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CALLIGRAPHY CONCERNING CASUALLY COMPILED CARDINAL CHARACTERISTIC COMPARISONS

THILO WEINERT

Abstract. We establish several inequalities between cardinal characteristics of the continuum. In particular, it is shown that the partition splitting number is not larger than the uniformity of the meagre ideal; not all sets of reals having the cardinality of an (the?) ε-almost bisecting number are of strong measure zero; no fewer sets of strong measure zero than indicated by the statistically reaping number suffice to cover the reals; the pair-splitting number is not smaller than the evasion number; the cofinally bisecting number is not larger than both the unbounding number and the splitting number; and the subseries number is neither smaller than the pair-splitting number nor than the minimum of the unbounding number and the unbisecting number.

Moreover we provide a diagram putting these results into context and give a brief historical account.

Contents

1. Preliminaries  2
1.1. Introduction  2
1.1.1. Prologue  2
1.1.2. History  2
1.2. Notation, Definitions, a Summary, and a Diagram  4
1.2.1. Notation and Definitions  4
1.2.2. A Summary and a Diagram  8
2. Results  10
2.1. Inequalities Involving Halfway New Cardinal Characteristics  10
2.1.1. Splitting  10
2.1.2. Reaping  12
2.2. Various Inequalities  15
2.2.1. Splitting of Partitions and the Uniformity of the Meagre Ideal  15
2.2.2. Pair-Splitting  16
References  18

2020 Mathematics Subject Classification. Primary 03E17; Secondary 40A05. The author thanks Andreas Blass, Shimon Garti, Michael Hrušák, and Diana Montoya for enlightening correspondence

by electronic mail and gratefully acknowledges support through the Singaporean Ministry of Education Tier 2 grant No. MOE2017-T2-2-125.

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2 THILO WEINERT

1. Preliminaries

1.1. Introduction.

1.1.1. Prologue. Cardinal characteristics of the continuum are infinite cardinals, usually uncountable and no larger than the continuum, given by a specific definition. They are often derived from considerations not belonging to set theory proper and sometimes from investigations in combinatorial set theory for their own sake. There are two kinds of possible results about them: Often one can show metamathematically that it is consistent with the axioms of set theory to assume that one particular characteristic is smaller than another; sometimes one can straightforwardly show that one is no larger than another. This paper contains seven results of the latter sort, in general related to work of Blass, Hrušák, Laflamme, Meza-Alcántara, A. Miller, Minami, Raghavan and Steprans, [9, 7, 1, 31, 32, 24, 33] and is particularly inspired by the more recent work of J. Brendle, Brian, Halbeisen, Hamkins, Klausner, Lischka, and Shelah, [14, 13]. Its purpose is twofold. On the one hand it shall disseminate the results presented among those specialising—among possibly other fields—in the theory of cardinal characteristics of the continuum. On the other hand it shall give a broader audience a small overview over parts of the subject and serve as a possible starting point for enquiries. While thus having aspects of a survey, it does not aim to be an exposition. For a more expository treatment, see [39, 5, 11, 8, 22]. For a thorough historical account, see [37]. Combining these objectives seems appropriate as, firstly, none of the results presented are of a metamathematical nature, so the reader does not need to know any mathematical logic, and, secondly, they are somewhat scattered about within the subject of cardinal characteristics of the continuum. A Hasse diagram is provided on page 8. It includes all cardinal characteristics involved in new results presented here—see the Summary on page 8—and then some. The general aim was to included as many cardinal characteristics as possible without rendering the resulting diagram incomprehensible. In particular, if a characteristic appears in the diagram, known relations to other characteristic in the diagram appear as well; moreover, lines are not crossing. These conditions explain why 𝔟 does appear in the diagram within both a minimum and a maximum but not on its own, as 𝔥_{ℚ} ⩽ 𝔟. No reader should infer that a cardinal characteristic is generally wider known or in some objective sense more interesting than another only because in contrast to the latter, the former appears in the diagram. A line between two characteristics means an inequality applies with the characteristic higher up on the page being no smaller than the one further down on it.

All proofs presented in this paper are by contradiction and written unwaveringly in this manner. This is to say, that the contradiction is not reached before the end of the proof. Their style might strike the readers as terse but the author hopes that they will (grow to) like it. It may be less suited to give readers a rough idea but it probably is all the more suited to thoroughly check for correctness which, in the absence of a coauthor, is an important feature.

1.1.2. History. It only gradually became a custom to name cardinal characteristics when they appeared in a theorem. This is, by the way, the reason that the legend on page 8 contains two columns with references: One to a place in which an inequality between characteristics can be said to have appeared earliest in a history of ideas, and one to a source where explicit definitions and proofs in modern terminology can

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C. CONCERNING C. C. CARDINAL CHARACTERISTIC COMPARISONS 3

be found, would this not be possible in the the original source. Many characteristics were named by van Douwen in [39] but appeared in mathematics earlier. It later became customary to depict characteristics thus named in a Hasse diagram called van Douwen’s diagram.

The unbounding number 𝔟, cf. Definition 1, goes back to Rothberger and Sierpiński, [35, 36], the splitting number 𝔰, cf. Definition 10, to Booth, [10] and the distributivity number 𝔥, cf. Definition 3, to Balcar, Pelant, and Simon, [2]. The study of cardinal characteristics related to measure and category became oriented around what, following Fremlin, [21], became known as Cichoń’s diagram. Again, investigations had begun before that, at the latest with A. Miller’s [30]. The groupwise density number 𝔤, cf. Definition 2 was introduced by Blass and Laflamme, [9]. In [7], Blass defined the evasion number 𝔢, cf. Definition 19 and the linear evasion number 𝔢_{ℓ}, and used them to analyse what he termed the Specker phenomenon thus continuing earlier investigations by Eda, [20]. A subgroup G of the Baer-Specker-group ℤ^{ω}, exhibits the Specker phenomenon if it contains a sequence ⟨a_{i} | i < ω⟩ of linearly independent elements such that for every homomorphism h : G ⟶ ℤ the set {i < ω | h(a_{i}) ≠ 0} is finite. Blass showed that there is a group no larger than min(𝔟, 𝔢) exhibiting the Specker phenomenon and that the smallest group exhibiting it can be no smaller than 𝔢_{ℓ}. Brendle, cf. [15], went on to show that the smallest group exhibiting it is in fact of cardinality 𝔢_{ℓ} which he proved to equal min(𝔟, 𝔢).

Another development starting in the nineties of the last century with a paper by Majcher, [29], is the dualisation of van Douwen’s diagram. The notion of duality here is the same as in the dual form of Ramsey’s Theorem, proved by Carlson and Simpson, cf. [17]: One switches from subsets of a set to partitions of it and as subsets are induced by ranges of injections, partitions are induced by domains of surjections. These dual characteristics were subsequently analysed by several people: In [32], Minami analysed the dual splitting number and the dual reaping number 𝔯_{d} and to that end introduced the notions of pair-splitting and pair-reaping.

The characteristic 𝔢𝔮 was introduced officially by A. Miller and Steprans, [31, p. 57, Def. 16], but one could argue that it was in the air since Miller proved that it is a combinatorial description of the minimal size of a set of reals failing to have strong measure zero, cf. [30, p. 97, Thm. 2.3]. Note, however, that this characterisation holds for ℝ and the Cantor space but fails for the Baire space as discussed by Hrušák, Wohofsky, and Zindulka, cf. [25].

The cardinal characteristics in Definition 22 were first considered under different names in [16] and subsequently analysed in [23, 34, 33].

Another variation of van Douwen’s diagram comes from studying the structure (Dense(ℚ), ⊂_{nwd}) instead of (P(ω), ⊂^{∗}). This was done by Balcar, Brendle, Cichoń, Hernández-Hernández, and Hrušák, cf. [19, 1, 12].

Topological selection principles inspired the definition of the linear refinement number, [38, Definition 61] and the linear excluded middle number, [28, Definition 1.2].

Most recently, Blass, Brendle, Brian, Hamkins, Hardy, and Larson analysed the number of manipulations needed to change the limit of a conditionally convergent series, by respectively rearranging, cf. [6], and dropping some summands, cf. [13]. Meanwhile Brendle, Halbeisen, Klausner, Lischka, and Shelah

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investigated variants of the splitting and unsplitting numbers derived from requiring splitting to happen equitably in various ways.

1.2. Notation, Definitions, a Summary, and a Diagram.

1.2.1. Notation and Definitions. For a set A, we write P(A) for the power set of A; [A]^{κ} for the families of subsets of A having cardinality κ; and [A]^{<κ} for the families of subsets of A having fewer than κ elements. For sets A and B, by writing ^{A}B, we refer to the collection of all functions from A to B. We employ von-Neumann-ordinals, which means that every ordinal number is equal to the set of its predecessors. So 0 denotes the number zero but also the empty set and ω denotes the smallest transfinite ordinal but also the set of all natural numbers. Moreover, this means that for two ordinals α and β, the set α \ β is the half-open interval from β (and if β < α including it) to α but excluding it. By writing X ⊂ Y, we mean that X is a subset of Y, not necessarily a proper subset. Writing X ⊂^{∗} Y means that Y \ X is finite while for sets X and Y of rational numbers, X ⊂_{nwd} Y means that Y \ X is nowhere dense in ℚ.

We denote by M the ideal of meagre (or first category) sets, by N the ideal of sets of Lebesgue measure 0, and by SN the ideal of sets of strong measure zero. Recall that a set X is of strong measure zero if for every sequence ⟨ε_{i} | i < ω⟩ there is a sequence ⟨I_{n} | n < ω⟩ of open intervals with I_{n} having length at most ε_{n} for every natural number n such that X ⊂ ⋃_{n<ω} I_{n}. Finally, Z_{0} denotes the ideal of sets of natural numbers that have asymptotic density 0.

Definition 1. The unbounding number (sometimes called the bounding number) 𝔟 is the minimal cardinality of a family F of functions f ∈ ^{ω}ω such that there is no function g ∈ ^{ω}ω satisfying f ⩽^{∗} g for all f ∈ F.

A family G ⊂ [ω]^{ω} is groupwise dense if it is open and for every interval partition ⟨J_{i} | i < ω⟩ there is an infinite set A of natural numbers such that ⋃_{i∈A} J_{i} ∈ G.

Definition 2. The groupwise density number 𝔤 is the minimal cardinality of a collection C of groupwise dense families such that ⋂ C = 0.

A family F ⊂ [ω]^{ω} is called open if for every X ∈ F we have Y ∈ F for all Y ⊂^{∗} X. It is called dense if for every X ∈ [ω]^{ω} there is a Y ∈ P(X) ∩ F.

Definition 3. The distributivity number 𝔥 is the minimal cardinality of a collection C of dense open families of sets of natural numbers such that ⋂ C = 0.

A family F ⊂ Dense(ℚ) is called ℚ-open if for every X ∈ F we have Y ∈ F for all Y ⊂_{nwd} X. Moreover, a family D ⊂ Dense(ℚ) is called ℚ-dense if for every X ∈ Dense(ℚ) there is a Y ∈ P(X) ∩ D.

Definition 4. The distributivity number of (Dense(ℚ), ⊂_{nwd}), written 𝔥_{ℚ} is the minimal cardinality of a collection C of ℚ-dense ℚ-open families of sets of rationals such that ⋂ C = 0.

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C. CONCERNING C. C. CARDINAL CHARACTERISTIC COMPARISONS 5

Definition 5. For infinite sets S and X of natural numbers and ε ∈]0,^{1}/2[, we say

S splits X if both X ∩ S and X \ S are infinite,
S bisects X if lim_{n↗∞} \frac{|S ∩ X ∩ n|}{|X ∩ n|} = \frac{1}{2},
S infinitely often^{1} bisects X if \frac{|S ∩ X ∩ n|}{|X ∩ n|} = \frac{1}{2} for infinitely many natural numbersn,
S ε-almost bisects X if \{n < ω | \frac{|S ∩ X ∩ n|}{|X ∩ n|} ∉ ]\frac{1}{2} − ε, \frac{1}{2} + ε[ \} is finite.

A set X of natural numbers is called moderate if

0 < lim inf_{n↗∞} \frac{|X ∩ n|}{n} ⩽ lim sup_{n↗∞} \frac{|X ∩ n|}{n} < 1.

We also say that a set A of pairs of natural numbers is unbounded if A ⊄ [ω]^{2} \ [ω \ k]^{2} for every natural number k. Moreover, an infinite set X of natural numbers pair-splits A if A \ ([X]^{2} ∪ [ω \ X]^{2}) is infinite.

Definition 6. A family F of infinite sets of natural numbers is called

splitting if for every infinite set X of natural numbers there is a member of F splitting X, pair-splitting if for every unboundedA ⊂ [ω]^{2} there is a member of F pair-splitting it. unbisected if there is no set S of natural numbers bisecting every member ofF, statistically refining^{2} if there is no moderate S ∈ [ω]^{ω} such that lim_{n↗∞} \frac{n · |S ∩ X ∩ n|}{|S ∩ n| · |X ∩ n|} = 1 for all X ∈ F, bisecting if for every infinite set X of natural numbers there is a member of F bisecting X, cofinally bisecting if for every X ∈ [ω]^{ω} there is a member of F cofinally bisecting X, ε-almost bisecting for an ε ∈]0,^{1}/2[ if for every X ∈ [ω]^{ω} there is a member of F ε-almost bisecting X.

The following notion of splitting number was introduced by Raghavan, cf. [33].

Definition 7. Let P be an infinite partition of ω and a an infinite set of natural numbers. We say that P splits a if a ∩ x is infinite for every x ∈ P. An family F of partitions of ω is called a splitting family of partitions if for every a ∈ [ω]^{ω} there is a P ∈ F splitting a. The partition splitting number 𝔰(𝔭𝔯) is the minimal cardinality of a splitting family of partitions.

It is currently open whether 𝔰(𝔭𝔯) can be different from 𝔰, cf. [33, p. 2].

Definition 8. For sets X, Y ∈ Dense(ℚ) we say that X ℚ-splits Y if both Y ∩ X and Y \ X are dense in ℚ. A family F of dense sets of rational numbers is called ℚ-splitting if for every dense set X of rational numbers there is a member of F which ℚ-splits X.

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Definition 9. The statistically refining^{3} number 𝔯_{∗} is the minimal cardinality of a statistically refining family. Definition 10. The splitting number 𝔰 is the minimal cardinality of a splitting family. Definition 11. The ℚ-splitting number 𝔰_{ℚ} is the minimal cardinality of a ℚ-splitting family. Definition 12. The pair-splitting number 𝔰_{pair} is the minimal cardinality of a pair-splitting family. Definition 13. The bisecting number 𝔰_{_{1}/_{2}} is the minimal cardinality of a bisecting family. Definition 14. The unbisecting (or semirefining^{4}) number 𝔯_{_{1}/_{2}} is the minimal cardinality of an unbisected family. Definition 15. The cofinally bisecting number 𝔰_{^{1}/2}^{∞} is the minimal cardinality of a cofinally bisecting family. Definition 16. The ε-almost bisecting number 𝔰_{^{1}/2±ε} is the minimal cardinality of an ε-almost bisecting family.

Recall that a series of real numbers ∑_{i↗∞} r_{i} is conditionally convergent if it is convergent but there is a set A of natural numbers such that ∑_{i∈A} r_{i} fails to be. Moreover we say that ∑_{i↗∞} r_{i} diverges by oscillation if for every natural number k and every real number r there are ℓ, m, n ∈ ω \ k such that

∑_{i<ℓ} r_{i} ∉]r −^{1}/k, r +^{1}/k[, (1)

∑_{i<m} r_{i} > −k, (2)

∑_{i<n} r_{i} < k. (3)

Definition 17. The subseries number ß is the minimal cardinality of a family I of sets of natural numbers such that for every conditionally convergent series ∑_{i↗∞} r_{i} there is an index set I ∈ I such that ∑_{j∈I} r_{j} fails to converge to a real number. Definition 18. The oscillating subseries number ß_{o} is the minimal cardinality of a family I of sets of natural numbers such that for every conditionally convergent series ∑_{i↗∞} r_{i} there is an index set I ∈ I such that ∑_{j∈I} r_{j} diverges by oscillation.

We call a pair π = (D, ⟨π_{n} | n ∈ D⟩) where D is an infinite set of natural numbers and π_{n} : ^{n}ω ⟶ ω for every n ∈ D, a predictor. We say that π predicts a sequence s ∈ ^{ω}ω if {n ∈ D | π_{n}(s ↾ n) ≠ s(n)} is finite, otherwise s evades π.

3or statistically reaping 4or semireaping

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C. CONCERNING C. C. CARDINAL CHARACTERISTIC COMPARISONS 7

Definition 19. The evasion number 𝔢 is the minimal cardinality of a family E of functions in ^{ω}ω such that every predictor is evaded by a member of E.

Definition 20. 𝔢𝔮 is the minimal cardinality of a bounded family F ⊂ ^{ω}ω such that for every g ∈ ^{ω}ω there is an f ∈ F such that f(n) ≠ g(n) for all natural numbers n.

Definition 21. Given a proper ideal I on a set X,

add(I) is the minimal cardinality of a subfamily ofI whose union is not in I,
cov(I) is the minimal cardinality of a subfamily ofI whose union is X,
non(I) is the minimal cardinality of a an element of P(X) \ I,
cof(I) is the minimal cardinality of a base of I.

We have add(I) ⩽ cov(I). If I contains all singletons then, clearly, add(I) ⩽ non(I), also, the union of any base is then X, hence cov(I) ⩽ cof(I). Also, non(I) ⩽ cof(I) for otherwise one could inductively in step α pick an x_{α} ∈ X \ (B_{α} ∪ {x_{β} | β < α}) for any base {B_{α} | α < cof(I)} ∈ [I]^{cof(I)} which would yield {x_{α} | α < cof(I)} ∈ [X]^{cof(I)} \ I.

Definition 22. Given an ideal I on the set of natural numbers,

add^{∗}(I) is the minimal size of an F ⊂ I such that for allx ∈ I there is a y ∈ F with |x \ y| = ℵ_{0}.
non^{∗}(I) is the minimal size of an F ⊂ [ω]^{ω} such that for allx ∈ I there is a y ∈ F with |x ∩ y| < ℵ_{0}.
cov^{∗}(I) is the minimal size of an F ⊂ I such that for allx ∈ [ω]^{ω} there is a y ∈ F with |y \ x| < ℵ_{0}.
cof^{∗}(I) is the minimal size of an F ⊂ I such that for allx ∈ I there is a y ∈ F with |x ∩ y| < ℵ_{0}.

We now consider partitions of ω. We say that a partition R almost refines another partition X if

{A ∈ X | ∀B ⊂ R : ⋃ B ≠ A} is finite.

We say that S dual-splits X if S is not almost refining X yet there is an infinite partition of the natural numbers almost refining both S and X.

Definition 23. The dual reaping number 𝔯_{d} is the minimal cardinality of a family F of partitions of ω for which there is no single partition of ω dual-splitting all members of F.

A family of sets is called linear, if it is linearly ordered by ⊂^{∗} and a family G is a refinement of a family F if for every A ∈ F there is a B ∈ G with B ⊂^{∗} A.

Definition 24. The linear refinement number 𝔩𝔯 is the minimal cardinality of a family F ⊂ [ω]^{ω} without a linear refinement such that for every finite A ⊂ F the set ⋂ A is infinite.

Definition 25. The linear excluded middle number 𝔩𝔵 is the minimal cardinality of a family F ⊂ ^{ω}ω such that for every h ∈ ^{ω}ω for which S := { {n < ω | f(n) ⩽ h(n)} | f ∈ F} ⊂ [ω]^{ω}, the family S has no linear refinement.

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1.2.2. A Summary and a Diagram.

Summary. The following statements are being proved:

𝔰_{^{1}/2}^{∞} ⩽ max(𝔟, 𝔰)  Theorem 1
𝔢𝔮 ⩽ inf_{ε↗^{1}/2}(𝔰_{^{1}/2±ε}) Theorem 2
𝔯_{∗} ⩽ cov(SN)  Theorem 3
min(𝔟, 𝔯_{_{1}/_{2}}) ⩽ ß  Theorem 4
𝔰(𝔭𝔯) ⩽ non(M)  Theorem 5
𝔢 ⩽ 𝔰_{pair}  Theorem 6
𝔰_{pair} ⩽ ß  Theorem 7

Note that the Mathias model witnesses both the consistency of 𝔥 = ℵ_{2}, [8, Section 11.8], and the one of the Borel conjecture,[5, Thm. 8.3.4]. So in particular, the Mathias model believes in 𝔢𝔮 = non(SN) = ℵ_{1} thus witnessing that the inequalities in Theorems 2, and 6 are consistently strict. Analogously to the proof of Theorem 5 and even more straightforwardly it is possible to show that 𝔰(𝔭𝔯) ⩽ 𝔡. See [8, Thm. 3.3] and [18, Thm. 2.2] to get an idea about proving it. Furthermore, the Random model witnesses that max(𝔟, 𝔡, non(N)) < cov(N) is consistent, [8, Section 11.4], so a fortiori the inequalities in Theorems 4, 5, and 7 are consistently strict.

Legend.

Inequalities  Origin  Proof
cov(N) ⩽ ß ⩽ ß_{o}  ⩽ non(M)  [13]
cov(N) ⩽ 𝔯_{_{1}/_{2}} ⩽ 𝔯_{∗}  ⩽ non(M)  [14]
𝔰 ⩽ 𝔰_{^{1}/2±ε} ⩽ 𝔰_{_{1}/_{2}} ⩽ non(N)  [14]
𝔰 ⩽ 𝔰_{^{1}/2}^{w} ⩽ 𝔰_{^{1}/2}^{∞} ⩽ non(N)  [14]
𝔰 ⩽ 𝔰_{pair} ⩽ non(N)  [32]
𝔯_{d}  ⩽ non(M), non(N) J. Brendle [32, p. 503, Prop. 6]
𝔰_{ℚ}  ⩽ add(M), 𝔰  [12, p. 57]
𝔥_{ℚ} ⩽ 𝔰_{ℚ}  [1, p. 73]
𝔢  ⩽ cov(M)  [26]
𝔥  ⩽ 𝔤  [9, p. 53] [8, p. 433, Prop. 6.27]
𝔥  ⩽ 𝔰  [3, p. 354] [8, p. 427, Thm. 6.9]
cov(M) ⩽ 𝔢𝔮  [4]  [8, p. 420, Thm. 5.9]
𝔟  ⩽ non(M)  [4, p. 237] [8, p. 419, Prop. 5.5]
𝔰  ⩽ non(M)  Folklore [8, p. 424, Thm. 5.19]
max(𝔟, 𝔰, 𝔩𝔯) ⩽ 𝔩𝔵  [28, Cor. 2.13]
cov^{∗}(Z_{0}) ⩽ max (𝔟, 𝔰(𝔭𝔯)) [33, Cor. 30]

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ß ß_{o} 𝔰𝔰_{ℚ} 𝔰_{^{1}/2}^{∞} non(N) max(𝔰_{^{1}/2}^{∞}, 𝔰_{_{1}/_{2}}) max(𝔟, 𝔰) max(𝔟, 𝔰(𝔭𝔯)) cov^{∗}(Z_{0}) 𝔩𝔵 𝔩𝔯 non(M) cov(SN) 𝔯_{d} cov(N) cov(M) 𝔥 𝔤 𝔥_{ℚ} 𝔢𝔢𝔮 𝔰_{^{1}/2±ε} 𝔰_{_{1}/_{2}} 𝔰_{^{1}/2}^{w} min(𝔟, 𝔯_{_{1}/_{2}}) 𝔯_{_{1}/_{2}} 𝔯_{∗} 𝔢𝔰_{pair}

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2. Results

2.1. Inequalities Involving Halfway New Cardinal Characteristics.

2.1.1. Splitting.

Theorem 1. 𝔰_{^{1}/2}^{∞} ⩽ max(𝔟, 𝔰).

This proof is to a certain degree inspired by the work of Kamburelis and Węglorz, [27].

Proof. Assume towards a contradiction that max(𝔟, 𝔰) < 𝔰_{^{1}/2}^{∞} and let S ⊂ [ω]^{ω}, U ⊂ ^{ω}ω be families which are respectively splitting, of size 𝔰, and unbounded, of size 𝔟. We may assume without loss of generality that all f ∈ U are properly increasing and satisfy f(0) > 0. For any function f : ω ⟶ ω, let g_{f} : ω ⟶ ω be the function given inductively by g_{f} (0) := 0 and g_{f} (n + 1) := f(g_{f} (n)). Now we define

b(S, f) := ⋃_{n∈S} g_{f} (n + 1) \ g_{f} (n) for S ∈ S and f ∈ U,
B :={b(S, f) | {S, ω \ S} ⊄ [ω]^{ω} \ S ∧ f ∈ U}.

Clearly, |B| ⩽ 2 · 𝔟 · 𝔰 = max(𝔟, 𝔰). As max(𝔟, 𝔰) < 𝔰_{^{1}/2}^{∞}, the family B cannot be infinitely often bisecting, so let X ∈ [ω]^{ω} witness this, i.e. X is not infinitely often bisected by any member of B. Now define a function h : ω ⟶ ω by n ⟼ min({i < ω | |X ∩ i \ n| > n}) and pick an A ∈ U which is unbounded by h. Let this be witnessed by Y ∈ [ω]^{ω}, i.e. h(n) < A(n) for all n ∈ Y. Let Z ∈ S split Y. As B := b(Z, A) fails to infinitely bisect X, there is a natural number k such that 2|B ∩ X ∩ n| ≠ |X ∩ n| for all n ∈ ω \ k. Note that under these circumstances, 2|B ∩ X ∩ k| < |X ∩ k| implies 2|B ∩ X ∩ n| < |X ∩ n| for all n ∈ ω \ k and 2|B ∩ X ∩ k| > |X ∩ k| implies 2|B ∩ X ∩ n| > |X ∩ n| for all n ∈ ω \ k, in other words, the density of B in X up to n cannot, as n gets larger, change between being larger than one half and smaller than one half without assuming the value of one half itself. As {Ξ | ω \ Ξ ∈ B} = B, we may suppose without loss of generality that

2|B ∩ X ∩ n| < |X ∩ n| for all n ∈ ω \ k. (4)

Now let ℓ ∈ Y ∩Z be minimal such that g_{A}(ℓ) ⩾ k. Note that g_{A}(ℓ+1)\g_{A}(ℓ) ⊂ B and h(g_{A}(ℓ)) < g_{A}(ℓ+1). We have

g_{A}(ℓ) < |X ∩ h(g_{A}(ℓ)) \ g_{A}(ℓ)|,
X ∩ g_{A}(ℓ + 1) ⊂ (B ∩ X ∩ g_{A}(ℓ + 1)) ∪ g_{A}(ℓ) and
X ∩ h(g_{A}(ℓ)) \ g_{A}(ℓ) ⊂ B ∩ X ∩ g_{A}(ℓ + 1),

contradicting (4) □

Theorem 2. 𝔢𝔮 ⩽ inf_{ε↗^{1}/2}(𝔰_{^{1}/2±ε}).

The following proof uses an partition of the natural numbers into intervals whose lengths are growing sufficiently quickly such that one can define an eventually different sequence from a bisecting sequence.

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C. CONCERNING C. C. CARDINAL CHARACTERISTIC COMPARISONS 11

Proof. Suppose towards a contradiction that 𝔰_{^{1}/2±ε} < 𝔢𝔮 for some ε ∈ ]0,^{1}/2[ and let S be an ε-almost bisecting family of cardinality 𝔰_{^{1}/2±ε}. Let ⟨I_{k} | k < ω⟩ be the interval partition of the natural numbers defined by min(I_{0}) = 0, min(I_{k+1}) = max(I_{k}) + 1 and max(I_{k}) = k + 2⌈2 min(I_{k})/(1 − 2ε)⌉ for all natural numbers k. Note that every interval has an odd length. For every natural number k, choose a function h_{k} : P(I_{k}) ⟶ 2^{max(I_{k})−min(I_{k})} such that h_{k}(A) = h_{k}(B) if and only if A ∈ {B, I_{k} \ B}. Now we can define

E := {e ∈ ^{ω}ω | ∃S ∈ S ∃k ∈ ω ∀ℓ ∈ ω \ k : e(ℓ) = h_{ℓ}(S ∩ I_{ℓ})} .

We have |E| = ℵ_{0} · |S| = ℵ_{0} · 𝔰_{^{1}/2±ε} = 𝔰_{^{1}/2±ε} < 𝔢𝔮. As E is bounded, there is an f ∈ ^{ω}ω such that for all e ∈ E there is a natural number ℓ such that e(ℓ) = f(ℓ). Now, for every natural number k, let g_{k} be the right-inverse of h_{k} such that |g_{k}(ℓ)| > |I_{k} \ g_{k}(ℓ)| for all ℓ < 2^{max(I_{k})−min(I_{k})}. We consider the set

X := ⋃_{ℓ<ω} g_{ℓ}(max(f(ℓ), 2^{max(I_{ℓ})−min(I_{ℓ})} − 1)).

Now there is an S ∈ S which is ε-almost bisecting X. Let m be a natural number such that

\frac{|S ∩ X ∩ n|}{|X ∩ n|} ∈ ]\frac{1}{2} − ε, \frac{1}{2} + ε[ for all n ∈ ω \ m.

Let k be the least natural number such that I_{k} ⊂ ω \ m. We define a function,

d : ω ⟶ ω
n ⟼ \begin{cases}f(n) + 1 & if n < k, \\ h_{n}(S ∩ I_{n}) & else. \end{cases}

Clearly, d ∈ E. Now there is a natural number ℓ such that d(ℓ) = f(ℓ). By definition of d we have ℓ ∈ ω \k, moreover

f(ℓ) = d(ℓ) = h_{ℓ}(S ∩ I_{n}) < 2^{max(I_{ℓ})−min(I_{ℓ})}.

Next we distinguish two cases. First assume that 2|S ∩ I_{ℓ}| < |I_{ℓ}|. Then

X ∩ I_{ℓ} = g_{ℓ}(max(f(ℓ), 2^{max(I_{ℓ})−min(I_{ℓ})} − 1)) = g_{ℓ}(f(ℓ)) = g_{ℓ}(d(ℓ)) = g_{ℓ}(h_{ℓ}(I_{ℓ} ∩ S)) = I_{ℓ} \ S,

and therefore S ∩ X ∩ min(I_{ℓ+1}) ⊂ min(I_{ℓ}). We have

\frac{|X ∩ S ∩ min(I_{ℓ+1})|}{|X ∩ min(I_{ℓ+1})|} = \frac{|X ∩ S ∩ min(I_{ℓ})|}{|X ∩ min(I_{ℓ+1})|} ⩽ \frac{min(I_{ℓ})}{|X ∩ min(I_{ℓ+1})|}

< \frac{2 min(I_{ℓ})}{min(I_{ℓ+1})} < \frac{2 min(I_{ℓ})}{max(I_{ℓ})} ⩽ \frac{2 min(I_{ℓ})(1 − 2ε)}{ℓ(1 − 2ε) + 4 min(I_{ℓ})} ⩽ \frac{1}{2} − ε,

implying ℓ < k, a contradiction.

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Now we assume |I_{ℓ}| < 2|S ∩ I_{ℓ}|, we have

X ∩ I_{ℓ} = g_{ℓ}(max(f(ℓ), 2^{max(I_{ℓ})−min(I_{ℓ})} − 1)) = g_{ℓ}(f(ℓ)) = g_{ℓ}(d(ℓ)) = g_{ℓ}(h_{ℓ}(I_{ℓ} ∩ S)) = I_{ℓ} ∩ S,

and therefore S ∩ I_{ℓ} ⊂ X ∩ S. Then

\frac{|X ∩ S ∩ min(I_{ℓ+1})|}{|X ∩ min(I_{ℓ+1})|} ⩾ \frac{|S ∩ I_{ℓ}|}{|X ∩ min(I_{ℓ+1})|} ⩾ \frac{|S ∩ I_{ℓ}|}{min(I_{ℓ}) + |X ∩ I_{ℓ}|} = \frac{|S ∩ I_{ℓ}|}{min(I_{ℓ}) + |S ∩ I_{ℓ}|} and as,

for nonnegative values, the function \frac{x}{a + x} is monotonically increasing in x,

⩾ \frac{|I_{ℓ}|}{2 min(I_{ℓ}) + |I_{ℓ}|} = \frac{min(I_{ℓ+1}) − min(I_{ℓ})}{min(I_{ℓ+1}) + min(I_{ℓ})} > \frac{max(I_{ℓ}) − min(I_{ℓ})}{max(I_{ℓ}) + min(I_{ℓ})}

⩾ \frac{4 min(I_{ℓ}) + (1 − 2ε)(ℓ − min(I_{ℓ}))}{4 min(I_{ℓ}) + (1 − 2ε)(ℓ + min(I_{ℓ}))} and as,

for nonnegative values, the function \frac{a + b(x − 1)}{a + b(x + 1)} is monotonically increasing in x,

⩾ \frac{3 + 2ε}{5 − 2ε} > \frac{1}{2} + ε, implying ℓ < k, a contradiction.

□

2.1.2. Reaping.

Theorem 3. 𝔯_{∗} ⩽ cov(SN).

The following proof uses the fact that a bisecting real can be used to code avoidance of a sequence of sufficiently small open intervals.

Proof. Assume towards a contradiction that cov(SN) < 𝔯_{∗} and let C be a family of cov(SN) sets of strong measure zero covering the Cantor space. Let ⟨J_{i} | i < ω⟩ be an interval partition defined by J_{i} := 1 + i + (i + 2)! \ ⋃_{k<i} J_{k}. Note that all parts of this partition are of odd length! Furthermore, let

⟨P_{i} | i < ω⟩ be a partition of ω into infinitely many infinite parts with P_{i} ⊂ ω \ i for all natural numbers i. For every Z ∈ C let ⟨s_{j}^{Z} | j < ω⟩ be a sequence of s_{j}^{Z} ∈ ^{⋃_{i⩽j} J_{i}}2 such that for all natural numbers k and all x ∈ Z there is a j ∈ P_{k} such that x ↾ ⋃_{i⩽j} J_{i} = s_{j}^{Z}. This is possible as Z is of strong measure zero. Now we can define

D := {X_{Z} | Z ∈ C}
where X_{Z} := ⋃_{i<ω} F_{i}^{Z}
and F_{i}^{Z} is the larger of the two sets {k ∈ J_{i} | s_{i}^{Z}(k) = 0} and {k ∈ J_{i} | s_{i}^{Z}(k) = 1} .

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As |D| ⩽ |C| = cov(SN) < 𝔯_{∗}, there is a moderate S ∈ [ω]^{ω} such that lim_{n↗∞} \frac{n · |S ∩ X ∩ n|}{|S ∩ n| · |X ∩ n|} = 1 for all X ∈ D. As C covers the Cantor space, we can fix a Z ∈ C containing χ_{S}. Let ε ∈ ]0,^{1}/2[ be such that

ε < lim inf_{n↗∞} \frac{|S ∩ n|}{n} and lim sup_{n↗∞} \frac{|S ∩ n|}{n} < 1 − ε.
By the choice of S we then also have ε < lim inf_{n↗∞} \frac{|S ∩ X_{Z} ∩ n|}{|X_{Z} ∩ n|} and lim sup_{n↗∞} \frac{|S ∩ X_{Z} ∩ n|}{|X_{Z} ∩ n|} < 1 − ε.

Now choose a natural number k > ^{3}/ε large enough such that \frac{|S ∩ X_{Z} ∩ n|}{|X_{Z} ∩ n|} ∈ ]ε, 1 − ε[ for all n ∈ ω \ k. Choose a natural number j such that P_{j} ⊂ ω \ k and let ℓ ∈ P_{j} be such that χ_{S} ↾ ⋃_{i⩽ℓ} J_{i} = s_{ℓ}^{Z}. Note that 2|X_{Z} ∩ ⋃_{i⩽ℓ} J_{i}| > | ⋃_{i⩽ℓ} J_{i}|. Also note that, as P_{j} ∋ ℓ consists of natural numbers no smaller than k,

we have ℓ ⩾ k > ^{3}/ε. We now distinguish two cases. First assume, that F_{ℓ}^{Z} = {k ∈ J_{ℓ} | s_{ℓ}^{Z}(k) = 0}. Then S ∩ X_{Z} ∩ ⋃_{i⩽ℓ} J_{i} ⊂ ⋃_{i<ℓ} J_{i} and hence

\frac{|S ∩ X_{Z} ∩ ⋃_{i⩽ℓ} J_{i}|}{|X_{Z} ∩ ⋃_{i⩽ℓ} J_{i}|} ⩽ \frac{2| ⋃_{i<ℓ} J_{i}|}{| ⋃_{i⩽ℓ} J_{i}|} = \frac{2(ℓ + (ℓ + 1)!)}{1 + ℓ + (ℓ + 2)!} < \frac{2(ℓ + (ℓ + 1)!)}{ℓ + (ℓ + 2)!} = \frac{2(1 + (ℓ + 1)(ℓ − 1)!)}{1 + (ℓ + 1)(ℓ + 2)(ℓ − 1)!}

< \frac{2(1 + (ℓ + 1)(ℓ − 1)!)}{(ℓ + 1)(ℓ + 2)(ℓ − 1)!} ⩽ \frac{2(ℓ + 2)(ℓ − 1)!}{(ℓ + 1)(ℓ + 2)(ℓ − 1)!} = \frac{2}{ℓ + 1} < \frac{2}{ℓ} < \frac{2}{^{3}/ε} = \frac{2ε}{3} < ε.

Now assume, that F_{ℓ}^{Z} = {k ∈ J_{ℓ} | s_{ℓ}^{Z}(k) = 1}. Then X_{Z} ∩ J_{ℓ} ⊂ S ∩ X_{Z} ∩ ⋃_{i⩽ℓ} J_{i} and hence

\frac{|S ∩ X_{Z} ∩ ⋃_{i⩽ℓ} J_{i}|}{|X_{Z} ∩ ⋃_{i⩽ℓ} J_{i}|} ⩾ \frac{|X_{Z} ∩ J_{ℓ}|}{|X_{Z} ∩ ⋃_{i⩽ℓ} J_{i}|} = \frac{|X_{Z} ∩ J_{ℓ}|}{|X_{Z} ∩ ⋃_{i<ℓ} J_{i}| + |X_{Z} ∩ J_{ℓ}|}.
As X_{Z} ∩ J_{ℓ} = F_{ℓ}^{Z} and 2|F_{ℓ}^{Z}| > |J_{ℓ}| and for nonnegative values, the function

\frac{x}{a + x} is monotonically increasing in x,

\frac{|X_{Z} ∩ J_{ℓ}|}{|X_{Z} ∩ ⋃_{i<ℓ} J_{i}| + |X_{Z} ∩ J_{ℓ}|} > \frac{|J_{ℓ}|}{2|X_{Z} ∩ ⋃_{i<ℓ} J_{i}| + |J_{ℓ}|} ⩾ \frac{|J_{ℓ}|}{2| ⋃_{i<ℓ} J_{i}| + |J_{ℓ}|}

= \frac{1 + (ℓ + 2)! − (ℓ + 1)!}{(ℓ + 2)! + (ℓ + 1)! + 2ℓ + 1} > \frac{(ℓ + 2)! − (ℓ + 1)!}{(ℓ + 2)! + 2(ℓ + 1)!} = \frac{ℓ + 1}{ℓ + 4} > \frac{3 + ε}{3 + 4ε} > 1 − ε.

In both cases we reach a contradiction thus concluding the proof. □

Theorem 4. min(𝔟, 𝔯_{_{1}/_{2}}) ⩽ ß.

The idea of the following proof is to use a bound for the distances between elements of the index set of a subseries and observe that a bisecting sequence implies convergence of the respective subseries.

Proof. Assume towards a contradiction that ß < min(𝔟, 𝔯_{_{1}/_{2}}). Let I be a ß-sized family of infinite index sets of natural numbers such that for every conditionally convergent series ∑_{i→∞} r_{i} there is an index set

I ∈ I such that ∑_{i∈I} r_{i} is diverging. As |I| = ß < 𝔯_{_{1}/_{2}}, there is an S ∈ [ω]^{ω} bisecting all members of I. We

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define a family

B := {f_{I} | I ∈ I} , where
f_{I} := ⟨n_{j}^{I} | j < ω⟩, and
n_{j}^{I} := min (\{k < ω | ∀ℓ ∈ ω \ k : \frac{|S ∩ I ∩ ℓ|}{|I ∩ ℓ|} ∈ ]\frac{1}{2} − \frac{1}{2j} , \frac{1}{2} + \frac{1}{2j} [ \}) .

As |B| ⩽ |I| = ß < 𝔟, there is a g ∈ ^{ω}ω such that f ⩽^{∗} g for all f ∈ B. We may suppose without loss of generality that g(i)2(i + 1) ⩽ g(i + 1) for all natural numbers i. We define a conditionally convergent series

∑_{k→∞} t_{k}

with t_{k} = \begin{cases}\frac{(−1)^{i}}{g(i)(i+1)(2i+1)} for k ∈ g(i)2(i + 1) \ g(i), \\ 0 else. \end{cases}

By assumption, there is an index set I ∈ I such that ∑_{j∈I} t_{j} is diverging. Let k ∈ ω \ 30 be such that

f_{I}(ℓ) ⩽ g(ℓ) for all ℓ ∈ ω \ k. Since

∑_{j∈I} t_{j} = ∑_{j↗∞} u_{j} with
u_{j} := ∑_{ℓ∈I∩g(j)2(j+1)\g(j)} t_{ℓ},

and u_{j} < 0 if and only if j is odd, the Leibniz criterion implies that there has to be a j ∈ ω \ k such that |u_{j}| ⩽ |u_{j+1}|. But then

∑_{ℓ∈I∩g(j)2(j+1)\g(j)} |t_{ℓ}| = |u_{j}| ⩽ |u_{j+1}| = ∑_{ℓ∈I∩g(j+1)2(j+2)\g(j+1)} |t_{ℓ}|

⟺ ∑_{ℓ∈I∩g(j)2(j+1)\g(j)} \frac{1}{g(j)(j + 1)(2j + 1)} = |u_{j}| ⩽ |u_{j+1}| = ∑_{ℓ∈I∩g(j+1)2(j+2)\g(j+1)} \frac{1}{g(j + 1)(j + 2)(2j + 3)}
⟺\frac{|I ∩ g(j)2(j + 1) \ g(j)|}{g(j)(j + 1)(2j + 1)} = |u_{j}| ⩽ |u_{j+1}| = \frac{|I ∩ g(j + 1)2(j + 2) \ g(j + 1)|}{g(j + 1)(j + 2)(2j + 3)}

We have k ⩽ j and therefore f_{I}(j) ⩽ g(j), so by definition of f_{I}, the absolute values of u_{j} and u_{j+1} can be estimated as follows:

|u_{j}| ⩾\frac{(^{1}/2 −^{1}/2j) g(j)2(j + 1) − (^{1}/2 +^{1}/2j) g(j)}{g(j)(j + 1)(2j + 1)} = \frac{(j − 1)2(j + 1) − (j + 1)}{2j(j + 1)(2j + 1)} = \frac{2j^{2} − j − 3}{j(4j^{2} + 6j + 2)}
|u_{j+1}| ⩽\frac{(^{1}/2 +^{1}/2j) g(j + 1)2(j + 2) − (^{1}/2 −^{1}/2j) g(j + 1)}{g(j + 1)(j + 2)(2j + 3)} = \frac{2(j + 1)(j + 2) − (j − 1)}{2j(j + 2)(2j + 3)} = \frac{2j^{2} + 5j + 5}{j(4j^{2} + 14j + 12)}

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C. CONCERNING C. C. CARDINAL CHARACTERISTIC COMPARISONS 15

Therefore,

j^{2}(4j^{2} + 6j + 2)(4j^{2} + 14j + 12)|u_{j}| ⩽ j^{2}(4j^{2} + 6j + 2)(4j^{2} + 14j + 12)|u_{j+1}|

⟹(2j^{2} − j − 3)(4j^{2} + 14j + 12) ⩽ (2j^{2} + 5j + 5)(4j^{2} + 6j + 2)
⟺8j^{4} + 24j^{3} − 2j^{2} − 54j − 36 ⩽ 8j^{4} + 22j^{3} + 54j^{2} + 40j + 10
⟺2j^{3} − 56j^{2} − 94j − 46 ⩽ 0
⟺27j^{3} − 756j^{2} − 1269j − 621 ⩽ 0
⟹27j^{3} − 756j^{2} − 1269j − 1958 ⩽ 0
⟺(3j − 89)(9j^{2} + 15j + 22) ⩽ 0
⟺9j^{2} + 15j + 22 ⩽ 0
⟺36j^{2} + 60j + 88 ⩽ 0
⟺(6j + 5)^{2} ⩽ −63, a contradiction.
□

2.2. Various Inequalities.

2.2.1. Splitting of Partitions and the Uniformity of the Meagre Ideal.

Theorem 5. 𝔰(𝔭𝔯) ⩽ non(M).

The following proof uses Bartoszyński’s characterisation of non(M) via sequences of natural numbers equalling each other infinitely often. Such sequences are used in this proof to code sets of natural numbers thus guessing larger and larger parts of an infinite set of natural numbers thus enabling the definition of a splitting partition.

Proof. Assume towards a contradiction that non(M) < 𝔰(𝔭𝔯). Recall that non(M) is the minimal cardinality of an infinitely often equal family, cf. [8, Thm. 5.9]. So let E be an infinitely often equal family of size non(M). Let a bijection e : ω ⟶ [ω]^{<ω} encode all finite sets of natural numbers by a natural number. For every sequence s ∈ ^{ω}ω we inductively define an infinite partition P_{s} of ω. If i < j or |e(s(i))| ≠ (2i)!(i + 1) let Q_{s}(j, i) := 0, otherwise let ⟨f_{s}(i, j) | j < (2i)!(i + 1)⟩ be the ascending enumeration of e(s(i)) and

Q_{s}(j, i) := {f_{s}(i, m) | m ∈ (2i)!(j + 1) \ (2i)!j} \ ⋃_{k,ℓ<i} Q_{s}(ℓ, k).

Note that |Q_{s}(j, i)| ⩽ (2i)! for all natural numbers i and j. Having so defined pairwise disjoint Q_{s}(j, i) for all i, j < ω, let

P_{s}(j) := ⋃_{i<ω} Q_{s}(j, i) for all positive natural numbers j and P_{s}(0) := ω \ ⋃_{j∈ω\1} P_{s}(j),
we have P_{s}(0) ⊃ ⋃_{i<ω} Q_{s}(0, i).

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We now define a family of partitions P := {P_{s} | s ∈ E}. As |P| ⩽ |E| = non(M) < 𝔰(𝔭𝔯) there is an infinite set A of natural numbers which is not split by any member of P. Let ⟨a_{i} | i < ω⟩ be the inreasing enumeration of A. Now consider the sequence s := ⟨e^{−1}({a_{j} | j < (2i)!(i + 1)}) | i < ω⟩. As E is an infinitely often equal family, there is a t ∈ E infinitely often equalling s. As P_{t} does not split A there is a natural number n such that P_{t}(n) ∩ A is finite. Let k be a natural number such that A ∩ P_{t}(n) ⊂ k and ℓ be one such that B := {i ∈ ℓ \ (n + 1) | s(i) = t(i)} has more than k elements. We have

A ∩ ⋃_{i∈B} Q_{s}(n, i) = A ∩ ⋃_{i∈B} Q_{t}(n, i) ⊂ A ∩ ⋃_{i<ℓ} Q_{t}(n, i) ⊂ A ∩ ⋃_{i<ω} Q_{t}(n, i) = A ∩ P_{t}(n) ⊂ k

so by the pigeonhole principle there has to be an i ∈ B such that Q_{s}(n, i) ⊂ ω \ A. Remember that a_{j} = f_{s}(i, j) for all j ⩽ i < ω. So we have

{a_{j} | j ∈ (2i)!(n + 1) \ (2i)!n} \ ⋃_{m,j<i} Q_{s}(m, j) = Q_{s}(n, i) ⊂ ω \ A and hence
{a_{j} | j ∈ (2i)!(n + 1) \ (2i)!n} ⊂ ⋃_{m,j<i} Q_{s}(m, j), implying i ⩾ 1 and
(2i)! ⩽ ∑_{j,m<i} |Q_{s}(m, j)| ⩽ i ∑_{j<i}(2j)! ⩽ i^{2}(2i − 2)!.
Therefore (2i)(2i − 1) ⩽ i^{2},
so i(3i − 2) ⩽ 0 contradicting both factors of this product being no smaller than 1.
□

2.2.2. Pair-Splitting.

Theorem 6. 𝔢 ⩽ 𝔰_{pair}.

The following proof uses an interval partition with intervals of sufficiently quickly growing lengths such that, regardless of the behaviour of a set S of integers on previous intervals, a predictor will still narrow down the possibilities of that set on the next interval sufficiently to point to two elements in the current interval which are either both in S or both outside of S.

Proof. Assume towards a contradiction that 𝔰_{pair} < 𝔢 and let S be a pair-splitting family of cardinality 𝔰_{pair}. We inductively define a function f : ω ⟶ ω by f(0) := 3 and f(i + 1) := 2f(i) (1 + 2^{2^{f(i)}}) and use it to define an interval partition ⟨J_{i} | i < ω⟩ by setting J_{i} := f(i) \ ⋃_{k<i} J_{k} for all natural numbers i. Note

the all intervals in this partition have odd length. Let ⟨F_{i} | i < ω⟩ be an enumeration of all finite sets of natural numbers. We define a family

E := {g_{S} | S ∈ S}, where
g_{S} := ⟨m_{k}^{S} | k < ω⟩ for S ∈ S, and
m_{k}^{S} is such that F_{m_{k}^{S}} = S ∩ J_{k}.

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C. CONCERNING C. C. CARDINAL CHARACTERISTIC COMPARISONS 17

As |E| ⩽ |S| = 𝔰_{pair} < 𝔢, there is a predictor π = (B, ⟨π_{i} | i ∈ B⟩) predicting every member of E.

Claim. For every natural number ℓ there is a pair a_{ℓ} = {b_{2ℓ}, b_{2ℓ+1}} ⊂ J_{ℓ} (where b_{2ℓ} < b_{2ℓ+1}) such that for every x ∈ E, we have a_{ℓ} ∩ F_{π_{k}(x↾ℓ)} ∈ {0, a_{ℓ}}.

Proof of Claim. For ℓ = 0 the claim holds true by the pigeonhole principle. In the following we show the claim to holds for ℓ = k + 1. For any natural number k there are 2^{f(k)} subsets of f(k) and hence just as many initial segments x ↾ k for x ∈ E. Let ⟨t_{i} | i < 2^{f(k)}⟩ be an enumeration of these initial segments. Set K_{2^{f(k)}} := J_{k+1} and for every i ⩽ 2^{f(k)}, let K_{i} be the bigger one of the two sets K_{i+1} ∩ F_{π_{k}(t_{i})} and K_{i+1} \ F_{π_{k}(t_{i})}. Inductively we have |K_{i}| ⩾ 2^{i} + 1 for every i ⩽ 2^{f(k)}. Now we let b_{2k} := min(K_{0}) and b_{2k+1} := max(K_{0}). Then a_{k} := {b_{2k}, b_{2k+1}} provides what was demanded. (End of Proof of Claim.)

Now we let A := {a_{ℓ} | ℓ < ω}. The set A is an unbounded set of pairs, therefore there is an S ∈ S pair-splitting it. Let m be a natural number such that π_{i}(g_{S} ↾ i) = g_{S}(i) for all i ∈ B \ m and let n ∈ B \ m be such that a_{n} ∈ A \ [g(m)]^{2} is split by S, i.e. exactly one of b_{2n}, b_{2n+1} is in S, in other words, S ∩ a_{n} ∉ {0, a_{n}}. Then,

{0, a_{n}} ∋ a_{n} ∩ F_{π_{n}(g_{S}↾n)} = a_{n} ∩ F_{g_{S}(n)} = a_{n} ∩ F_{m_{n}^{S}} = a_{n} ∩ J_{n} ∩ S = a_{n} ∩ S ∉ {0, a_{n}}, a contradiction.

□

Theorem 7. 𝔰_{pair} ⩽ ß.

The following proof depends on the observation that for any unbounded set A of pairs of natural numbers one can define an alternating harmonic series padded with zeros(a technique employed in [6] and [13]) such that any index set rendering the respective subseries divergent will also pair-split A.

Proof. Assume towards a contradiction that ß < 𝔰_{pair} and let I be an ß-sized family of infinite index sets of natural numbers such that for every conditionally convergent series ∑_{i↗∞} r_{i} there is an index set I ∈ I

such that ∑_{i∈I} r_{i} is diverging. As |I| = ß < 𝔰_{pair}, there is an unbounded collection A of pairs of natural

numbers which is not pair-split by any member of the family S. Let ⟨a_{i} | i < ω⟩ be an enumeration of A and inductively define a B ∈ [A]^{ω} by setting

B_{0} :=0;
B_{i+1} := \begin{cases}B ∪ \{a_{i}\} if max(b) < min(a_{i}) for all b ∈ B_{i}, \\ B_{i+1} := B_{i} otherwise. \end{cases}

and finally B := ⋃_{i<ω} B_{i}. Clearly, B is an infinite unbounded subcollection of A, moreover we have a_{1} < a_{2} or a_{3} < a_{0} for {{a_{0}, a_{1}}_{<}, {a_{2}, a_{3}}_{<}} ∈ [B]^{2}. Let ⟨b_{i} | i < ω⟩ be the increasing enumeration of ⋃ B. Clearly, B = {{b_{2i}, b_{2i+1}} | i < ω}. Let n(i) := min({j < ω | b_{j} ⩾ i}) for all natural numbers i. We

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now consider the series

∑_{i<↗∞} r_{i},

where r_{i} := \begin{cases}\frac{(−1)^{n(i)}}{n(i)+1} & if i ∈ ⋃ A, \\ 0 & else. \end{cases}

This is just the alternating harmonic series padded with zeroes, and therefore it is conditionally convergent. Now let I ∈ I be such that ∑_{i∈I} r_{i} diverges. Let k be a natural number such that A ⊂ [k]^{2} ∪ [ω \ k]^{2} and

a ⊂ I or a ⊂ ω \ I for all a ∈ A \ [k]^{2}. As ∑_{i∈I} r_{i} diverges, so does ∑_{i∈I\k} r_{i}. But ∑_{i∈I\k} r_{i} = ∑ ; i∈I∩⋃ A\k r_{i}

which, being divergent, by the Leibniz criterion cannot be an alternating series. Then a ⊄ I or a ⊄ ω \ I for some a ∈ A \ [k]^{2}, a contradiction. □

Since together with Theorem 6 we now established the inequality 𝔢 ⩽ ß between characteristics originating in, respectively, algebra and analysis, let us point out the following immediate Corollary.

Corollary. For any cardinal κ < 2^{ℵ_{0}} at least one of the the following holds:

• For all families F ∈ [[ω]^{ω}]^{κ} there is a conditionally convergent series ∑_{i↗∞} r_{i} such that for all

index sets I ∈ F the series ∑_{i∈I} r_{i} converges.

• There is a subgroup of the Baer-Specker group no larger than κ exhibiting the Specker phenomenon.

References

[1] Bohuslav Balcar, Fernando Hernández-Hernández, and Michael Hrušák. Combinatorics of dense subsets of the rationals. Fund. Math., 183(1):59–80, 2004.

[2] Bohuslav Balcar, Jan Pelant, and Petr Simon. The space of ultrafilters on N covered by nowhere dense sets. Fund. Math., 110(1):11–24, 1980.

[3] Bohuslav Balcar and Petr Simon. Disjoint refinement. In Handbook of Boolean algebras, Vol. 2, pages 333–388. North-Holland, Amsterdam, 1989.

[4] Tomek Bartoszyński. Combinatorial aspects of measure and category. Fund. Math., 127(3):225–239, 1987.

[5] Tomek Bartoszyński and Haim Judah. Set theory. A K Peters Ltd., Wellesley, MA, 1995. On the structure of the real line.

[6] Andreas Blass, Jörg Brendle, Will Brian, Joel David Hamkins, Michael Hardy, and Paul B. Larson. The rearrangement number. Trans. Amer. Math. Soc., 373(1):41–69, 2020.

[7] Andreas Raphael Blass. Cardinal characteristics and the product of countably many infinite cyclic groups. J. Algebra, 169(2):512–540, 1994.

[8] Andreas Raphael Blass. Combinatorial cardinal characteristics of the continuum. In Handbook of set theory. Vols. 1, 2, 3, pages 395–489. Springer, Dordrecht, 2010.

[9] Andreas Raphael Blass and Claude Laflamme. Consistency results about filters and the number of inequivalent growth types. J. Symbolic Logic, 54(1):50–56, 1989.

[10] David Booth. A Boolean view of sequential compactness. Fund. Math., 85(2):99–102, 1974.

Preprint

[11] Jörg Brendle. Cardinal invariants of the continuum—a survey. Sūrikaisekikenkyūsho Kōkyūroku, (1202):7–32, 2001. Axiomatic set theory (Japanese) (Kyoto, 2000).

[12] Jörg Brendle. Van Douwen’s diagram for dense sets of rationals. Ann. Pure Appl. Logic, 143(1-3):54–69, 2006.

[13] Jörg Brendle, Will Brian, and Joel David Hamkins. The subseries number.

[14] Jörg Brendle, Lorenz J. Halbeisen, Lukas Daniel Klausner, Marc Lischka, and Saharon Shelah. Halfway new cardinal characteristics.

[15] Jörg Brendle and Saharon Shelah. Evasion and prediction. II. J. London Math. Soc. (2), 53(1):19–27, 1996.

[16] Jörg Brendle and Saharon Shelah. Ultrafilters on ω—their ideals and their cardinal characteristics. Trans. Amer. Math. Soc., 351(7):2643–2674, 1999.

[17] Timothy John Carlson and Stephen George Simpson. A dual form of Ramsey’s theorem. Adv. in Math., 53(3):265–290, 1984.

[18] William Chen, Shimon Garti, and Thilo Weinert. Cardinal characteristics of the continuum and partitions. Israel J. Math., 235(1):13–38, January 2020.

[19] Jacek Cichoń. On dense subsets of rational numbers. volume 42, pages 5–10. 2001. 29th Winter School on Abstract Analysis (Lhota nad Rohanovem/Zahrádky u České Lípy, 2001).

[20] Katsuya Eda. A note on subgroups of Z^{N}. In Abelian group theory (Honolulu, Hawaii, 1983), volume 1006 of Lecture Notes in Math., pages 371–374. Springer, Berlin, 1983.

[21] David Heaver Fremlin. Cichoń’s diagram. Publ. Math. Univ. Pierre Marie Curie 66, Sémin. Initiation Anal. 23ème Année-1983/84, Exp. No. 5, 13 p. (1984)., 1984.

[22] Lorenz J. Halbeisen. Combinatorial set theory. Springer Monographs in Mathematics. Springer, Cham, 2017. With a gentle introduction to forcing, Second edition of [ MR3025440].

[23] Fernando Hernández-Hernández and Michael Hrušák. Cardinal invariants of analytic P -ideals. Canad. J. Math., 59(3):575–595, 2007.

[24] Michael Hrušák, David Meza-Alcántara, and Hiroaki Minami. Pair-splitting, pair-reaping and cardinal invariants of F_{σ}-ideals. J. Symbolic Logic, 75(2):661–677, 2010.

[25] Michael Hrušák, Wolfgang Wohofsky, and Ondřej Zindulka. Strong measure zero in separable metric spaces and Polish groups. Arch. Math. Logic, 55(1-2):105–131, 2016.

[26] Masaru Kada. The Baire category theorem and the evasion number. Proc. Amer. Math. Soc., 126(11):3381–3383, 1998.

[27] Anastasis Kamburelis and Bogdan Zbigniew Węglorz. Splittings. Arch. Math. Logic, 35(4):263–277, 1996.

[28] Michał Machura, Saharon Shelah, and Boaz Tsaban. The linear refinement number and selection theory. Fund. Math., 234(1):15–40, 2016.

[29] Barbara Majcher. Orthogonal partitions. volume 31, pages 59–63. 1990. 18th Winter School on Abstract Analysis (Srní, 1990).

[30] Arnold William Miller. Some properties of measure and category. Trans. Amer. Math. Soc., 266(1):93–114, 1981.

[31] Arnold William Miller and Juris Steprāns. The number of translates of a closed nowhere dense set required to cover a Polish group. Ann. Pure Appl. Logic, 140(1-3):52–59, 2006.

[32] Hiroaki Minami. Around splitting and reaping for partitions of ω. Arch. Math. Logic, 49(4):501–518, 2010.

[33] Dilip Raghavan. The density zero ideal and the splitting number. Ann. Pure Appl. Logic, 171(7):102807, 15, 2020.

[34] Dilip Raghavan and Saharon Shelah. Two inequalities between cardinal invariants. Fund. Math., 237(2):187–200, 2017.

[35] Fritz Rothberger. Sur un ensemble toujours de premiere categorie qui est depourvu de la propriété λ. Fundam. Math., 32:294–300, 1939.

[36] Wacław Sierpiński. Sur un ensemble à propriété λ. Fundam. Math., 32:306–310, 1939.

Preprint

[37] Juris Steprāns. History of the continuum in the 20th century. In Sets and extensions in the twentieth century, volume 6 of Handb. Hist. Log., pages 73–144. Elsevier/North-Holland, Amsterdam, 2012.

[38] Boaz Tsaban. Selection principles and the minimal tower problem. Note Mat., 22(2):53–81, 2003/04.

[39] Eric Karel van Douwen. The integers and topology. In Handbook of set-theoretic topology, pages 111–167. North-Holland, Amsterdam, 1984.

Universität Wien, Institut für Mathematik, Kurt Gödel Research Centre, Kolingasse 14–16, 1090 Wien, AUTRICHE

Email address: thilo.weinert@univie.ac.at