1.1 Motivation

In this paper, we initiate a study of various aspects and forms of club guessing. Our definitions are quite general, and in order to motivate them we start with a brief survey of the various forms of club guessing that have appeared in the literature as well as their applications. All undefined notation can be found in Section 1.4, but we remind the reader right away that for a pair $\lambda < \kappa $ of infinite regular cardinals, $E^\kappa _\lambda := \{\delta < \kappa \ {\mathrel {|}}\ \operatorname {\mathrm {cf}}(\delta ) = \lambda \}$ , and $E^\kappa _{\neq \lambda }:= \{\delta < \kappa \ {\mathrel {|}}\ \operatorname {\mathrm {cf}}(\delta ) \neq \lambda \}$ .

Shortly after Jensen constructed Souslin trees in L [Reference Jensen34], he isolated a combinatorial principle named diamond, which is sufficient for the construction.

It is easy to see that for $S\subseteq \kappa $ as above, $\diamondsuit (S)$ implies that $2^{<\kappa } = \kappa $ , and hence diamond is not a consequence of $\textsf{{ZFC}}$ . In contrast, the following result of Shelah [Reference Shelah, Gabbay, Macintyre and Scott62], which is the most well-known club guessing result, is a theorem of $\textsf{{ZFC}}$ .

In particular, unlike the $\diamondsuit $ principle or its descendants the $\clubsuit $ and principles (see [Reference Broverman, Ginsburg, Kunen and Tall12, Reference Ostaszewski46]), the focus is not on predicting arbitrary or even just cofinal subsets of $\kappa $ , but rather only the closed and unbounded subsets of $\kappa $ . This makes the task of guessing easier, since the collection of club subsets of $\kappa $ generate a normal $\kappa $ -complete filter.

The most famous application of Fact 1.2 is Shelah’s PCF bound (see [Reference Shelah, Gabbay, Macintyre and Scott61] or [Reference Abraham, Magidor, Foreman and Kanamori1, Theorem 7.3]):

$$ \begin{align*}2^{\aleph_\omega} \leq \max\{2^{\aleph_0}, \aleph_{\omega_4}\}.\end{align*} $$

Apart from upper bounds on cardinal exponentiation, Fact 1.2 has many other uses in PCF theory. As an example from the basic theory, in obtaining exact upper bounds for sequences of ordinal functions (see [Reference Abraham, Magidor, Foreman and Kanamori1, Lemma 2.19]), in fact showing that there are stationary sets consisting of points of large cofinality in the approachability ideal $I[\lambda ]$ (see [Reference Eisworth, Foreman and Kanamori15, Section 3]). Outside of PCF theory, there are applications of Fact 1.2 to the universality spectrum of models [Reference Dzamonja13, Reference Kojman and Shelah37, Reference Shelah73], cardinal invariants of the continuum [Reference Zapletal84, Reference Zapletal85], cardinal invariants at uncountable cardinals [Reference Blass, Hyttinen and Zhang6, Reference Guzmán, Hrušák and Zapletal22], to the study of the Boolean algebra $\operatorname {\mathrm {P}}(\lambda )/[\lambda ]^{<\lambda }$ for $\lambda $ singular of countable cofinality [Reference Kojman and Shelah39], to showing the incompactness of chromatic number [Reference Shelah72], to obtain a refinement of the downwards Löwenheim–Skolem Theorem [Reference Foreman and Todorčević20], to study the saturation of the non-stationary ideal on $\operatorname {\mathrm {P}}_\lambda (\kappa )$ [Reference Shioya76], to obtain two-cardinal diamond principles in $\textsf{{ZFC}}$ [Reference Matet43, Reference Shelah74, Reference Shioya, Chong, Feng, Slaman, Woodin and Yang77, Reference Todorčević and Thomas79], to obtaining consequence of forcing axioms [Reference Shelah70], to obtain limitative results on forcing [Reference Rosłanowski and Shelah60, Reference Todorčević81], to constructing graphs with a prescribed rate of growth of the chromatic number of its finite subgraphs [Reference Lambie-Hanson40]. That Fact 1.2 is a theorem of $\textsf{{ZFC}}$ also imposes important limitations on the theory of forcing axioms at successors of uncountable cardinals (see, for example, [Reference Neeman45, Reference Shelah67, Reference Shelah68]).

While club guessing was motivated by finding a weak substitute for the diamond principle, in [Reference Shelah71], Shelah, using arguments that materialized through the development of the theory of club guessing, proved the next theorem on diamond, concluding a 40-year-old search for such a result (see the review in [Reference Rinot48]).

Apart from Fact 1.2, there are other, finer, forms of club guessing which are less well-known and yet altogether have a variety of applications. For instance, Fact 1.2 says nothing about the case $\kappa =\lambda ^+$ . For this, we have the following result of Shelah.

To compare the preceding with Fact 1.2, we see two differences in the corresponding Clause (ii). The first, here we require a single $\delta $ instead of stationarily many, however, this is easily seen to be equivalent.Footnote ¹ Second, which is more important, instead of requiring $C_\delta $ to be a subset of D, we now merely require that the intersection of D with ( $E^{\aleph _2}_{\aleph _1}$ and) the set $\operatorname {\mathrm {nacc}}(C_\delta )$ of all non-accumulation points of $C_\delta $ to be cofinal in $\delta $ . This choice is not arbitrary. For club many $\delta \in E^{\aleph _2}_{\aleph _1}$ , both the set $\operatorname {\mathrm {acc}}(C_\delta )$ of all accumulation points of $C_\delta $ and the set $D\cap \delta $ are clubs in $\delta $ , and hence $\operatorname {\mathrm {acc}}(C_\delta )\cap D$ is trivially cofinal in $\delta $ .

Consider now another example due to Shelah concerning the case $\kappa =\lambda ^+$ (see [Reference Soukup and Soukup78], for a short proof):

Comparing the two, we see that Fact 1.4 features a sequence where the guessing is measured against the ideal $J^{\mathrm {bd}}[\delta ]$ of bounded subsets of $\delta $ , whereas here we measure against the nonstationary ideal $\mathrm{NS}_\delta $ . However, Fact 1.4 features a sequence which guesses clubs relative to the set $E^{\aleph _2}_{\aleph _1}$ , and for this reason the two results are incomparable. In this paper, a join of the two results is obtained.

Note that so far we have always required that the clubs $C_\delta $ have the minimal possible ordertype of $\operatorname {\mathrm {cf}}(\delta )$ . The small ordertype requirement is trivially gotten for a sequence that guesses as in Clause (ii) of Fact 1.2. In other cases, however, obtaining that the local clubs have a small ordertype requires extra care (see, for example, [Reference Kojman36, Theorem 9], where the weaker form of Fact 1.4 is proved where one requires each of the $C_\delta $ to merely have size $\aleph _1$ ). As a sample application, we mention that in [Reference Hernández-Hernández and Szeptycki23], a strong form of club guessing at $\aleph _1$ with minimal ordertype is used to construct a small Dowker space.

However, attention should not be restricted only to guessing with minimal ordertypes. At the level of $\aleph _1$ , the ordertypes of guessing sequences play a crucial role in separating forcing axioms at $\aleph _1$ in [Reference Shelah64, Chapter XVII], and later in [Reference Asperó, Friedman, Mota and Sabok3] as well. At higher cardinals, guessing sequences $\vec C$ with very large ordertypes are useful for getting a pathological graph $G(\vec C)$ with maximal chromatic number [Reference Rinot53]. An open question concerning guessing sequences of maximal ordertype is stated in [Reference Rinot52, Question 2]. For an extended discussion, see the introduction to [Reference Rinot54].

At a cardinal $\kappa $ that is a limit or a successor of a limit, another type of relative club guessing has shown to be useful, where the guessing feature stipulates additional conditions on the sequence $\langle \operatorname {\mathrm {cf}}(\gamma ) \ {\mathrel {|}}\ \gamma \in \operatorname {\mathrm {nacc}}(C_\delta ) \rangle $ . In [Reference Hoffman25, Reference Shelah, Gabbay, Macintyre and Scott62], the additional condition is that this sequence is strictly increasing and converging to $|\delta |$ . This is used to construct colourings satisfying strong negative square bracket partition relations [Reference Eisworth and Shelah17, Reference Eisworth and Shelah18]. An earlier construction (see [Reference Shelah, Gabbay, Macintyre and Scott62] or [Reference Eisworth, Foreman and Kanamori15, Theorem 5.19]) requires that the sequence $\langle \operatorname {\mathrm {cf}}(\gamma ) \ {\mathrel {|}}\ \gamma \in \operatorname {\mathrm {nacc}}(C_\delta ) \rangle $ have cofinally many cardinals carrying a Jónsson algebra. This is used to construct Jónsson algebras at $\kappa $ . Note that the existence of a club guessing sequence of large ordertypes in $\textsf{{ZFC}}$ would give rise to such sequences, in particular, solving [Reference Eisworth and Shelah18, Question 2.4] in the affirmative.

We move on to the next example, this time a question of Shelah.

Compared to Fact 1.5, here we require that for stationarily many $\beta < \delta $ , two consecutive non-accumulation points following $\beta $ are in the club D. Shelah mentions (without proof) that this slight strengthening of Fact 1.5 combined with $\textsf{{GCH}}$ allows for the construction of a $\kappa $ -Souslin tree. This is related to the open problem of whether $\textsf{{GCH}}$ implies the existence of an $\aleph _2$ -Souslin tree (see [Reference Rinot56]), and the earlier work of Kojman and Shelah on that matter [Reference Kojman and Shelah38].

Here we shall prove that at the level of $\aleph _2$ , an affirmative answer to Shelah’s question follows from the existence of a sequence as in Fact 1.4 in which $D \cap E^{\aleph _2}_{\aleph _1}$ in Clause (ii) is replaced by $D \cap E^{\aleph _2}_{\aleph _0}$ . However, Asperó has answered Shelah’s question negatively [Reference Asperó, Geschke, Lowe and Schlicht2]. Getting this failure together with the $\textsf{{GCH}}$ remains open.

The feature of guessing consecutive points has other applications in the construction of Souslin trees: in [Reference Brodsky and Rinot11, Section 5], the feature of guessing with two consecutive points allowed to reduce a $\diamondsuit (\kappa )$ hypothesis from [Reference Brodsky and Rinot7] to just $\kappa ^{<\kappa }=\kappa $ . In [Reference Brodsky and Rinot8], a feature of guessing with $\omega $ -many consecutive points is used to construct Souslin trees with precise control over their reduced powers.

Returning to the discussion after Fact 1.5, there is another way to impose that the set of good guesses be ‘large’. Here is an example, again due to Shelah.

This result is used in [Reference Rinot51, Section 2] in producing a strong oscillation with $\omega $ many colours, sufficient to derive strong colourings [Reference Rinot51, Section 3] and transformations of the transfinite plane [Reference Rinot and Zhang58]. Any improvement of the above result that partitions the club guessing into $\theta $ many pieces immediately translates to getting a strong oscillation with $\theta $ colours. This also connects to our previous discussion on guessing sequences $\langle C_\delta \ {\mathrel {|}}\ \delta \in S\rangle $ with very large ordertypes, since the number of pieces into which $C_\delta $ may be partitioned is bounded by $|C_\delta |$ .

We shall later show that for the purpose of obtaining such partitioned club guessings, the move from the unbounded ideal to the non-stationary ideal as in Fact 1.4 to Fact 1.5 is beneficial. Sufficient conditions and applications to an even stronger form of partitioned club guessing in which there is global function $h:\kappa \rightarrow \theta $ such that $h_\delta =h{\mathbin \upharpoonright } C_\delta $ for all $\delta $ may be found in [Reference Shelah, Gabbay, Macintyre and Scott62, Section 3] and [Reference Lambie-Hanson and Rinot41, Theorem 4.20].

A very useful feature of club guessing sequences we have so far ignored is coherence. Coherent club guessing sequences have been applied to set theory of the real line [Reference Bartoszyński, Larson and Shelah5], and to cardinal invariants of the continuum [Reference Shelah and Zapletal75]. Coherent club guessing sequences were also used to show the non-existence of a natural forcing axiom [Reference Shelah69] and to construct strong colourings [Reference Rinot50, Reference Rinot and Zhang59]. Weakly coherent club guessing at the level of $\aleph _1$ has been used to define a pathological topology on the real line [Reference Zapletal83], and weakly coherent club guessing at the level of a successor $\lambda ^+$ of a singular cardinal $\lambda $ was used in [Reference Brodsky and Rinot9, Section 2.1] to prevent $\lambda $ -distributive $\lambda ^+$ -trees from having a cofinal branch, thus, yielding nonspecial $\lambda ^+$ -Aronszajn trees.

The above is hardly an exhaustive list of applications of club guessing but merely a selection biased by the themes of this paper. Additional key results, including those from [Reference Eisworth14, Reference Foreman and Komjath19, Reference Gitik and Shelah21, Reference Ishiu29] will be discussed in Part II of this series. Another caveat is that in this paper we shall only be concerned with getting club guessing results at $\kappa \ge \aleph _2$ . The behavior of club guessing at the level of $\aleph _1$ is entirely independent of $\textsf{{ZFC}}$ , and we refer the reader to [Reference Asperó and Krueger4, Reference Eisworth and Nyikos16, Reference Hirschorn24, Reference Ishiu30–Reference Ishiu and Larson32, Reference Moore44] for more on that matter.

1.2 The results

Throughout the paper, $\kappa $ stands for an arbitrary regular uncountable cardinal; $\theta ,\mu ,\chi $ are (possibly finite) cardinals $\le \kappa $ , $\lambda $ and $\nu $ are infinite cardinals $<\kappa $ , $\xi ,\sigma $ are nonzero ordinals $\le \kappa $ , and S and T are stationary subsets of $\kappa $ . We shall sometimes implicitly assume that S consists of nonzero limit ordinals.

Our first main result fulfills the promise of finding a join of Facts 1.4 and 1.5.

Theorem A. For every successor cardinal $\lambda $ , there exists a $\lambda $ -bounded C-sequence $\vec C=\langle C_\delta \ {\mathrel {|}}\ \delta \in E^{\lambda ^+}_\lambda \rangle $ satisfying the following. For every club $D \subseteq \lambda ^+$ , there is a $\delta \in E^{\lambda ^+}_\lambda $ such that the following set is stationary in $\delta :$

$$ \begin{align*}\{\beta <\delta \ {\mathrel{|}}\ \min(C_\delta \setminus(\beta+1))\in D\cap E^{\lambda^+}_\lambda\}.\end{align*} $$

Our next result deals with coherent guessing sequences.

The next two results address the problem of partitioning a given club guessing sequence into $\theta $ many pieces as in Fact 1.7.Footnote ²

Theorem C. Suppose that for each $\delta \in E^\kappa _\lambda $ , $J_\delta $ is some $\lambda $ -complete ideal over $\delta $ , where $\lambda $ is regular uncountable, and suppose that $\vec C=\langle C_\delta \ {\mathrel {|}}\ \delta \in E^\kappa _\lambda \rangle $ is a given $\lambda $ -bounded C-sequence satisfying that for every club $D\subseteq \lambda $ , there exists $\delta \in E^\kappa _\lambda $ such that

$$ \begin{align*}\{\beta<\delta\ {\mathrel{|}}\ \min(C_\delta\setminus(\beta+1))\in D\cap T\}\in J_\delta^+.\end{align*} $$

Any of the following hypotheses imply that there exists a map $h:\lambda \rightarrow \theta $ such that for every club $D\subseteq \lambda $ , there is a $\delta \in E^\kappa _\lambda $ such that, for every $\tau <\theta $ ,

$$ \begin{align*}\{\beta<\delta\ {\mathrel{|}}\ h(\operatorname{\mathrm{otp}}(C_\delta\cap\beta))=\tau\ \&\ \min(C_\delta\setminus(\beta+1))\in D\cap T\}\in J_\delta^+.\end{align*} $$

1. (i) $\theta =\lambda =\lambda ^{<\lambda }$ and $\lambda $ is a successor cardinal $;$

2. (ii) $\theta =\lambda $ , $\diamondsuit (\lambda )$ holds, and $\lambda $ is not Mahlo $;$

3. (iii) $\theta =\lambda $ , $\diamondsuit ^*(\lambda )$ holds, and each $J_\delta $ is normal $;$

4. (iv) $\theta <\lambda $ is regular and $\lambda $ is not greatly Mahlo $;$

5. (v) $\theta <\lambda $ is regular and there is a $\lambda $ -Aronszajn tree.

Theorem D. Under the same setup of the previous theorem, any of the following hypotheses imply that there exists a sequence of maps $\langle h_\delta :\lambda \rightarrow \theta \ {\mathrel {|}}\ \delta \in E^\kappa _\lambda \rangle $ such that for every club $D\subseteq \lambda $ , there is a $\delta \in E^\kappa _\lambda $ such that, for every $\tau <\theta $ ,

$$ \begin{align*}\{\beta<\delta\ {\mathrel{|}}\ h_\delta(\operatorname{\mathrm{otp}}(C_\delta\cap\beta))=\tau\ \&\ \min(C_\delta\setminus(\beta+1))\in D\cap T\}\in J_\delta^+.\end{align*} $$

1. (i) $\diamondsuit ^*(\lambda )$ holds and $\theta =\lambda ;$

2. (ii) $\lambda $ is not strongly inaccessible, and $\theta $ is the least to satisfy $2^\theta \ge \lambda ;$

3. (iii) $\lambda $ is strongly inaccessible, not ineffable, and $\theta <\lambda ;$

4. (iv) $\lambda $ is not greatly Mahlo, and $\theta <\lambda $ .

Our last result fulfills the promise to show that at the level of $\aleph _2$ an affirmative answer to Question 1.6 follows from the existence of a sequence as in Fact 1.4 in which $D \cap E^{\aleph _2}_{\aleph _1}$ in Clause (ii) is replaced by $D \cap E^{\aleph _2}_{\aleph _0}$ .

Theorem E. For every successor cardinal $\lambda $ , if there exists a $\lambda $ -bounded C-sequence $\vec C=\langle C_\delta \ {\mathrel {|}}\ \delta \in E^{\lambda ^+}_{\lambda }\rangle $ such that for every club $D\subseteq \lambda ^+$ , there is a $\delta \in E^{\lambda ^+}_\lambda $ such that $\sup (\operatorname {\mathrm {nacc}}(C_\delta )\cap D\cap E^{\lambda ^+}_{<\lambda })=\delta $ , then there exists a $\lambda $ -bounded C-sequence $\vec C=\langle C_\delta \ {\mathrel {|}}\ \delta \in E^{\lambda ^+}_{\lambda }\rangle $ such that for every club $D\subseteq \lambda ^+$ , there is a $\delta \in E^{\lambda ^+}_\lambda $ such that the following set is stationary in $\delta :$

$$ \begin{align*}\{\beta <\delta \ {\mathrel{|}}\ \beta_1, \beta_2 \in D\text{ where }\beta_1&:=\min(C_\delta \setminus(\beta+1))\ \&\\ \beta_2&:=\min(C_\delta \setminus (\beta_1+1))\}.\end{align*} $$

1.3 Organization of this paper

The aim of Section 2 is to give the reader a tour of the basic methods for proving club guessing theorems. The purpose is introductory, and with one exception, all the results we prove are not new, though some of them are not widely known. In particular, we give a proof of Fact 1.2. In Section 2.1 our main definition, Definition 2.2, can be found.

In Section 3, our theme is to obtain club guessing sequences with additional coherence properties. This is done by starting with an arbitrary C-sequence with some degree of coherence, and then improving it to make it guess clubs as well, all the while preserving the coherence. This allows us to obtain ‘coherent forms’ of known results such as Fact 1.4. At the end of the section, we record the results which can be obtained by the same proofs, but without any assumptions of coherence on the initial C-sequence. A proof of Theorem B can be found in this section.

In Section 4, we consider partitioned club guessing. We show how the colouring principles of [Reference Inamdar and Rinot26, Reference Inamdar and Rinot27] allow us to not just obtain partitioned club guessing, but in fact partition club guessing (recall footnote 2). Furthermore, using these colouring principles allows us to separate the combinatorial content from the club guessing content in previous results about partitioned club guessing (see [Reference Shelah63, Lemma 3.10]). A proof of Theorem C can be found here.

In Section 5, we turn to the problem of guessing many consecutive non-accumulation points as in the discussion surrounding Question 1.6. We show how a sequence guessing clubs relative to points of small cofinality can be modified for this purpose.

In the last section, Section 6, our focus is on improving the quality of the guessing calibrated against the ideal. Mainly, our focus is moving from the unbounded ideal to the non-stationary as in the move from Fact 1.4 to Fact 1.5. Similar ideas also allow us to improve some results from Section 4. The proofs of Theorems A, D, and E can be found here.

1.4 Notation and conventions

We have already listed some conventions in the beginning of Section 1.2. Here, we list some more. Let $\log _\chi (\lambda )$ denote the least cardinal $\theta \le \lambda $ such that $\chi ^\theta \ge \lambda $ . For sets of ordinals $A,B$ , we denote $A\circledast B:=\{(\alpha ,\beta )\in A\times B\ {\mathrel {|}}\ \alpha <\beta \}$ and we identify $[B]^2$ with $B\circledast B$ . For $\theta>2$ , $[\kappa ]^\theta $ simply stands for the collection of all subsets of $\kappa $ of size $\theta $ . For sets of ordinals $A,B$ , we write $A\sqsubseteq B$ iff there exists an ordinal $\delta $ such that $A=B\cap \delta $ ; we write $A \sqsubseteq ^* B$ iff there exists a pair of ordinals $\epsilon <\delta $ such that $A\setminus \epsilon =B\cap [\epsilon ,\delta )$ .

Let $E^\kappa _\theta :=\{\alpha < \kappa \ {\mathrel {|}}\ \operatorname {\mathrm {cf}}(\alpha ) = \theta \}$ , and define $E^\kappa _{\le \theta }$ , $E^\kappa _{<\theta }$ , $E^\kappa _{\ge \theta }$ , $E^\kappa _{>\theta }$ , $E^\kappa _{\neq \theta }$ analogously. For a stationary $S\subseteq \kappa $ , we write $\operatorname {\mathrm {Tr}}(S):= \{\alpha \in E^\kappa _{>\omega }\ {\mathrel {|}}\ S\cap \alpha \text { is stationary in }\alpha \}$ .

For a set of ordinals A, we write $\operatorname {\mathrm {ssup}}(A) := \sup \{\alpha + 1 \ {\mathrel {|}}\ \alpha \in A\}$ , $\operatorname {\mathrm {acc}}^+(A) := \{\alpha < \operatorname {\mathrm {ssup}}(A) \ {\mathrel {|}}\ \sup (A \cap \alpha ) = \alpha> 0\}$ , $\operatorname {\mathrm {acc}}(A) := A \cap \operatorname {\mathrm {acc}}^+(A)$ , $\operatorname {\mathrm {nacc}}(A) := A \setminus \operatorname {\mathrm {acc}}(A)$ , and $\operatorname {\mathrm {cl}}(A):= A \cup \operatorname {\mathrm {acc}}^+(A)$ . A function $f:A\rightarrow \operatorname {\mathrm {Ord}}$ is regressive iff $f(\alpha )<\alpha $ for every nonzero $\alpha \in A$ . A function $f:[A]^2\rightarrow \operatorname {\mathrm {Ord}}$ is upper-regressive iff $f(\alpha ,\beta )<\beta $ for every pair $(\alpha ,\beta )\in [A]^2$ .