Tiltan and superclub

We force superclub with an arbitrary large value of cov($\mathscr{M}$). We force tiltan with an arbitrary large value of add($\mathscr{M}$). Finally, we obtain a negative square bracket relation from superclub.


Introduction
The diamond principle ♦ κ is a prediction principle discovered by Jensen in [Jen72].Suppose that κ is regular and uncountable.A ♦ κ -sequence is a sequence of sets (A α : α ∈ κ) such that A α ⊆ α whenever α ∈ κ and for every A ⊆ κ the set S A = {α ∈ κ : A ∩ α = A α } is a stationary subset of κ.We say that ♦ κ holds iff there exists a ♦ κ -sequence.Notice that ♦ κ + implies 2 κ = κ + .
The club principle ♣ κ is a weaker prediction principle, discovered by Ostaszewski in [Ost76].Denote the set of limit ordinals of κ by lim(κ).We say that (T δ : δ ∈ lim(κ)) is a ♣ κ -sequence iff every T δ is an unbounded subset of δ and for every A ∈ [κ] κ the set T A = {δ ∈ lim(κ) : T δ ⊆ A ∩ δ} is a stationary subset of κ.
Motivated by a stubborn open problem of Juhasz about the connection between ♣ ℵ 1 and the existence of Suslin trees, Primavesi defined in [Pri11] an intermediate principle, dubbed as superclub.We phrase the definition in the generalized form.Definition 0.1.Superclub.Assume that κ = cf(κ) > ℵ 0 .
Comparing superclub to diamond and club it is clear from the definition that diamond at κ implies superclub at κ which implies, in turn, the club principle at κ.Both implications are irreversible.It has been proved in [Gar18] that superclub at κ + implies Galvin's property at κ + while ♣ κ + is consistent with the failure of Galvin's property at κ + .It has been proved in [Che17] that superclub at ℵ 1 is consistent with 2 The forcing of [Che17] is based on the existence of a strongly inaccessible cardinal in the ground model.A natural question is whether superclub with a large continuum has some consistency strength.We shall give a negative answer.Specifically, in the model of [FSS97] the continuum hypothesis fails and superclub holds.An immediate consequence is the consistency of superclub with a large value of some cardinal characteristics like the dominating number d.
A more interesting challenge is the bounding number b and the category invariant add(M ).We do not know whether superclub is consistent with a large value of these characteristics, even if the desired value is just ω 2 .However, we shall force club with an arbitrarily large value of b and cov(M ) simultaneously, thus we will have a large value of add(M ) as well.
From the definition it seems that superclub has more affinities with the club principle.The prediction of superclub applies only to [κ] κ and based on inclusion since if B ∩ α = S α then S α ⊆ A ∩ α and possibly S α = A ∩ α.Hence in the above mentioned features which distinguish club from diamond, superclub behaves like club and unlike diamond.But there is another feature in which superclub is similar to diamond.
If A ⊆ κ and S A is the stationary set of guesses given by ♦ κ then the corresponding elements of the diamond sequence are coherent in the following sense.If γ, δ ∈ S A and γ < δ then A δ end-extends A γ .This is a consequence of the equality which forms the prediction of diamond sequences.If one assumes only club at κ then this coherence fades away.Actually, one can choose a club sequence and then shrink every element in the sequence by taking subsets of order-type ω, in which case A δ never extends A γ .
From this point of view, but only from this one, superclub is similar to diamond.Given A ∈ [κ] κ and choosing B ∈ [A] κ , the superclub sequence acts on B as a diamond sequence with equality as the predicate for prediction.Consequently, the elements of the superclub sequence are coherent, despite the fact that they are only included in the original set A. In other words, a superclub sequence is a coherent club sequence.This idea stands behind the proof of Galvin's property from superclub, and it will be exploited in the last section.
Our notation is (hopefully) standard.We shall use the Jerusalem forcing notation, so p ≤ q means that p is weaker than q.Occasionally we shall use the word tiltan instead of club when refering to the club principle.The paper contains three additional sections.In the first part we force superclub with large continuum without an inaccessible cardinal.Actually, cov(M ) assumes the value of the continuum in the generic extension, hence d = 2 ω as well.In the second part we show that one can force club with add(M ) = c where c is arbitrarily large.In the last section we deal with a graph-theoretic statement which follows from the continuum hypothesis and here we show that it follows merely from superclub.
We thank the anonymous referee of the paper for a careful reading of the article and for the helpful comments, all of them are integrated within the manuscript.We also thank Thilo Weinert for a helpful conversation concerning Cichoń's diagram and the bounding number.Finally, we thank Jörg Brendle and Lajos Soukup for several important comments and useful information.

A takeaway theorem
In this section we prove that the method of [FSS97] for enlarging the continuum while preserving club sequences from the ground model works equally well with respect to superclub sequences.The main conclusion is that one can force superclub with an arbitrarily large value of the continuum without the assumption that there is an inaccessible cardinal in the ground model.Another conclusion is that one can force superclub with a large value of cov(M ), a result which will be useful in the next section.
Let us describe the basic forcing notion of [FSS97] in the spirit of Prikry forcing, namely we define the forcing order which is quite incomplete and we also define a pure order which is ω 1 -complete.
q ∈ P then p ≤ q iff p ⊆ q and u pq is finite.We define p ≤ * q iff p ⊆ q and u pq is empty.
The following statement is proved in [Pri11, Proposition 5.1.4].If f ˜is a P-name, p ∈ P and p forces that f ˜: ˜→ ω 1 such that g p,f ˜∈ V and for every ordinal η ∈ ω V 1 there is a condition q η ∈ P such that p ≤ q η and q η g p,f Similar assertions appear in [FSS97], see for example Lemma 3.9 there.
It follows from this statement that ℵ 1 is preserved, and moreover stationary subsets of ℵ 1 remain stationary in the generic extension as follows from [FSS97, Theorem 2.6].To see that ℵ 1 is preserved consider h In the above definition the domain of any condition p is contained in ω 2 , but one can replace ω 2 by an arbitrarily large cardinal κ.If κ ℵ 0 = κ in the ground model then 2 ω = κ in the generic extension.

Proof.
By the comment before the statement of the theorem we know that ℵ 1 is preserved.Cardinals above ℵ 1 are preserved since P is ℵ 2 -cc (here we use the fact that V models GCH).It is easy to infer from density arguments that 2 ω = κ in V [G], as the generic set G adds a function from κ into {0, 1} which can be sliced into κ distinct functions from ω into {0, 1}.
Suppose that and let f ˜be a name for the increasing enumeration of the elements of A. Fix a condition p which forces this fact.Let A p,f ˜and g p,f ˜be as described ahead of the proof, so A p,f ˜∈ V and for every δ ∈ ω V 1 one can find q δ ≥ p such that q δ rang(g p,f ˜↾ A p,f ˜) ⊆ A. Applying superclub in the ground model to g ′′ p,f ˜Ap,f ˜, let B be an uncountable subset of g ′′ p,f ˜Ap,f ˜on which the superclub sequence (S α : α ∈ lim(ω 1 )) acts like diamond. Let Since this is true for every α ∈ S B and the stationarity of S B is preserved, one can see that superclub holds in the generic extension.It is easy to verify that cov(M ) equals c in the generic extension, see [Bre06] for details, so we are done.
1.2 Remark that the main result of this section already shows that in order to force superclub with a large value of the continuum there is no need of an inaccessible cardinal in the ground model.In the next section we will show that the above result can be strengthened if one replaces superclub by tiltan.The idea is that the forcing notion of [FSS97] can be composed with Hechler forcing, thus making both cov(M ) and b large.We do not know how to preserve superclub sequences from the ground model in this construction, but if one begins with a diamond sequence in the ground model then it remains a club sequence in the generic extension.

The additivity of the meager ideal
In this section we prove the consistency of tiltan (the club principle) with an arbitrarily large value of add(M ).This confirms a conjecture of Brendle, see [Bre06,Conjecture 9.4].The context of this result is nicely displayed by Cichoń's diagram: cov(N ) A result of Truss from [Tru83] says that ♣ ℵ 1 implies add(N ) = ℵ 1 , so the smallest invariant in the above diagram becomes ℵ 1 under the club principle.Nevertheless, we shall see that ♣ ℵ 1 is consistent with an arbitrarily large value of add(M ).The idea is to increase both cov(M ) and b while keeping the club principle.By a result of Miller from [Mil81] we know that add(M ) = min(b, cov(M )), so our theorem will follow.
A natural way to increase cov(M ) is by using [FSS97], as described in the previous section.A natural way to increase b is Hechler forcing from [Hec74].We need, therefore, to amalgamate these two forcing notions.Moreover, we would like to do it while maintaining tiltan.For this end, we shall use a strong version of ccc, called sweetness.The concept of sweet forcing notions comes from [She84], and a good background is contained in [RaS04].Let us recall the formal definition.
Definition 2.1.Sweet forcing notions.A forcing notion P is sweet iff there are a dense subset D of P and a sequence Ē = (E n : n ∈ ω) of equivalence relations on D such that the following requirements are met: for every n ∈ ω there is a condition q n ≥ r such that q n E n r and ∀i ≥ n, p i ≤ q n .(d) If p, q ∈ D and n ∈ ω then there is some m ∈ ω so that for every r ∈ [p] Em one can find t ∈ [q] En such that r ≤ t.
There is another concept of sweetness, defined by Stern in [Ste85] and based on topological considerations.For the formal definition of topological sweetness and the concept of an iterably sweet forcing notion we refer to [RaS04,Section 4].Here we just mention the fact that if P is sweet and every pair of compatible conditions has a least upper bound then P is iterably sweet.The forcing notion of [FSS97] mentioned in the previous section is sweet and also iterably sweet.It has been proved in [She84, Section 7] that if P is sweet and D ˜is a P-name of Hechler forcing then P * D ˜is sweet as well.In what follows we describe a class of forcing notions which generalize the forcing of [FSS97].Define K 0 as the class of objects K which are essentially an iteration of ccc forcing notions.So ℓg(K) is an ordinal (the length of the iteration) and P α is a forcing notion for every α ≤ ℓg(K).If α ≤ β ≤ ℓg(K) then we require P α ⋖ P β , and for every β < ℓg(K) we have a P β -name Q ˜β of a ccc forcing notion.
Assuming CH in the ground model, if K ∈ K 0 then P K is ℵ 2 -cc.The pure order is ℵ 1 -complete, and moreover if (p n : n ∈ ω) is ≤ pr K -increasing then n∈ω p n is an upper bound, which will be called the canonical upper bound.One can verify that P K is proper.Assume that (p i : i ∈ ω 1 ) is ≤ pr K -increasing and continuous, that is p j is the canonical upper bound of (p i : i ∈ j) whenever j ∈ ω 1 is a limit ordinal.Assume further that p i+1 ≤ ap K q i+1 for every i ∈ ω 1 .Then one can find i < j < ω 1 such that q i+1 q j+1 .It can be verified that the forcing notion of [FSS97] used in the previous section belongs to K 0 .Let K be the subclass of K 0 which consists of iterations K which are Suslin ccc (this means that the forcing and its order are Σ 1 1 -definable) and iterably sweet.We shall use this class in order to prove the following: Theorem 2.2.Assume that ♦ ℵ 1 holds in V, K ∈ K and Q ˜K,β is a name of Hechler forcing for every β ≤ ℓg(K).Then ♣ ℵ 1 holds in the generic extension by P K .Concomitantly, b = ℓg(K) in the generic extension by P K , provided that ℓg(K) is a regular and uncountable cardinal.

Proof.
The fact that b = ℓg(K) in the generic extension by P K follows from the properties of Hechler forcing.We must show, therefore, that ♣ ℵ 1 holds in this generic extension.Let (A α : α ∈ ω 1 ) be a diamond sequence in V .Since ℓg(K) is typically large, we get the negation of CH in the generic extension and hence diamond fails.However, we will show that (A α : α ∈ ω 1 ) exemplifies ♣ ℵ 1 in the generic extension.Suppose that A ˜is a P K -name of an element of [ω 1 ] ℵ 1 .Fix a generic set G ⊆ P K and a condition p such that p A ˜∈ [ω 1 ] ℵ 1 .By induction on i ∈ ω 1 we choose a triple (p i , q i , α i ) such that the following requirements are met: (a) (e) q i forces that the ith member of A ˜is αi .(f ) p i+1 ≤ ap K q i .Let χ be a sufficiently large regular cardinal.For every ε ∈ ω 1 choose and the following property holds.For every first order formula ϕ(x, ȳ) and every ā ∈ ℓg(ȳ) N δ and every large enough n ∈ ω, We define a condition q as follows: (ℵ) dom(q) = N δ ∩ ℓg(K).
Here we use the fact that (A α : α ∈ ω 1 ) is a diamond sequence in V .Since q δn ≤ q for every n ∈ ω we see that q a = A α ⊆ A ˜for every α ∈ S, so we are done.
2.2 We can derive now the main result of this section: Corollary 2.3.Tiltan and add(M ).It is consistent that ♣ ℵ 1 holds and add(M ) is arbitrarily large.

Proof.
Assume ♦ ℵ 1 in V and choose an arbitrarily large regular cardinal λ which satisfies λ ℵ 1 = λ.Let K ∈ K be such that λ = ℓg(K) and Q ˜K,β is Hechler forcing for every β ∈ λ.Let G ⊆ P K be V -generic.
By the definition of K we use a finite support iteration in K and hence Cohen reals are added at limit stages thus cov , so the proof is accomplished.
2.3 There is an interesting upshot which follows from the above corollary.We shall phrase and prove it in Corollary 2.5 below, but we need some background.Let f be a function from κ into P(κ).We say that f is C(λ, µ) ∈ f (y) whenever {x, y} ⊆ A. Without the assumption that f is κ-reasonable, one can find functions f : κ → P(κ) for which there are no infinite free sets (and even no two-element free set, see [Mut15]).However, the mere assumption of κ-reasonability is insufficient for getting infinite free sets, as shown in the above mentioned paper.
It has been proved in [Gar16] that if f is κ-reasonable and κ ω → κ ω 2 then there exists an infinite free set for f .A natural question is whether this assumption is necessary.We shall give a negative answer to this question.Recall that λ κ → λ κ 2 holds iff for every coloring c : λ × κ → 2 there are A ⊆ λ and B ⊆ κ such that |A| = λ, |B| = κ and c ↾ (A × B) is constant.Our first statement says that one can replace the assumption κ ω → κ ω 2 by the weaker assumption κ κ → κ ω 2 .Theorem 2.4.Infinite free subsets.Assume that κ κ → κ ω 2 and f : κ → P(κ) is κ-reasonable.Then there exists an infinite free subset for f .Proof.We define a coloring c : If there exists such H 1 then we are done, since it would be a free set for f by the definition of the coloring c, so assume towards contradiction that there is no H 1 as above.
It follows that there is a 0-monochromatic set H 0 of size κ.We decompose it into H 0 = A ∪ B such that A ∩ B = ∅ and |A| = |B| = κ.Now we separate the cartesian product A × B into two disjoint collections: ∈ f (b) and all the members are taken from H 0 ), so B 0 ⊆ {f (a) : a ∈ A 0 }, contradicting the assumption that f is C(ω, κ).
2.4 Question 2.8 from [Gar16] is whether the assumption κ ω → κ ω 2 is necessary for proving that every κ-reasonable function has an infinite free set.In order to give a negative answer we will show that the positive relation  [CGW20].On the other hand, tiltan implies stick and by Proposition 3.3 of [CGW20] we have κ ω 1 → κ ω 2 and a fortiori κ κ → κ ω 2 .From Theorem 2.4 we infer that every κ-reasonable function f : κ → P(κ) has an infinite free set, so we are done.
2.5 We conclude this section with two problems.The first one is about the relationship between superclub and add(M ).We do not know whether club can be replaced by superclub in the main result of this section.The essential property that we used in the previous section in order to guarantee the preservation of superclub sequences seems problematic in the context of Hechler forcing.
As mentioned at the beginning of this section, club and hence superclub imply add(N ) = ℵ 1 , so the results of this section show that this cannot be stressed further in the horizontal dimension of Cichońs diagram.Brendle proved in [Bre06] that club is consistent with cov(N ) = ℵ 2 , so in some sense we are covered in the vertical dimension as well.In fact, it follows from the paper of Brendle that tiltan is consistent with an arbitrarily large value of cov(N ), see [Bre06, Section 5].

Independent subsets and complete subgraphs
Suppose that κ, µ ≤ λ.The negative square brackets relation λ [κ] 2 µ is the assertion that one can find a coloring c : [λ] 2 → µ such that for every A ∈ [λ] κ one has c ′′ [A] 2 = µ.Namely, every subset of λ of size κ assumes all the colors.In the language of graph theory one can describe this relation as follows.If G = (V, E) is the complete graph with λ vertices and one defines a decomposition (E α : α ∈ µ) of E then every subset A ⊆ V of size κ has at least one edge in every E α , that is The fact that G is the complete graph is helpful here, since there are many edges and hence the probability that each A of size κ will intersect any E α increases.A graph theoretic generalization of this relation would be the same statement with respect to a larger class of graphs.One has, however, to insert sufficiently many edges into G.Recall that I ⊆ V is independent iff I is edge-free, that is [I] 2 ∩ E = ∅.A necessary requirement for our generalization is the absence of independent sets of size κ.Definition 3.1.Q(λ, κ, µ).Assume that κ, µ ≤ λ.The property Q(λ, κ, µ) means that for every graph G = (λ, E) with no independent subset of size κ there exists a partition (E α : α ∈ µ) of E such that for every α ∈ µ the graph G α = (λ, E α ) has no independent subset of size κ.
The definition of Q(λ, κ, µ) comes from [GHK95], and the fact that it is a strengthening of the square brackets relation with the same parameters is mentioned there.Let us spell-out the proof.
3.3 We do not know whether ♣ κ + is consistent with the failure of the principle Q(κ + , κ + , κ + ).It is clear, however, that the strategy of [FSS97] cannot yield such a result, since it cannot separate club from superclub.On the other hand, the method of [She98] is promising in this context.The idea is to force some property at ℵ 2 while keeping ♣ S at S = S ω 2 ω and then to collapse ℵ 1 .The collapse secures the club principle at ω 1 but destroys superclub.This method is limited to properties which can be forced at ℵ 2 and one also has to make sure that the pertinent property is preserved by the collapse.Question 3.4.Is it consistent that ♣ κ + holds but Q(κ + , κ + , κ + ) fails for some infinite cardinal κ?
Let us indicate that the only property which separates tiltan from superclub, as far as we know, is Galvin's property.Thus, a positive answer to the above question would be interesting.