − α ∪ q + β and a color j α,β < λ ′ such that r α,β (cid:13) “ ˙ c ( α, β ) = j α,β ” . Let v α,β = dom( r α,β ). Repeating the above process with h r α,β | ( α, β ) ∈ [ A ∗ ] i inplace of h q α,β | ( α, β ) ∈ [ θ ] i , we can find ordinals i < λ ′ and ξ < λ , a function d : ξ →
2, and an unbounded A ⊆ A ∗ such that(8) for all ( α, β ) ∈ [ A ] , we have(a) j α,β = i ;(b) otp( v α,β ) = ξ ;(c) letting v α,β be enumerated in increasing order as { γ ζ | ζ < ξ } , wehave that r α,β ( γ ζ ) = d ( ζ ) for all ζ < ξ ;(9) for all α ∈ A , the set { v α,β | β ∈ A \ ( α + 1) } is a ∆-system, with root v α ;(10) the set { v α | α ∈ A } is a ∆-system, with root v ∅ .As above, we define conditions h r α | α ∈ A i by letting r α = r α,β ↾ v α for some β ∈ A \ ( α + 1), and we define r ∅ = r α,β ↾ v ∅ for some ( α, β ) ∈ [ A ] . Again, thesedefinitions are independent of our choices. NOTE ON HIGHLY CONNECTED AND WELL-CONNECTED RAMSEY THEORY 11
We thin out our set A one final time in the following way. Define a function e : [ A ] → e ( α, β ) = 0 if q α,β and r β are compatible in P , and letting e ( α, β ) = 1 otherwise. Using the weak compactness of θ , find an unbounded A ⊆ A such that e is constant on [ A ] . Claim 4.3. e “[ A ] = { } .Proof. Since e is constant on [ A ] , it suffices to find a single pair ( α, β ) ∈ [ A ] suchthat e ( α, β ) = 0.Fix β ∈ A such that | A ∩ β | ≥ λ . The set { u α,β | α ∈ A ∩ β } forms a ∆-system with root u − β . Moreover, by construction, we have v β ⊇ u − β and r β ≤ q − β .We can therefore find α ∈ A ∩ β such that ( u α,β \ u − β ) ∩ v β = ∅ . Then we have q α,β ↾ u − β = q − β ≥ r β , so q α,β and r β are compatible in P and hence e ( α, β ) = 0. (cid:3) Now make the following assignments: • r = r ∅ ; • ˙ G is the canonical P -name for the P -generic filter; • ˙ X is a P -name for the set { α ∈ A | q + α ∈ ˙ G } ; • ˙ X is a P -name for the set { β ∈ A | r β ∈ ˙ G } ; • ˙ X is a P -name for ˙ X ∪ ˙ X ; • ˙ E is a P -name for ˙ c − ( { i , i } ) ∩ [ ˙ X ] .Notice that each q α,β ≤ p so also q ∅ ≤ p . It similarly follows that r ≤ p . We willend the proof by showing that r forces ( ˙ X, ˙ E ) to be a highly connected graph ofcardinality θ . Claim 4.4. r (cid:13) “ | ˙ X | = θ ” .Proof. We will show that r (cid:13) “ | ˙ X | = θ ”, which suffices. A similar proof will infact show that r (cid:13) “ | ˙ X | = θ ”, as well.Fix an arbitrary condition s ≤ r and an η < θ . It suffices to find α ∈ A \ η suchthat q + α and s are compatible in P . Since s ≤ r , we have dom( s ) ⊇ u ∅ , and s ≤ q ∅ .Recall that the set { u + α | α ∈ A \ η } is a ∆-system with root u ∅ . We can thereforefind α ∈ A such that ( u + α \ u ∅ ) ∩ dom( s ) = ∅ . Since q + α ↾ u ∅ = q ∅ ≥ s , it followsthat q + α and s are compatible in P , as desired. (cid:3) To show that r forces ( ˙ X, ˙ E ) to be highly connected, we prove a couple ofpreliminary claims. Claim 4.5. r (cid:13) “ ∀ ( α, β ) ∈ [ ˙ X ] ∀ η < θ ∃ γ ∈ ˙ X \ η [ { q α,γ , q β,γ } ⊆ ˙ G ]” .Proof. Fix an ordinal η < θ , a condition s ≤ r and ( α, β ) ∈ [ A ] such that s forcesboth α and β to be in ˙ X . Without loss of generality, we can assume that s ≤ q + α and s ≤ q + β and that η > β . It will suffice to find γ ∈ A \ η such that the conditions s , q α,γ , q β,γ , and r γ are all pairwise compatible, since then the union of these fourconditions would itself be a condition extending s and forcing γ to be as desired.Note also that, for all γ ∈ A \ η , we know that q α,γ and q β,γ are compatible, since { α, γ } and { β, γ } are aligned. We also know that r γ is compatible with each of q α,γ and q β,γ by Claim 4.3. It therefore suffices to find γ ∈ A \ η such that s iscompatible with each of q α,γ , q β,γ , and r γ .By assumption, we know that dom( s ) ⊇ u + α ∪ u + β ∪ v ∅ and that s extends eachof q + α , q + β , and r ∅ . Recall also that the sets { u α,γ | γ ∈ A \ η } , { u β,γ | γ ∈ A \ η } , and { v γ | γ ∈ A \ η } are ∆-systems with roots u + α , u + β , and v ∅ , respectively. Wecan therefore find γ ∈ A \ η such that each of the sets ( u α,γ \ u + α ), ( u β,γ \ u + β ), and v γ \ v ∅ is disjoint from dom( s ). But then we have • q α,γ ↾ dom( s ) = q + α ; • q β,γ ↾ dom( s ) = q + β ; and • r γ ↾ dom( s ) = r ∅ .Therefore, since s extends each of q + α , q + β , and r ∅ , it follows that s is compatiblewith each of q α,γ , q β,γ , and r γ , as desired. (cid:3) Claim 4.6. r (cid:13) “ ∀ α ∈ ˙ X ∀ η < λ ∃ β ∈ ˙ X \ η [ r α,β ∈ ˙ G ]” .Proof. Fix an ordinal η < θ , a condition s ≤ r and α ∈ A such that s (cid:13) “ α ∈ ˙ X ”.Without loss of generality, assume that η > α and s ≤ r α , and hence dom( s ) ⊇ v α .The set { v α,β | β ∈ A \ η } is a ∆-system with root v α ; we can therefore find β ∈ A \ η such that ( v α,β \ v α ) ∩ dom( s ) = ∅ . We know that r α,β ↾ dom( s ) = r α ≥ s , so itfollows that s and r α,β are compatible. Recall that r α,β ≤ q + β and therefore forces β to be in ˙ X . Therefore, the union of s and r α,β forces β to be as desired. (cid:3) Let G be a P -generic filter over V with r ∈ G . Let c , X , X , X , and E bethe realizations of ˙ c , ˙ X , ˙ X , ˙ X , and ˙ E , respectively, in V [ G ]. By the definitionof ˙ E , we know that c “[ E ] ⊆ { i , i } . By Claim 4.4, we know that | X | = θ . Itthus remains to show that, for all Y ∈ [ X ] <θ , the graph ( X \ Y, E ∩ [ X \ Y ] ) isconnected.Fix Y ∈ [ X ] <θ , and let Z = X \ Y . Also fix ( α, β ) ∈ [ Z ] . Since θ is regular and | Y | < | X | = θ , there is η < θ such that Y ⊆ η , and hence X \ η ⊆ Z . By increasing η if necessarily, we may assume that β < η . There are now a number of cases, notnecessarily mutually exclusive, to consider. Case 1: α, β ∈ X . In this case, Claim 4.5 implies that there is γ ∈ X \ η suchthat { q α,γ , q β,γ } ⊆ G . It follows that c ( α, γ ) = c ( α, γ ) = i , so h α, γ, β i is a pathfrom α to β in ( Z, E ∩ [ Z ] ). Case 2: α ∈ X and β ∈ X . By Claim 4.6, we can find γ ∈ X \ η such that r β,γ ∈ G . Then, by Claim 4.5, we can find δ ∈ X \ ( γ +1) such that { q α,δ , q γ,δ } ⊆ G .It follows that c ( α, δ ) = c ( γ, δ ) = i and c ( β, γ ) = i , so h α, δ, γ, β i is a path from α to β in ( Z, E ∩ [ Z ] ). Case 3: α ∈ X and β ∈ X . This is symmetric to Case 2.
Case 4: α, β ∈ X . By Claim 4.6, we can first find γ ∈ X \ η such that r α,γ ∈ G and then δ ∈ X \ ( γ + 1) such that r β,δ ∈ G . Then, by Claim 4.5, we can find ǫ ∈ X \ ( δ + 1) such that { q γ,ǫ , q δ,ǫ } ⊆ G . It follows that c ( α, γ ) = c ( β, δ ) = i and c ( γ, ǫ ) = c ( δ, ǫ ) = i , so h α, γ, ǫ, δ, β i is a path from α to β in ( Z, E ∩ [ Z ] ).This exhausts all possible cases, so we have shown that, in V [ G ], ( X, E ) is highlyconnected, thus finishing the proof. (cid:3)
Remark 4.7.
We have seen that Theorem 4.2 is sharp in the sense that the con-clusion cannot be improved to 2 λ → hc (2 λ ) λ . It is also sharp in the sense that the“ hc ” subscript cannot be dropped. For example, it is easily seen that the coloring∆ : [ λ → λ defined by letting ∆( f, g ) be the least i < λ such that f ( i ) = g ( i )witnesses the negative square bracket relation2 λ [ ℵ ] λ,< ℵ , NOTE ON HIGHLY CONNECTED AND WELL-CONNECTED RAMSEY THEORY 13 and, more generally, if ν ≤ λ , µ ≤ λ , and 2 χ < µ for all χ < ν , then ∆ witnesses2 λ [ µ ] λ,<ν . Open questions
A number of questions remain open. We first want to reiterate a question ofBergfalk, Hruˇs´ak, and Shelah.
Question 5.1 (Bergfalk-Hruˇs´ak-Shelah [2]) . Is ℵ → hc ( ℵ ) ℵ consistent? We also want to emphasize the analogous question at ℵ ω +1 , which was also askedin [1]. Question 5.2 (Bergfalk [1]) . Is ℵ ω +1 → hc ( ℵ ω ) ℵ consistent? If so, what about ℵ ω +1 → hc ( ℵ ω +1 ) ℵ ? A more open-ended, speculative question involves generalizations of the partitionrelations being studied to higher dimensions. In the case of the classical partitionrelation ν → ( µ ) λ , it is clear how to generalize to ν → ( µ ) kλ for k >
2. In thecase of ν → hc ( µ ) λ or ν → wc ( µ ) λ , however, such a generalization would requireisolating the correct definition(s) of “highly connected” and “well-connected” in thecontext of k -uniform hypergraphs. There are a number of different approaches onemight take to this generalization, but it is presently unclear, at least to us, which, ifany, of these approaches yields an interesting theory of higher-dimensional partitionrelations. We therefore ask the following deliberately vague problem. Problem 5.3.
Isolate the correct definition(s) for a generalization (or generaliza-tions) of highly connected or well-connected Ramsey theory to higher dimensions.
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Department of Mathematics and Applied Mathematics, Virginia Commonwealth Uni-versity, Richmond, VA 23284, United States
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