KNASTER AND FRIENDS III: SUBADDITIVE COLORINGS

Abstract We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals 
$\theta < \kappa $
 , the existence of a strongly unbounded coloring 
$c:[\kappa ]^2 \rightarrow \theta $
 is a theorem of 
$\textsf{ZFC}$
 . Adding the requirement of subadditivity to a strongly unbounded coloring is a significant strengthening, though, and here we see that in many cases the existence of a subadditive strongly unbounded coloring 
$c:[\kappa ]^2 \rightarrow \theta $
 is independent of 
$\textsf{ZFC}$
 . We connect the existence of subadditive strongly unbounded colorings with a number of other infinitary combinatorial principles, including the narrow system property, the existence of 
$\kappa $
 -Aronszajn trees with ascent paths, and square principles. In particular, we show that the existence of a closed, subadditive, strongly unbounded coloring 
$c:[\kappa ]^2 \rightarrow \theta $
 is equivalent to a certain weak indexed square principle 
$\boxminus ^{\operatorname {\mathrm {ind}}}(\kappa , \theta )$
 . We conclude the paper with an application to the failure of the infinite productivity of 
$\kappa $
 -stationarily layered posets, answering a question of Cox.


Introduction
For infinite regular cardinals θ < κ, the positive partition relation κ → (κ) 2 θ , which asserts that every coloring c : [κ] 2 → θ has a homogeneous set of cardinality κ, is equivalent to κ being weakly compact.For non-weakly-compact cardinals κ, though, one can seek to measure the incompactness of κ by asking whether certain strengthenings of the negative relation κ (κ) 2 θ hold.One natural such strenthening is to require that there exist colorings c : [κ] 2 → θ witnessing certain strong unboundedness properties.In [LHR18], which forms Part I of this series of papers, the authors introduce the following coloring principle, which asserts the existence of such strongly unbounded colorings, and use it to answer questions about the infinite productivity of the κ-Knaster condition for uncountable κ.
Definition 1.1.U(κ, µ, θ, χ) asserts the existence of a coloring c : [κ] 2 → θ such that for every σ < χ, every pairwise disjoint subfamily A ⊆ [κ] σ of size κ, and every i < θ, there exists B ∈ [A] µ such that min(c[a × b]) > i for all (a, b) Much of [LHR18] is devoted to analyzing situations in which U(. ..) necessarily holds and, moreover, is witnessed by closed or somewhere-closed colorings (see Definition 2.1 below).In Part II of this series [LHR21], we studied Cspec(κ), the C-sequence spectrum of κ (see Definition 5.5 below), which is another measure of the incompactness of κ, and found some unexpected connections between Cspec(κ) and the validity of instances of U(κ, . ..).
In this paper, which can be read largely independently of [LHR18,LHR21], we investigate subadditive witnesses to U(. ..).
Adding the requirement of subadditivity significantly strengthens the coloring principle, and we prove that the existence of closed, subadditive witnesses to U(. ..) is equivalent to a certain indexed square principle.Our first main result improves Clause (1) of [LHR18, Theorem A].
We also prove that a version of square with built-in diamond for a singular cardinal λ gives rise to somewhere-closed subadditive witnesses to U(λ + , . ..), which in turn imply that the C-sequence spectrum of λ + is rich: Theorem B. Suppose that λ is a singular cardinal, f is a scale for λ in some product λ, and ♦( λ) holds.Let Σ denote the set of good points for f .
For a pair of infinite regular cardinals θ < κ and a coloring c : [κ] 2 → θ, an interesting facet of the study of the unboundedness properties of c is the set ∂(c) of its levels of divergence (see Definition 3.22 below).Any coloring c for which ∂(c) is stationary is automatically a somewhere-closed witness to U(κ, κ, θ, θ).We prove that the existence of a (fully) closed witness c to U(κ, κ, θ, θ) for which ∂(c) is stationary is equivalent to the existence of a nonreflecting stationary subset of E κ θ , and that the existence of a nonreflecting stationary subset of E κ θ does not suffice to yield a subadditive witness to U(κ, 2, θ, 2).We have three main consistency results concerning the characteristic ∂(c): (1) For any pair of infinite regular cardinals θ < κ, there is a κstrategically closed, θ + -directed closed forcing notion that adds a subadditive witness c to U(κ, κ, θ, θ) for which ∂(c) is stationary; (2) For any pair of infinite regular cardinals θ < κ, there is a κ-strategically closed, θ-directed closed forcing notion that adds a closed subadditive witness c to U(κ, κ, θ, θ) for which ∂(c) is stationary; (3) For regular uncountable cardinals θ < λ < κ such that λ is supercompact and κ is weakly compact, there is a forcing extension in which (κ, θ) fails, yet, there is a closed, subadditive witness c to U(κ, κ, θ, θ) for which ∂(c) is stationary.
On the Ramsey-theoretic side, we prove that in the presence of large cardinals, for many pairs of infinite regular cardinals θ < κ, κ → [κ] 2 θ,finite holds restricted to the class of subadditive colorings (in particular, refuting subadditive instances of U(κ, 2, θ, 2)), and that similar results hold at small cardinals in forcing extensions or in the presence of forcing axioms.
On the anti-Ramsey-theoretic side, we have a result reminiscent of the motivating result of [LHR18] concerning the infinite productivity of strong forms of the κ-chain condition.When combined with Theorem A, the next theorem shows that (κ) yields a gallery of counterexamples to productivity of κ-stationarily layered posets, answering a question of Cox [Cox17].
Theorem D. Suppose that θ ≤ χ < κ are infinite, regular cardinals, κ is (<χ)inaccessible, and there is a closed and subadditive witness c to U(κ, 2, θ, 2).Then there is a sequence of posets P i | i < θ such that: (1) for all i < θ, P i is well-met and χ-directed closed with greatest lower bounds; (2) for all j < θ, i<j P i is κ-stationarily layered; (3) i<θ P i is not κ-cc.If, in addition, ∂(c) ∩ E κ χ is stationary, then the sequence P i | i < θ can be made constant.
As a corollary, we get that Magidor's forcing for changing the cofinality of a measurable cardinal λ to a regular cardinal θ < λ adds a poset P whose θ th power is not λ + -cc, but all of whose lower powers are λ + -stationarily layered.
In Section 3, we review the notion of subadditivity and some of its variations and prove that any subadditive witness to U(κ, 2, θ, 2) is in fact a witness to U(κ, µ, θ, χ) for all µ < κ and all χ ≤ cf(θ) (and, under certain closure assumptions, even stronger principles).Subsection 3.1 contains results connecting subadditive strongly unbounded colorings to narrow systems and trees with ascent paths.In Subsection 3.2, we discuss locally small colorings of the form c : [λ + ] 2 → cf(λ), focusing in particular on the case in which λ is a singular cardinal.Locally small colorings are necessarily witnesses to U(λ + , 2, cf(λ), cf(λ)), and retain this property in any outer model with the same cardinals.In Subsection 3.3, we introduce a subset ∂(c) ⊆ κ associated with a coloring c : [κ] 2 → θ that is useful in the analysis of U(κ, µ, θ, χ), particularly in the context of subadditive colorings.We then introduce a forcing notion that establishes Clause (1) of Theorem C. Subsection 3.4 contains a number of results indicating the extent to which various compactness principles place limits on the existence of certain subadditive witnesses to U(κ, µ, θ, χ).In particular, it is shown that simultaneous stationary reflection, the existence of highly complete or indecomposable ultrafilters, and the P-ideal dichotomy all have such an effect.
In Section 4, we introduce an indexed square principle ⊟ ind (κ, θ) and prove that it is equivalent to the existence of a closed, subadditive witness to U(κ, 2, θ, 2), thereby establishing the first part of Theorem A. We also prove a consistency result indicating that ⊟ ind (κ, θ) is a proper weakening of ind (κ, θ) and does not even imply (κ, θ), in the process proving Clause (3) of Theorem C. Section 4 also contains the proof of the second part of Theorem A and the proof of Clause (2) of Theorem C.
Section 5 is concerned with successors of singular cardinals.We begin by proving Theorem B, showing that a certain square with built-in diamond sequence on a singular cardinal λ entails the existence of a subadditive witness to U(λ + , λ + , θ, λ) for all θ ∈ Reg(λ)\ (cf (λ)+ 1).We then present an improvement upon a result from Part I of this series proving the existence of closed witnesses to U(λ + , λ + , θ, cf(λ)) for all singular λ whose cofinality is not greatly Mahlo and all θ ≤ cf(λ).
Section 6 deals with the infinity productivity of κ-stationarily layered posets and contains our proof of Theorem D. 1.2.Notation and conventions.Throughout the paper, κ denotes a regular uncountable cardinal, and χ, θ, and µ denote cardinals ≤ κ. λ will always denote an infinite cardinal.We say that κ is χ-inaccessible iff, for all ν < κ, ν χ < κ, and say that κ is (<χ)-inaccessible iff, for all ν < κ and µ < χ, ν µ < κ.We denote by H Υ the collection of all sets of hereditary cardinality less Υ, where Υ is a regular cardinal sufficiently large to ensure that all objects of interest are in H Υ .
For a set of ordinals a, we write ssup(a , nacc(a) := a \ acc(a), and cl(a) := a ∪ acc + (a).For sets of ordinals a and b, we write a < b if, for all α ∈ a and all β ∈ b, we have α < β.For a set of ordinals a and an ordinal β, we write a < β instead of a < {β} and β < a instead of {β} < a.
For any set A, we write of all unordered pairs from A. In some scenarios, we will also be interested in ordered pairs from A. In particular, if A is either a set of ordinals or a collection of sets of ordinals, then we will abuse notation and write (a, b) ∈ [A] 2 to mean {a, b} ∈ [A] 2 and a < b.

Preliminaries
In this brief section, we recall a key definition and present a few useful facts about U(. ..).We start by recalling the following definition from [LHR18] concerning closed colorings.
(1) For all β < κ and i ≤ θ, we let The following fact is a useful tool for proving that certain colorings satisfy strong instances of U(. ..).
(1) Towards a contradiction, suppose that A is a counterexample.Then, for every ǫ < κ, there exists a large enough ≤i * (β).(4) Towards a contradiction, suppose that S and H form a counterexample.For every ǫ ∈ S, fix a function h ǫ ∈ H and a club The following is a corollary to a result from [LHR18] which we never took the time to derive.
Here we deal with a pseudo-inverse: Proof.In V , let c : [κ] 2 → θ be a witness to U(κ, 2, θ, χ).Let G be Add(θ, 1)generic over V , and work for now in V Add(θ,1) .Let g : θ → θ be the Cohen-generic function, and set d := g •c.To see that d witnesses Pr 1 (κ, κ, θ, χ), let A be a κ-sized pairwise disjoint subfamily of [κ] <χ , and let τ < θ; we need to find (a, b) Ȧ ∈ V be an Add(θ, 1)-name for A, we can fix for each a ∈ A a condition p a ∈ G such that p a "a ∈ Ȧ.As χ ≤ θ <θ < κ, and hence | Add(θ, 1)| < κ, by passing to a subfamily if necessary, we may assume that there is a fixed p * ∈ G such that p a = p * for all a ∈ A, and therefore A = {a ∈ [κ] <χ | p "a ∈ Ȧ"} is in V .We therefore move back to V and run a density argument.Let p : i → θ be an arbitrary condition in Add(θ, 1) below p * .By the hypothesis on c, pick (a, b) Let j := ssup(x), and let q : j → θ be some condition extending p and satisfying q(ξ) = τ for all ξ ∈ x.Clearly, q(c(α, β)) = τ for all (α, β) ∈ a × b, so q forces that d[a × b] = {τ }, as sought.
We conclude this short section with another simple fact worth recording.
Proof.For each i < θ, we define an ordering < i on κ by letting α < i β iff α < β and c(α, β) ≤ i.The fact that < i is transitive follows from the fact that c is subadditive of the first kind.
In addition, by the hypothesis on c, for every ⊠ (1) Fix µ ≤ κ, and suppose that we are given some A ∈ [κ] κ and i < θ.We would like to find B ∈ [A] µ such that c(α, β) > i for all (α, β) ∈ [B] 2 .In particular, if there exists B ∈ [κ] µ which is an antichain with respect to < i , then we are done.Hereafter, suppose this is not the case.
◮ If µ < κ, then (A, < i ) is a tree of size κ all of whose levels have size < µ.Since µ < κ and κ is regular, a result of Kurepa [Kur77] implies that (κ, < i ) has a branch of size κ, contradicting the preceding claim.
Proof.Suppose not.Let T := D ∩ Σ, and note that, for all γ ∈ T and all a ∈ A with a > γ, for some β ∈ a, we have c(γ, β) ≤ i.Consider the tree (T, < i ).We claim that it has no antichains of size χ.To see this, fix an arbitrary X ⊆ T of order type χ, and let δ := sup(X).Fix an arbitrary a ∈ A with a > δ.For all γ ∈ X, since a > γ, we may find some β ∈ a with c(γ, β) ≤ i.
(2) By subadditivity of the first kind.
We thus arrive at the following Ramsey-theoretic result.
Next, we prove a pair of lemmas linking U subadditive (. ..) to the existence of trees with ascent paths but no cofinal branches.We first recall the following definition.Definition 3.8.Suppose that T = (T, < T ) is a tree of height κ.
Proof.Otherwise, by a standard argument (e.g., the proof of [Rin14, Corollary 2.6]), there exists i < θ for which ρ i 2 admits a homogeneous set of size κ.Fix such an i.By [Tod07, Theorem 6.3.2],then, we may fix a club C ⊆ κ such that, for every α < κ, there exists β < κ such that C ∩ α ⊆ C i β .By the definition of C i , for every α < κ with otp(C ∩ α) > θ, it follows that there exist j < θ and β < κ such that . By the pigeonhole principle, we may now fix j < κ such that for every α < κ, there exists To see that f α | α < κ forms a θ-ascent path through our tree, fix arbitrary α < β < κ.Write γ := min(S\α) and δ := min(S\β).
Proof.Given A and i as above, fix a large enough j < cf(λ) such that j ≥ i and >λj , and pick any a ∈ A j with γ < a.
The following result, due independently to Shani and Lambie-Hanson, shows that the hypothesis of λ cannot be weakened to λ,2 in Lemma 3.15(2).(We note that GCH is not explicitly mentioned in the quoted results, but it is evident from their proofs that, if GCH holds in the relevant ground models, then it continues to hold in the forcing extensions witnessing the conclusion of the result.)Fact 3.16 (Shani, [Sha16, Theorem 1], Lambie-Hanson, [LH17a, Corollaries 5.13 and 5.14]).Relative to the existence of large cardinals, it is consistent with GCH that there is an uncountable cardinal λ such that λ,2 holds, and, for every θ < λ, U subadditive (λ + , 2, θ, 2) fails.λ can be either regular or singular here, though attaining the result for singular λ requires significantly larger cardinals than attaining it for regular λ.
To this end, fix arbitrary γ ∈ E λ + >cf(λ) and a stationary s ⊆ γ.As cf(γ) > cf(λ), there exists a large enough i < cf(λ) such that S γ := D c ≤i (γ) ∩ s is stationary in γ.Since c is subadditive of the second kind, for any pair (α, Corollary 3.18.If λ is a singular strong limit and there exists a locally small and subadditive coloring c : Proof.By the preceding proposition, the hypothesis imply that E λ + >cf(λ) ∈ I[λ + ; λ].Now, given that λ is a strong limit, we moreover get that λ meaning that AP λ holds.
3.3.Forms of coherence and levels of divergence.We will also be interested in variants of subadditivity, as captured by the next definitions.
(1) c is ℓ ∞ -coherent iff for all γ < δ < κ, there is j < θ such that, for all i < θ, Motivated by the proof of Lemma 3.11, we introduce the following definition.
We will be particularly interested in situations in which ∂(c) is stationary in κ; one reason for this is the following lemma, indicating that colorings c for which ∂(c) is stationary automatically witness an instance of U(. ..).Lemma 3.24.Suppose that c : [κ] 2 → θ is a coloring for which ∂(c) is stationary.Then c witnesses U(κ, κ, θ, cf(θ)).
Proof.Suppose that there exists a uniformly coherent κ-Souslin tree.This means that there exists a downward closed subfamily T ⊆ <κ 2 such that: is in T .
We now show that the existence of a coloring c for which ∂(c) is stationary is in fact equivalent to the existence of a nonreflecting stationary subset of E κ θ .
Remark 3.32.By Fact 3.16, the preceding lemma cannot be strengthened to assert that the existence of a nonreflecting stationary subset of E κ θ gives rise to a subadditive coloring c : [κ] 2 → θ for which ∂(c) is stationary.In fact, a nonreflecting stationary subset of E κ θ is not even enough to imply the existence of a coloring c : [κ] 2 → θ such that ∂(c) is stationary and c is weakly subadditive of the first kind.This is because, by Theorem 3.45 below, PFA implies that, for example, any witness to U(ω 3 , 2, ω, 2) is not weakly subadditive of the first kind, whereas, by a result of Beaudoin (see the remark at the end of [Bea91, §2]), PFA is consistent with the existence of a nonreflecting stationary subset of E ω3 ω .
By Lemma 3.15(1) and Lemma 3.27(2), for every infinite regular cardinal λ, there exists a locally small coloring c : [λ + ] 2 → λ that is λ-coherent.We shall now prove that for every singular cardinal λ, a locally small coloring c : [λ + ] 2 → cf(λ) is never cf(λ)-coherent.Assuming that c is subadditive of the first kind (which is indeed possible, by Lemma 3.14), even weaker forms of coherence are not feasible.
(1) If λ is regular or if c is subadditive of the first kind, then for every cardinal Proof.The proof is similar to that of [Kön03, Theorem 3.7].Suppose for sake of contradiction that c is θ-coherent for some fixed cardinal θ < λ and that either c is subadditive of the first kind or θ ≤ cf(λ).
We will define a forcing notion P whose generic object will generate a coloring c as above.Our poset P consists of all subadditive colorings of the form p : • for all β ≤ γ p and all i < θ, we have |D p ≤i (β)| < λ i .P is ordered by reverse inclusion.We also include the empty set as the unique maximal condition in P.
Proof.Note that P is tree-like, i.e., if p, q, r ∈ P and r extends both p and q, then p and q are ≤ P -comparable.It therefore suffices to prove that P is θ + -closed.To this end, suppose that ξ < θ + and p = p η | η < ξ is a decreasing sequence of conditions in P. We may assume without loss of generality that ξ is an infinite regular cardinal and p is strictly decreasing, i.e., γ η | η < ξ is strictly increasing, where γ η denotes γ pη .For all η < ξ, by possibly extending p η to copy some information from p η+1 , we may also assume that γ η is a successor ordinal.Let γ * := sup{γ η | η < ξ}, and let q * := η<ξ p η .Note that q * is not a condition in P, since its domain is not the square of a successor ordinal.We will extend it to a condition q : [γ * + 1] 2 → θ, which will then be a lower bound for p.To do so, it suffices to specify q(α, γ * ) for all α < γ * .There are two cases to consider: ◮ Assume that ξ < θ.We can then fix an i * < θ such that q * (γ η , γ η ′ ) ≤ i * for all η < η ′ < ξ.Now, given α < γ * , let η α < ξ be the least η for which α < γ η and then set q(α, γ * ) := max{i * , q * (α, γ ηα )}.It is straightforward to prove, using our choice of i * and the fact that each p η is subadditive, that the coloring q thus defined is also subadditive.
Since each p η is E γη+1 ≥θ -closed, in order to show that q is E γ * +1 ≥θ -closed, it suffices to prove that for all A ⊆ γ * and all i < θ such that A ⊆ D q ≤i (γ * ) and sup(A) ∈ E γ * ≥θ , we have sup(A) ∈ D q ≤i (γ * ).To this end, fix such an A and i.Let β := sup(A).By our choice of i * , we know that i ≥ i * and q(α, γ * ) = max{i * , p η β (α, γ η β )} for all α ∈ A ∪ {β}.By the fact that p η β is E γη β +1 ≥θ -closed, we know that p η β (β) ≤ i, so β ∈ D q ≤i (γ * ).To show that q is a condition, it remains only to verify that |D q ≤i (γ * )| < λ i for all i < θ.To this end, fix i < θ.By our construction, we have D q ≤i (γ * ) ⊆ η<ξ D pη ≤i (γ η ).Since ξ < λ i and λ i is regular, the fact that each p η is a condition in P then implies that |D q ≤i (γ * )| < λ i .◮ Assume that ξ = θ.Fix a strictly increasing sequence i η | η < θ of ordinals below θ such that, for all η < η ′ < θ, we have q * (γ η , γ η ′ ) ≤ i η ′ .Now, given α < γ * , let η α < θ be the least η for which α < γ η and then set q(α, γ * ) := max{i ηα , q * (α, γ ηα )}.It is again straightforward to prove that the coloring q thus defined is subadditive.The verification involves a case analysis; to illustrate the type of argument involved, we go through the proof of one of the required inequalities in one of the cases, leaving the other similar arguments to the reader.
We now verify that q is E γ * +1 ≥θ -closed.As in the previous case, we fix an A ⊆ γ * and an i < θ such that A ⊆ D q ≤i (γ * ) and β := sup(A) is in E γ * ≥θ .It will suffice to show that β ∈ D q ≤i (γ * ).To avoid triviality, assume that β / ∈ A. Since β is a limit ordinal we know that β is not equal to γ η for any η < θ.It follows that, by passing to a tail of A if necessary, we may assume that η α = η β for all α ∈ A. Then, for all α ∈ A ∪ {β}, we have q(α, γ ).The fact that |D q ≤i (γ * )| < λ i for all i < θ follows by exactly the same reasoning as in the previous case.⊠ Let ċ be the canonical P-name for the union of the generic filter.Then ċ is forced to be a subadditive function from an initial segment of [κ] 2 to θ (we will see shortly that its domain is forced to be all of [κ] 2 ).
Note that in the ξ = θ case of the above claim, we actually proved something stronger that will be useful later: if p = p η | η < θ is a strictly decreasing sequence of conditions in P and γ := sup{γ pη | η < θ}, then there is a lower bound q for p such that q P "γ ∈ ∂( ċ)".
The next claim will show that P is (<κ)-distributive.
Proof.In Case (1), denote χ := κ.In Case (2), denote χ := λ.We describe a winning strategy for Player II in χ (P).(Note that, if χ = λ, then it appears that we are just showing that P is λ-strategically closed, but the fact that P is θ + -closed will then show that P is in fact λ + 1-strategically closed.)We will arrange so that, if p η | η < χ is a play of the game in which Player II plays according to their prescribed strategy, then, letting γ η := γ pη for all η < χ, (a) γ η | η < χ is a nonzero even ordinal is a continuous, strictly increasing sequence; (b) for all even ordinals η < ξ < χ, we have p ξ (γ η , γ ξ ) = min{i < θ | ξ < λ i }.Now suppose that ξ < χ is an even ordinal and p η | η < ξ is a partial run of χ (P) that thus far satisfies requirements (a) and (b) above.We will describe a strategy for Player II to choose the next play, p ξ , while maintaining (a) and (b).
For all α < γ * , let η α < ξ be the least even ordinal η such that α < γ η , and then set p ξ (α, γ * ) := max{i * , p * (α, γ ηα )}.By the fact that the play of the game thus far satisfied (b), we know that p * (γ η , γ η ′ ) ≤ i * for all even ordinals η < η ′ < ξ, so this definition does in fact ensure that p ξ (γ η , γ ξ ) = i * , so we have satisfied (b).The fact that the play of the game thus far satisfied (a) and (b) also implies that p ξ is subadditive and E γ ξ +1 ≥θ -closed and that we have continued to satisfy requirement (a).Finally, to show that |D p ξ ≤i (γ ξ )| < λ i for all i < θ, note firstly that D p ξ ≤i (γ ξ ) = ∅ for all i < i * and, secondly, that for all i ∈ [i * , θ), we have D For each i ∈ [i * , θ), we know that ξ < λ i and λ i is regular, so the fact that p η is a condition for all η < ξ implies that |D This completes the description of Player II's winning strategy and hence the proof of the claim. ⊠ By the argument of the proof of the above claim, it follows that, for every α < κ, the set E α of p ∈ P for which γ p ≥ α is dense in P. Therefore, the domain of ċ is forced to be [κ] 2 .The definition of P also immediately implies that ċ is forced to be E κ ≥θ -closed and, in Case (2), ċ is also forced to be locally small.We now finish the proof of the theorem by showing that ∂( ċ) is forced to be stationary.(By Lemma 3.24, this will imply that ċ witnesses U subadditive (κ, κ, θ, θ).)To this end, fix a condition p and a P-name Ḋ forced to be a club in κ.We will find q ≤ p and γ < κ such that q P "γ ∈ Ḋ ∩ ∂( ċ)".
(3) If c is locally small, then c is not subadditive.
Proof.Suppose that c is subadditive of the first kind.For each α < κ, pick i α < θ, for which the following set is stationary: Next, using the pigeonhole principle, fix H ∈ [κ] κ and i < θ such that i α = i for all α ∈ H. Claim 3.35.1.For every A ∈ [H] θ , there is β A < κ above sup(A) such that sup α∈A c(α, β A ) < θ.
We next show that the existence of subadditive witnesses to U(. ..) is ruled out by the existence of certain ultrafilters.Definition 3.37.An ultrafilter U over κ is θ-indecomposable if it is uniform and, for every sequence of sets (1) If there exists a θ + -complete uniform ultrafilter over κ, then c is not weakly subadditive; (2) If there exists a θ + -complete uniform ultrafilter over κ and κ is θ-inaccessible, then c is not weakly subadditive of the first kind; (3) If there exists a θ-indecomposable ultrafilter over κ, then c is not subadditive of the second kind. Proof.
(1) Suppose that U is a θ + -complete ultrafilter over κ.For all α < κ and i < θ, let ).Since U is, in particular, a θ-indecomposable ultrafilter over κ, we may find some i α < θ such that A iα α ∈ U .Next, using the pigeonhole principle, let us fix H ∈ [κ] κ and i < θ such that i α = i for all α ∈ H.As U is closed under intersections of length θ, for every A ∈ [H] θ , we may let , contradicting the choice of A.
(2) Let H, i and the notation β A be as in the proof of Clause (1).Assuming that κ is θ-inaccessible, and using Lemma 2.4(1), we may find some A ∈ [H] θ such that, for cofinally many β < κ, {c(α, β) | α ∈ A} is unbounded in θ.In particular, we may find such a β < κ above β A .If c were weakly subadditive of the first kind, then we could find j < θ such that D c ≤i (β A ) ⊆ D c ≤j (β).However, for every j < θ, there exists α ∈ A such that c(α, β) > j, so that α ∈ D c ≤i (β A ) \ D c ≤j (β).This is a contradiction.
Complementary to Lemma 3.14, we obtain the following.
Corollary 3.42.Suppose that c : [κ] 2 → θ is a witness to U(κ, 2, θ, 2) and there exists a strongly compact cardinal in the interval (θ, κ]. (1) c is not weakly subadditive; (2) If κ is not the successor a singular cardinal of cofinality θ, then c is not weakly subadditive of the first kind; (3) c is not subadditive of the second kind.
Proof.The hypothesis entails the existence of a θ + -complete uniform ultrafilter over κ, and in particular, the existence of a θ-indecomposable uniform ultrafilter over κ.In addition, if κ is not the successor a singular cardinal of cofinality θ, then by Solovay's theorem that SCH holds above a strongly compact cardinal [Sol74], κ is θ-inaccessible.Now appeal to Lemma 3.38.
Corollary 3.44.It is consistent that all of the following hold simultaneously: • GCH; Proof.By Lemma 3.15(1) and Fact 3.16.
(1) If κ ≥ ℵ 2 , then c is not weakly subadditive; (2) If κ > 2 ℵ0 is not the successor of a singular cardinal of countable cofinality, then c is not weakly subadditive of the first kind.
Proof.Suppose not.For all X ∈ [κ] ≤ℵ0 and β < κ, define a function In this case, we shall denote such a set Γ by Γ X and min(Γ X ) by γ X .Now, let I be the collection of all X ∈ [κ] ≤ℵ0 such that, for every β < κ, f β X is finite-to-one.It is clear that I is an ideal.Claim 3.45.1.Let Z ∈ [κ] <κ .Then there exists an ordinal ǫ Z ∈ [ssup(Z), κ) such that, for every X ∈ [Z] ≤ℵ0 , X ∈ I iff there exists γ ∈ [ǫ Z , κ) for which f γ X is finite-to-one.
Proof.◮ If c is weakly subadditive, then set ǫ Z := ssup(Z).Towards a contradiction, suppose that there exist X ∈ [Z] ≤ℵ0 and γ ∈ [ǫ Z , κ) such that f γ X is finite-toone, yet X ∈ I. Fix β < κ such that f β X is not finite-to-one, and then fix i < ω for which is not the successor of a singular cardinal of countable cofinality, then by Viale's theorem [Via08] that PID implies SCH, the fact that |Z| < κ implies that and use weak subadditivity of the first kind to find j < θ such that ) is a pseudo-union for X.In addition, for every i < ω, Y ∩ D c ≤i (γ) is covered by the finite set n<i (X n ∩ D c ≤i (γ)), so, by the preceding claim, Y ∈ I. Finally, by PID, one of the following alternatives must hold: (1) There exists There exists B ∈ [κ] ℵ1 such that [B] ℵ0 ⊆ I.In Case (1), given A ∈ [κ] κ , pick some strictly increasing function g : κ → A such that g(α) > ǫ A∩ssup(g[α]) for all α < κ.In effect, A ′ := Im(g) is a cofinal subset of A such that ǫ A ′ ∩γ < γ for every γ ∈ A ′ .Next, by Lemma 2.4(2), we may fix On the other hand, since MM is preserved by ω 2 -directed closed set forcings, Theorem 3.34 implies that MM does not refute U subadditive (κ, κ, θ, θ) for regular uncountable cardinals θ < κ.
Corollary 3.46.In the model of [Tod00], for every regular uncountable cardinal κ, the following are equivalent: • There is a witness to U(κ, κ, ω, ω) that is subadditive of the first kind; • There is a witness to U(κ, 2, ω, 2) that is weakly subadditive of the first kind; • κ is the successor of a cardinal of countable cofinality.
Proof.By Lemma 3.14, if κ = λ + for an infinite cardinal λ of countable cofinality, then there exists a witness to U(κ, κ, ω, ω) that is subadditive of the first kind.For any other κ, since in the model of [Tod00], CH and PID both hold, Theorem 3.45(2) implies that no witness to U(κ, 2, ω, 2) is weakly subadditive of the first kind.
We conclude this section by pointing out a corollary to Theorem 3.45 and the arguments from the proofs of Lemma 3.33 and Corollary 3.41.
We now turn to prove Theorem A.
The proof of the preceding theorem together with Remark 4.2 makes it clear that the following holds as well.
Proof.Suppose that ⊟ ind (κ, ω) holds.By Theorem 4.3, we may fix a closed, subadditive coloring c : Clearly, a δ is a cofinal subset of δ of order-type ω.
Next, suppose that β ∈ acc(κ)\Γ.Then the very same argument as before shows that for any pair γ < δ of successive elements of D c ≤i (β), if the interval (γ, δ) ∩ C β,i is nonempty, then it is covered by a δ .Moreover, by the definition of C β,i , we have ≤i (β)), so that i(α) ≤ i, and it is clear from the definition that C α,i = C β,i ∩ α.
◮ If α / ∈ Γ, then i(α) = 0 ≤ i, and it is clear from the definition that C α,i = C β,i ∩ α. ⊠ The following claim will now finish our proof.
Proof.By forcing with the Laver preparation forcing if necessary, we may assume that the supercompactness of λ is indestructible under λ-directed closed forcing.Following [LS21, §4], let C := Coll(λ, <κ), and let Ṁ be a C-name for the Magidor forcing that turns λ into a singular cardinal of cofinality θ.
In V C , M has the λ + -cc and therefore preserves λ + .Therefore, applying Theorem 4.7 to the models V C and V C * Ṁ shows that ⊟ ind (κ, θ) holds in V C * Ṁ.It remains to show that (κ, θ) fails in V C * Ṁ.In fact, we will show that (κ, τ ) fails for every τ < λ.
Suppose for sake of contradiction that τ < λ and Ḋ = Ḋα | α ∈ acc(κ) is a C * Ṁ-name for a (κ, τ )-sequence.This is a Π 1 1 statement about the structure (V κ , ∈, C * Ṁ, Ḋ) (the sole universal quantification over subsets of V κ is the assertion that there exists no C * Ṁ-name for a thread through Ḋ).Therefore, by the weak compactness of κ, we can find an inaccessible δ ∈ C such that Ḋ * := Ḋ ∩ V δ is a C δ * Ṁδ -name for a (δ, τ )-sequence.
The Mapping Reflection Principle (MRP), introduced by Moore in [Moo05], is a useful consequence of PFA.
(1) This follows directly from the definition of Γ.
Suppose now that β ∈ Γ\{min(Γ)} and we have defined C α,i | α ∈ Γ∩β, i(α) ≤ i < θ satisfying all relevant instances of Clauses (2) and (3) of Definition 4.1 as well as our recursive requirement.The construction breaks into a number of different cases based on the identity of β.In all cases, the verification that our sequence satisfies our recursive requirement and Clauses (2) and (3) of Definition 4.1 at β is routine and therefore largely left to the reader.
Our next goal is to improve the following fact from Part II, and present a weaker sufficient condition for Cspec(λ + ) to cover Reg(cf(λ)).
Let us first remind the reader that the definition of the ideal I goes through first fixing a stationary subset ∆ ⊆ E λ + cf(λ) and a sequence e = e δ | δ ∈ ∆ such that • for every δ ∈ ∆, e δ is a club in δ of order type cf(λ); • for every δ ∈ ∆, cf(γ) | γ ∈ nacc(e δ ) is strictly increasing and converging to λ; • for every club D in λ + , there exists δ ∈ ∆ such that e δ ⊆ D.
As successor cardinals are non-Mahlo, the following indeed improves Fact 5.7.
Proof.By [LHR21, Corollary 5.21], to show that an infinite regular cardinal θ is in Cspec(λ + ), it suffices to prove that there exists a closed witness to U(λ + , λ + , θ, θ).This is the content of the preceding theorem.

Stationarily layered posets
In Part I of this project [LHR18], we motivated the study of U(κ, µ, θ, χ) by showing that it places limits on the infinite productivity of the κ-Knaster condition.Here, we present an analogous result, indicating that closed witnesses to U subadditive (κ, 2, θ, 2) place limits on the infinite productivity of the property of being κ-stationarily layered, which is a strengthening of the κ-Knaster condition.Definition 6.1 ([Cox18]).A partial order P is κ-stationarily layered if the collection of regular suborders of P of size less than κ is stationary in P κ (P).
◮ If ∂(c) ∩ E κ χ is stationary, then we fix β ∈ ∂(c) ∩ E κ χ with κ ∩ M β = β > ǫ.Note that by the continuity of the sequence M , <χ M β ⊆ M β .Now, by Fact 6.2, it suffices to prove that Q ∩ M β is a regular suborder of Q.To this end, we will define, for each p ∈ Q, a reduction of p to M β , i.e., a condition p|M β ∈ Q ∩ M β such that, for all q ≤ Q p|M β with q ∈ M β , q is compatible with p.