MAD Families and the Ramsey Property

Carlos A. Di Prisco

1. Almost Disjoint Families It is well known that the axiom of choice or its equiv- The study of collections of sets of natural numbers poses alents can be used to produce badly behaved sets of real many interesting questions connected, sometimes surpris- numbers, for example non-Lebesgue-measurable or with- ingly, to other areas of mathematics. Identifying sets of out the Baire property. One of the leading ideas in the natural numbers with real numbers is often a way to estab- development of set theory has been to show that those lish those connections. pathological sets are not definable in simple terms, and their existence cannot be proved without the use of some Carlos A. Di Prisco is a professor of mathematics at the Universidad de Los form of the axiom of choice. We will illustrate this with an Andes. His email address is [email protected]. example related to almost disjoint families. Two infinite sets of natural numbers are almost disjoint Communicated by Notices Associate Editor Antonio Montalb´an. if their intersection is finite. A family of infinite subsets of For permission to reprint this article, please contact: ℕ is said to be almost disjoint if its elements are pairwise [email protected]. almost disjoint. DOI: https://doi.org/10.1090/noti2314

AUGUST 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1081 Given any infinite partition of ℕ into infinite sets, say, property, called the Ramsey property of collections of infi- 퐴0, 퐴1,… , a new infinite set 퐴 of natural numbers can be nite sets of natural numbers, using certain ideals of sets of formed taking, for example, the first element of each 퐴푛. natural numbers. We say more about this below. This new set and the elements of the partition form a larger family which is almost disjoint. Finite partitions of ℕ can- 2. Ideals and Their Complements not be extended like this. A finite partition of ℕ into infi- Before defining the Ramsey property and presenting itsre- nite sets is thus maximal in the sense that it is not properly lation to MAD families, we will consider collections of sub- contained in a larger almost disjoint family. sets of ℕ called ideals, and also their complements. Any infinite almost disjoint family of infinite subsets of An ideal of sets is a collection of sets closed under sub- ℕ is contained in a maximal almost disjoint family. Ap- sets and under finite unions. The elements of an ideal can plying Zorn’s lemma to the collection of all almost dis- be considered, in some sense, small sets: a subset of a joint families that contain the given almost disjoint fam- small set is also small; and the union of two small sets ily we obtain a maximal extension. The question now is if is still a small set. we can prove the existence of infinite maximal almost dis- The collection of all finite subsets of ℕ is clearly an ideal. joint families of subsets of ℕ without using Zorn’s lemma More elaborate examples appear in different guises. Con- ∞ or some form of the axiom of choice. sider a divergent series of real numbers ∑푖=0 푎푖 (you could Infinite maximal almost disjoint families of subsets of ℕ take 푎푖 = 1/(푖+1)); then, the collection {퐴 ⊆ ℕ ∶ ∑푖∈퐴 푎푖 < are usually called MAD families, and they have been stud- ∞} is an ideal of subsets of ℕ. ied extensively. Another example is this: let 푋 be a topological space A standard diagonalization argument shows that a and 푥 ∈ 푋; then, the collection {푌 ⊂ 푋 ∶ 푥 ∉ 푌 ⧵ {푥}} is an countable almost disjoint family of subsets of ℕ is not a ideal of subsets of 푋. MAD family. If {퐴푛 ∶ 푛 ∈ ℕ} is an almost disjoint fam- Other interesting examples related to our topic can be ily of infinite subsets of ℕ, define inductively a sequence constructed from almost disjoint families. Given an infi- {푎푛 ∶ 푛 ∈ ℕ} of natural numbers taking 푎0 ∈ 퐴0, 푎1 ∈ nite family 풜, the ideal generated by 풜, that is, the family 퐴1 ⧵ 퐴0, and 푎0 < 푎1. If 푎0 < ⋯ < 푎푛 have been defined, ℐ(풜) = {퐴 ⊆ ℕ ∶ 퐴 is almost contained in a finite union let 푎푛+1 ∈ 퐴푛+1⧵(퐴0∪⋯∪퐴푛) be such that 푎푛 < 푎푛+1. This 1 is possible since the intersection of 퐴푛+1 ∩ (퐴0 ∪ ⋯ ∪ 퐴푛) of elements of 풜}, is finite. The set 퐴 = {푎 , 푎 ,…} is infinite and has finite 0 1 has interesting properties that will be analyzed below. intersection with each 퐴 . So, if a family is MAD, it must 푛 A family ℋ of subsets of ℕ is a coideal if its complement be uncountable. in 풫(ℕ) is an ideal. Thus, ℋ is a coideal if it satisfies: An example of an almost disjoint family of uncount- able cardinality can be constructed as follows. Consider (i) If 퐴 ∈ ℋ and 퐴 ⊆ 퐵, then 퐵 ∈ ℋ. the (countable) set 2<∞ of all finite sequences of 0’s and (ii) If 퐴 ∪ 퐵 ∈ ℋ, then 퐴 ∈ ℋ or 퐵 ∈ ℋ. 1’s. Ordering by extension gives this set the structure of We will only consider here ideals of subsets of ℕ that the complete binary tree. Each infinite branch through the contain all finite subsets of ℕ; and we will be specifically tree is an infinite set of finite sequences linearly ordered interested in ideals with a rich complement. This richness by extension, and corresponds to an infinite sequence of often means closure under certain operations. For our pur- 0’s and 1’s. So, the set of infinite branches is in one-one poses, we will be interested in coideals closed under diag- ℕ correspondence with 2 , the set of all functions from ℕ onalizations in the following sense. If 퐴0 ⊇ 퐴1 ⊇ 퐴2 ⊇ ⋯ to {0, 1}, which has the cardinality of the real line ℝ. No- is a decreasing sequence of infinite subsets of ℕ, 퐴 is a tice that two different branches coincide only on a finite diagonalization of the sequence ⟨퐴푛 ∶ 푛 ∈ ℕ⟩ if 퐴/푛 ∶= initial segment, so the set of branches is an almost dis- {푚 ∈ 퐴 ∶ 푛 < 푚} ⊆ 퐴푛 for every 푛 ∈ 퐴. Equivalently, joint family of finite binary sequences of cardinality |ℝ|. 푓(푛 + 1) ∈ 퐴푓(푛) for every 푛 ∈ ℕ, where 푓 ∶ ℕ → ℕ is the This family fails to be maximal, as, for example, the set increasing enumeration of 퐴. {(0, 1), (0, 0, 1), (0, 0, 0, 1), … } meets each infinite branch in Notice that if 퐴 is a diagonalization of the sequence at most one element. Using any bijection between ℕ and ⟨퐴푛 ∶ 푛 ∈ ℕ⟩, then 퐴 is almost contained in each 퐴푛, i.e., <∞ 2 , this example can be turned into an almost disjoint 퐴 ⧵ 퐴푛 is finite for every 푛 and if 푛 ∈ 퐴, all elements of 퐴 family of infinite subsets of ℕ. above 푛 are in 퐴푛. The use of some form of the axiom of choice is un- A coideal ℋ ⊆ ℕ[∞] closed under diagonalizations is avoidable to prove the existence of MAD families, as it was called selective. In this case, every decreasing sequence 퐴0 ⊇ shown by A. R. D. Mathias in the nineteen seventies in an 퐴1 ⊇ 퐴2 ⊇ ⋯ of elements of ℋ has a diagonalization in article titled “Happy families” ([5]). Mathias explored the ℋ. relationship between MAD families and a combinatorial 1퐴 is almost contained in 퐵 if 퐴 ⧵ 퐵 is finite.

1082 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 7 Selective coideals were called happy families by Mathias (i) If 푀 accepts 푎, then every 푁 ∈ ℋ|푀 accepts 푎. in his study of the Ramsey property and maximal almost (ii) If 푀 rejects 푎, then every 푁 ∈ ℋ|푀 rejects 푎. disjoint families of subsets of ℕ ([5]). (iii) For every 푀 ∈ ℋ and every 푎, there is 푁 ∈ ℋ|푀 It will be convenient to introduce some notation. 퐴[∞] that decides 푎. denotes the collection of infinite subsets of the set 퐴; and (iv) If 푀 accepts 푎, then 푀 accepts 푎 ∪ {푛} for every 퐴[<∞] denotes the collection of its finite subsets. If 푛 ∈ ℕ, 푛 ∈ 푀/푎. 퐴[푛] is the set of subsets of 퐴 with exactly 푛 elements. (v) If 푀 rejects 푎, then {푛 ∈ 푀/푎 ∶ 푀 accepts 푎∪{푛}} ∉ [∞] It is easy to verify that ℕ is a selective coideal. If 퐴0 ⊇ ℋ. 퐴1 ⊇ ⋯ ⊇ 퐴푛 ⊇ ⋯ is a decreasing sequence of elements of ℕ[∞], one can construct a diagonalization inductively Claim 3. There is 퐴 ∈ ℋ that decides each of its finite subsets. taking 푎0 the first element of 퐴0, and for each 푖 > 0 taking Proof. Construct a decreasing sequence 퐴0 ⊇ 퐴1 ⊇ ⋯ ⊇ an element 푎푖 ∈ 퐴푎 with 푎푖−1 < 푎푖. 푖−1 퐴푛 ⊇ ⋯ of elements of ℋ such that if 푎푖 = min(퐴푖), then Another example is the complement of the ideal ℐ(풜) 푎0 < 푎1 < ⋯ and 퐴0 decides ∅, and for every 푘, 퐴푘+1 defined above for an infinite almost disjoint family 풜. decides every 푠 ⊆ {0, … , 푎푘}. This can be done applying Theorem 1 (Mathias, [5]). If 풜 is an infinite almost disjoint (iii). As ℋ is selective, there is a diagonalization 퐴 of the [∞] family, the coideal ℕ ⧵ ℐ(풜) is a selective coideal. sequence 퐴0, 퐴1,… in ℋ. If 푎 ⊆ 퐴 and 푛 is the first element of 퐴 above 푎, then 풜 Notice that if is a MAD family, then a set is in the 퐴/푛 ⊆ 퐴 , and so 퐴/푛 decides 푎. It follows that 퐴 decides ℕ[∞] ⧵ ℐ(풜) 푛 associated coideal if and only if it has infinite 푎. □ intersection with infinitely many elements of 풜. This ob- servation can be verified with little effort and will beused We now finish the proof of the lemma. Let 푀 ∈ ℋ below when discussing the Ramsey property. decide each of its finite subsets. If 푀 accepts ∅, then every The following lemma is a deep result proved by infinite subset of 푀 has an initial segment in ℱ. [∞] F. Galvin, originally for the coideal ℕ , which will be Suppose that 푀 rejects ∅. Construct a decreasing se- used in the next section to show that selective coideals have quence 푀 = 푀 ⊇ 푀 ⊇ 푀 ⊇ ⋯ of elements of ℋ such [2] 0 1 2 homogeneous sets for all colorings of ℕ . that if 푚 = min(푀 ), then 푚 < 푚 < ⋯ and 푀 rejects [∞] 푖 푖 0 1 푖+1 First, we give some definitions. For a family ℋ ⊆ ℕ 푎 ∪ {푛} for every 푎 ⊆ 푀 ∩ {0, ⋯ , 푚 } and every 푛 ∈ 푀 . If [∞] 푖 푖+1 and 퐴 ∈ ℕ , let ℋ|퐴 ∶= {퐵 ∈ ℋ ∶ 퐵 ⊆ 퐴}. 퐴 is a diagonalization of the sequence 푀 , 푀 ,… , then 퐴 [<∞] [∞] 0 1 Given 푎 ∈ ℕ and 푀 ∈ ℕ , and 푀 = rejects each of its finite subsets. Then 퐴[<∞] ∩ ℱ = ∅, be- {푚0, 푚1, 푚2,…} the increasing enumeration of 푀, we say cause if 푎 ∈ ℱ is contained in 퐴, then 퐴/푎 rejects 푎 (since that 푎 is an initial segment of 푀 if there is 푘 such that 퐴 rejects 푎), but this is impossible since any 푋 ∈ [푎, 퐴] has 푎 = {푚0, … , 푚푘}. Also, an initial element in ℱ, namely 푎, and so 퐴/푎 accepts 푎, a □ [푎, 푀] ∶= {푋 ∈ ℕ[∞] ∶ 푎 is an initial segment of 푋 and contradiction. 푋 ⊆ 푎 ∪ 푀}. 3. The Ramsey Property Lemma 2 (Galvin’s lemma for selective coideals). Let ℋ F. P. Ramsey proved in 1930 a theorem that originated a be a selective coideal, and ℱ a collection of finite subsets of ℕ. whole area of combinatorics now called Ramsey theory. Then, for every 퐴 ∈ ℋ there is 퐵 ∈ ℋ|퐴 such that either This theorem was proved to solve a problem in mathemat- (i) no finite subset of 퐵 belongs to ℱ (i.e., 퐵[<∞] ∩ℱ = ∅), ical logic, and it has many interesting applications in sev- or eral areas of mathematics. (ii) every infinite subset of 퐵 has an initial segment in ℱ. Theorem 4 (Ramsey, 1930; see [12], Theorem 1.3). Given [푛] The proof of this lemma uses a technique called com- positive integers 푛 and 푟, for every function 푐 ∶ ℕ → binatorial forcing that has been important in the develop- {0, 1, … , 푟 − 1} there is an infinite 퐻 ⊆ ℕ such that 푐 is con- [푛] ment of combinatorial set theory. We include it here for stant on 퐻 . its beauty and to give a good sample of a combinatorial A set 퐻 as in the theorem is said to be homogeneous for argument. The reader eager to get ahead can skip it. 푐. It is customary to call such a function, and the partition Proof. Fix a collection ℱ of finite subsets of ℕ for the rest of ℕ[푛] it induces, a coloring of ℕ[푛], and to say that the set of the proof. Let 푀 ∈ ℋ and 푎 ∈ ℕ[<∞]. We say that 푀 퐻[푛] is monochromatic for this coloring. accepts 푎 if every 푋 ∈ [푎, 푀] has an initial segment in ℱ; Consider the simplest non-trivial version of Ramsey’s and 푀 rejects 푎 if there is no 푁 ∈ ℋ|푀 that accepts 푎. We theorem, for 푛 = 푟 = 2. This version implies, for exam- say that 푀 decides 푎 if it either accepts or rejects 푎. ple, that every sequence ⟨푟푖 ∶ 푖 ∈ ℕ⟩ of real numbers with It follows from the definitions that: infinite range has a monotone subsequence.

AUGUST 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1083 [<∞] In terms of graphs, this version says that any infinite containing the basic sets 푈푠 (푠 ∈ ℕ ) and closed under graph has an infinite complete subgraph or an infinite in- countable unions and complements. dependent set of vertices. [∞] Sometimes it is needed to find a homogeneous set 퐻 Theorem 7 (Galvin-Prikry, [4]). Every Borel subset 풜 of ℕ for a coloring in a certain class of subsets of ℕ. For this is Ramsey. reason, it is interesting to determine what conditions on Proof. Notice that if 풜 ⊆ ℕ[∞] is open (in the topology a family ℋ of infinite subsets of ℕ imply that it contains inherited from the product topology of 2ℕ), then there is [2] homogeneous sets of every 푐 ∶ ℕ → {0, 1}. a family ℱ of finite subsets of ℕ such that 푋 ∈ 풜 if and A coideal ℋ on ℕ that contains homogeneous sets for only if 푋 has an initial segment in ℱ. Then, the theorem [2] every coloring 푐 ∶ ℕ → {0, 1} is called a Ramsey coideal. follows for open sets from Galvin’s lemma applied to the The following is a consequence of Lemma 2. coideal ℕ[∞]. Theorem 5. Every selective coideal is a Ramsey coideal.2 To complete the proof it is enough to show that the col- lection of Ramsey subsets of ℕ[∞] is closed under comple- Proof. Let ℋ be a selective coideal, and 푐 ∶ ℕ[2] → {0, 1}. −1 ments and under countable unions, and therefore it con- Let ℱ be 푐 {0}, and apply Lemma 2 to this family of pairs. tains all the Borel sets. This is by no means a trivial task Then there is 퐴 ∈ ℋ such that 퐴 does not contain any and we will skip it. □ element of ℱ and therefore 푐 takes constant value 1 on 퐴[2]; or every infinite subset of 퐴 starts with a pair in ℱ and so This theorem can be extended to the ℋ-Ramsey prop- 푐 takes constant value 0 on 퐴[2]. □ erty for ℋ a semiselective coideal. Moreover, any analytic4 set is Ramsey (Silver, [10]), and ℋ-Ramsey for every selec- Having considered homogeneous sets for finite parti- tive coideal ℋ (Mathias, [5]), or even for semiselective ℋ tions of ℕ[푛], it is natural to consider homogeneous sets for (Farah, [3]). Ellentuck gave an elegant topological charac- partitions of ℕ[∞]. A set 풳 ⊆ ℕ[∞] determines a partition terization of the Ramsey property. The reader can consult of ℕ[∞] in two pieces, 풳 and its complement; an infinite Todorcevic’s book ([12], Chapter 7) for a complete presen- set 퐻 is homogeneous for this partition if all of its infinite tation of this theory. subsets belong to the same side of the partition. This mo- The following remarkable relation between MAD fami- tivates the following definitions. lies and the Ramsey property was established by Mathias. Definition 6. We say that 풳 ⊆ ℕ[∞] is Ramsey (or has the Lemma 8 (Mathias, [5]). Let 풜 be an infinite almost disjoint Ramsey property) if there is 퐻 ⊆ ℕ[∞] such that 퐻[∞] ⊆ 풳 family. Then 풜 is maximal if and only if the associated coideal or 퐻[∞] ∩ 풳 = ∅. ℋ = ℕ[∞] ⧵ ℐ(풜) is not ℋ-Ramsey. If ℋ is a coideal and 풳 ⊆ ℕ[∞], we say that 풳 is ℋ- Ramsey if there is 퐻 ∈ ℋ such that [퐻][∞] ⊆ 풳 or [퐻][∞] ∩ Proof. If 풜 is a MAD family, then, by the observation made 풳 = ∅. after Theorem 1, every 퐻 ∈ ℋ contains an infinite subset not in ℋ. In other words, a set 풳 is ℋ-Ramsey if ℋ contains a [∞] homogeneous set for the partition of ℕ[∞] determined by And if 풜 is not maximal, take 푋 ∈ ℕ almost disjoint 풳 풳 ℕ[∞] from every element of 풜. Then, every infinite subset of 푋 ; and is Ramsey if it is -Ramsey. □ Using the axiom of choice one can find a set 풜 ⊆ ℕ[∞] is in the coideal ℋ. that is not Ramsey, but a set like this is not definable in From Lemma 8 and the fact that every analytic set is ℋ- [∞] simple terms. If ℕ is given the topology inherited from Ramsey for every selective coideal ℋ, Mathias concluded ℕ the product topology of 2 (where 2 = {0, 1} is taken with that there are no analytic MAD families. the discrete topology), every topologically simple subset of ℕ[∞] is Ramsey.3 Let us make this more precise. The 4. All Sets Can Be Ramsey [∞] collection of subsets of ℕ of the form 푈푠 ={푋 ⊆ℕ∶푠 is an In this section we examine the possibility of all sets of reals initial segment of the increasing enumeration of 푋}, where having the Ramsey property. The axiom of choice rules this 푠 is a finite subset of ℕ, is a basis for this topology. The out since it implies the existence of non-Ramsey sets, but collection of Borel subsets of ℕ[∞] is the smallest collection there are models of set theory which do not satisfy the ax- [∞] 2 iom of choice and where all subsets of ℕ have the Ram- Farah, in [3], introduced the notion of semiselective coideals. Semiselec- 5 tivity of a coideal is a property weaker than selectivity, but strong enough sey property. so that Lemma 2 holds for ℋ semiselective; therefore every semiselective co- ideal is a Ramsey coideal. A comprehensive presentation of these results appears 4A set of real numbers is analytic if it is the continuous image of a Borel set. in [12]. There are Ramsey coideals that are not semiselective, and semiselective There are analytic sets that are not Borel. coideals that are not selective. Nevertheless, for ultrafilters on ℕ, these proper- 5Axiomatic set theory was developed by Zermelo and Fraenkel to circumvent the ties are equivalent, since any Ramsey ultrafilter is selective. paradoxes discovered in the early 20th century. This axiomatic theory is known 3ℕ[∞] with this topology is (homeomorphic to) the Baire space. as ZF, Zermelo-Fraenkel set theory. If the axiom of choice is added, it is denoted

1084 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 68, NUMBER 7 Solovay ([11]) constructed a model of set theory where cardinal, and then considering the inner model 퐿(ℝ), gives every set of real numbers is Lebesgue measurable, has the a model of set theory where every 풜 ⊆ ℕ[∞] is ℋ-Ramsey property of Baire, and, if uncountable, contains a perfect for every selective coideal ℋ of the wide model. As a con- subset. Of course, the axiom of choice does not hold in sequence, there are no MAD families in this inner model. this model, although a weak form of the axiom called The usual axioms of set theory hold in this inner model dependent choice (DC) is true in the model. The con- but not the axiom of choice. It follows that the existence struction of this model starts assuming the existence of an of a MAD family cannot be proved from the axioms of set inaccessible cardinal,6 and combines the use of forcing7 theory without the axiom of choice. with another important technique for constructing mod- A natural question is then if there are MAD families in els of set theory, due to Gödel, which consists in taking Solovay’s original model. This question was answered re- the sets of a model which are definable in some precise cently by Törnquist [13]. Using a combinatorial argument sense. With this in mind, we can describe Solovay’s con- he showed that there are no MAD families in the tradi- struction as this: given a model with an inaccessible car- tional Solovay model. dinal 휅, the forcing technique is used to construct a wider Neeman and Norwood [6] proved a strengthening of model where all infinite cardinals below 휅 become count- Törnquist’s result by showing that in Solovay’s model (ob- able, making 휅 the first uncountable cardinal of the new tained using an inaccessible cardinal) every set of reals is model; then, within this new model, take the class 퐿(ℝ) of ℋ-Ramsey for every selective coideal ℋ. From this follows, all sets constructible from the reals. as we have seen, that there are no MAD families in this Solovay asked if the hypothesis of the existence of an in- model. accessible cardinal could be avoided to construct a model Other interesting partition properties hold in Solovay’s where all sets of real numbers are Lebesgue measurable, model. For example, the following parametrization of the but it was shown by Shelah [9] fourteen years later that the Ramsey property (see [1, 2]). existence of inaccessible cardinals is necessary to obtain Let ℕℕ denote the Baire space, that is, the set of all func- such a model. This result revealed a deep and unexpected tions from ℕ to ℕ with the product topology. relation between a property of sets of real numbers and For every 푐 ∶ ℕℕ × ℕ[∞] → 2 there is a perfect9 subset large cardinals. Surprisingly, this hypothesis is not nec- 푃 ⊆ ℕℕ and 퐻 ∈ ℕ[∞] such that 푐 is constant on 푃 × [퐻][∞]. essary to get a model of set theory where all sets of real A function 푐 ∶ ℕℕ × ℕ[∞] → 2 determines for each 푥 ∈ numbers have the Baire property (see [9]). ℕℕ a partition 푐(푥, ⋅) of ℕ[∞] into two parts. The property In [5], Mathias proved that in Solovay’s model every above means that there is a perfect set 푃 ⊆ ℕℕ and a set subset of ℕ[∞] is Ramsey. It has remained open if the inac- 퐻 ∈ ℕ[∞] that is homogeneous simultaneously for all the cessibility hypothesis is necessary in this case. partitions determined by the elements of 푃. Using a stronger large cardinal hypothesis Mathias As a final comment, we mention without any details a reached a stronger conclusion. Assuming the existence of couple of points about the axiom of determinacy (AD) and a Mahlo cardinal8 휅 a similar construction, first using forc- the Ramsey property. ing to obtain a wide model where 휅 is the first uncountable For every set 퐴 of infinite sequences of natural numbers define a game 퐺퐴 as follows. Two players 퐼 and 퐼퐼 alternate by ZFC, and it provides a basis for conventional mathematics. Some of the ax- playing natural numbers. Player 퐼 plays 푛0, then player 퐼퐼 ioms postulate the existence of certain sets; for example, there exists an infinite plays 푛1, then player 퐼 plays 푛2, and so on. Player 퐼 wins set; some other axioms are of the form “If A is a set, then there is a set B with if the infinite sequence ⟨푛 , 푛 , 푛 ,…⟩ they form belongs to certain properties” (for example, the power set axiom: if A is a set, then there 0 1 2 is a set whose elements are the subsets of A); other axioms have a more techni- 퐴; otherwise, 퐼퐼 wins. A winning strategy for 퐼 is a func- <∞ cal character. A model of set theory is a collection of sets that satisfies the ax- tion 푓 ∶ ℕ → ℕ, from the set of finite sequences of ioms. Not all mathematical questions can be settled from the axioms. A state- natural numbers to ℕ such that 퐼 wins any run of the game ment that is true in some models of set theory and false in some other models in which 푛0 = 푓(∅) and 푛2푘+2 = 푓(⟨푛0, 푛1, … , 푛2푘+1⟩) for cannot be proved nor disproved from the axioms. The Continuum Hypothesis is 푘 ∈ ℕ 퐼퐼 such a statement. every . A winning strategy for player is defined 6A cardinal 휅 is inaccessible if it is uncountable, it cannot be reached by unions analogously. of less than 휅 sets each of them of size less than 휅, and if 훼 < 휅 then 2훼 < 휅. A set 퐴 is determined if one of the players has a winning The existence of an inaccessible cardinal cannot be proved from the axioms of strategy for the game 퐺퐴. ZFC. 7 The axiom of determinacy, AD, states that every set Forcing is a technique used to prove independence results in set theory. It was 퐴 ⊆ ℕℕ introduced by Paul Cohen in 1963 to prove the independence of the continuum is determined. AD has become a central topic hypothesis and the axiom of choice from the axioms of Zermelo-Fraenkel set of contemporary set theory due to its interesting conse- theory. quences and its correlation to large cardinals. 8The existence of a Mahlo cardinal is a stronger large cardinal hypothesis than the existence of an inaccessible cardinal. A Mahlo cardinal is inaccessible and 9A subset of ℕℕ is perfect if it is non-empty, closed, and has no isolated points. has many inaccessible cardinals below it. Any perfect subset of ℕℕ has the cardinality of the continuum.

AUGUST 2021 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1085 AD implies that every set of reals is Lebesgue measur- [8] David Schrittesser and Asger Törnquist, The Ram- able and has other regularity properties, and so it is incom- sey property implies no mad families, Proc. Natl. Acad. patible with the axiom of choice. It is an open problem if Sci. USA 116 (2019), no. 38, 18883–18887, DOI AD implies that every set of real numbers is Ramsey. Prikry 10.1073/pnas.1906183116. MR4012549 [9] Saharon Shelah, Can you take Solovay’s inaccessible [7] proved that this is true for AD , a strong form of deter- ℝ away?, Israel J. Math. 48 (1984), no. 1, 1–47, DOI minacy. Neeman and Norwood give in [6] a partial answer 10.1007/BF02760522. MR768264 to the question if AD implies that there are no MAD fami- + [10] Jack Silver, Every analytic set is Ramsey, J. Symbolic Logic lies [13]. They show that a form of AD called AD implies 35 (1970), 60–64, DOI 10.2307/2271156. MR332480 that there are no MAD families. It is not known if AD and [11] Robert M. Solovay, A model of set-theory in which every set AD+ are equivalent. Schrittesser and Törnquist [8] have of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), shown, using a weak choice principle, that if all sets are 1–56, DOI 10.2307/1970696. MR265151 Ramsey, there are no MAD families. [12] Stevo Todorcevic, Introduction to Ramsey spaces, Annals of Mathematics Studies, vol. 174, Princeton University References Press, Princeton, NJ, 2010, DOI 10.1515/9781400835409. [1] Joan Bagaria and Carlos A. Di Prisco, Parameterized par- MR2603812 tition relations on the real numbers, Arch. Math. Logic 48 [13] Asger Törnquist, Definability and almost dis- (2009), no. 2, 201–226, DOI 10.1007/s00153-009-0121-y. joint families, Adv. Math. 330 (2018), 61–73, DOI MR2487224 10.1016/j.aim.2018.03.005. MR3787540 [2] C. A. Di Prisco and S. Todorcevic, Souslin partitions of prod- ucts of finite sets, Adv. Math. 176 (2003), no. 1, 145–173, DOI 10.1016/S0001-8708(02)00064-6. MR1978344 [3] Ilijas Farah, Semiselective coideals, Mathematika 45 (1998), no. 1, 79–103, DOI 10.1112/S0025579300014054. MR1644345 [4] Fred Galvin and Karel Prikry, Borel sets and Ramsey’s theorem, J. Symbolic Logic 38 (1973), 193–198, DOI 10.2307/2272055. MR337630 [5] A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), no. 1, 59–111,DOI 10.1016/0003-4843(77)90006- 7. MR491197 Carlos A. Di Prisco [6] Itay Neeman and Zach Norwood, Happy and mad fami- Credits lies in 퐿(ℝ), J. Symb. Log. 83 (2018), no. 2, 572–597, DOI 10.1017/jsl.2017.85. MR3835078 Opening image is courtesy of Dmitry Volkov via Getty. [7] Karel Prikry, Determinateness and partitions, Proc. Amer. Photo of Carlos A. Di Prisco is courtesy of Carlos A. Di Prisco. Math. Soc. 54 (1976), 303–306, DOI 10.2307/2040805. MR453540

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