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The Ramsey Property Implies No Mad Families

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The Ramsey Property Implies No Mad Families The Ramsey property implies no mad families David Schrittessera,1 and Asger Tornquist¨ b,1,2 aKurt Godel¨ Research Center, University of Vienna, 1090 Vienna, Austria; and bDepartment of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, Denmark Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved July 30, 2019 (received for review April 10, 2019) We show that if all collections of infinite subsets of N have the and Baire measurable. It was Mathias, in his “Happy fami- Ramsey property, then there are no infinite maximal almost dis- lies” paper (1) (drafts of which circulated already in the late joint (mad) families. The implication is proved in Zermelo–Fraenkel 1960s), who connected forcing to Erdos˝ and Rado’s question set theory with only weak choice principles. This gives a positive and to mad families. Mathias did this by introducing what is solution to a long-standing problem that goes back to Mathias now known as Mathias forcing, which he used to show that in 1 [A. R. D. Mathias, Ann. Math. Logic 12, 59–111 (1977)]. The proof Solovay’s model all sets S ⊆ [N] are Ramsey. Mad families, exploits an idea which has its natural roots in ergodic theory, and their connection to Mathias forcing, play a central role in topological dynamics, and invariant descriptive set theory: We use this proof. that a certain function associated to a purported mad family is Mathias asked 2 central questions, which his methods did invariant under the equivalence relation E0 and thus is constant not allow him to answer at the time: 1) Are there infinite mad 1 on a “large” set. Furthermore, we announce a number of addi- families in Solovay’s model? 2) If all sets S ⊆ [N] are Ram- tional results about mad families relative to more complicated sey, does it follow that there are no infinite mad families? A Borel ideals. positive answer to question 2 would give a negative answer to question 1. Ramsey property j maximal almost disjoint families j invariant descriptive There was only modest progress on these questions until very set theory j Borel ideals recently, when suddenly the research in mad families and forc- ing experienced a renaissance. Question 1 was solved in 2014 in ref. 5, and shortly after, Horowitz and Shelah showed in ref. n his seminal paper, Mathias (1) established a connection 6 that a model of ZF in which there are no mad families can MATHEMATICS Ibetween 3 different ideas in mathematics: the combinato- be achieved without using an inaccessible cardinal, which is rial set theory of maximal almost disjoint families, infinite- otherwise a crucial ingredient in the construction of Solovay’s dimensional Ramsey theory, and Cohen’s method of forcing. model. Neeman and Norwood in ref. 7 and independently, Bakke He asked whether the combinatorial statement “all sets have Haga in joint work with the present authors in ref. 8 proved the Ramsey property” implies that there are no infinite maxi- a number of further results, among them that V = L(R) + mal almost disjoint (mad) families. In this paper we answer this AD implies there are no mad families. Horowitz and She- in the affirmative, working in the theory ZF + DC + R-Unif lah also solved a number of related questions that had been (Definition 2). formulated over the years; in particular, they showed the exis- Let us recall the key notions: An almost disjoint family (on the tence of a Borel “med” family in ref. 9; see also ref. 10 for a natural numbers N) is a family A of infinite subsets of N such simpler proof. that if x, y 2 A, then either x = y or x \ y is finite. A maximal We denote by R-Unif the principle of uniformization on Ram- almost disjoint family (“mad family”) is an almost disjoint family sey positive sets (Definition 2). R-Unif is a weak choice principle, which is not a proper subset of an almost disjoint family. Finite which is weak enough that it holds in Solovay’s model, and this mad families are easily seen to exist, e.g., fE, Og, where E is the can be seen quite easily. In this paper we give the following set of even numbers and O is the set of odd numbers. The exis- positive solution to Mathias’ question. tence of infinite mad families follows easily from Zorn’s lemma (equivalently, the axiom of choice). Theorem 1. ZF + DC + R-Unif If all sets have the Ramsey k ( ) Given a set X and a natural number k 2 N, let [X ] denote property, then there are no infinite mad families. the set of all subsets of X with exactly k elements. The classical k infinite Ramsey theorem in combinatorics says that if S ⊆ [N] , k Significance then there is an infinite set B ⊆ N such that either [B] ⊆ S or [B]k \ S = ;. Motivated by a question of Erdos˝ and Rado, infinite-dimensional generalizations of this theorem were dis- Certain infinite combinatorial structures in modern mathe- covered in the 1960s and 1970s. In this paper, we denote by matics, called mad families, are known to exist only due to [X ]1 the set of countably infinite subsets of X . Moreover, given indirect, nonconstructive methods arising from a fundamen- 1 tal principle of mathematics, with many paradoxical conse- S ⊆ [N] , we will say that S has the Ramsey property, or sim- 1 1 quences, called the axiom of choice. This paper shows that ply is Ramsey, if there is B 2 [N] such that either [B] ⊆ S or [B]1 \ S = ;. Erdos˝ and Rado showed that the axiom of if we replace the axiom of choice with a natural assumption 1 of universal combinatorial regularity, a principle known as choice implies that not all sets S ⊆ [N] are Ramsey. Later, in 1 the Ramsey property for all sets, then no infinite mad fam- refs. 2 and 3, it was shown that Borel and analytic S ⊆ [N] are Ramsey, and finally Ellentuck (4) in 1974 characterized the Ram- ilies can exist. This solves a problem that has been open in sey property in terms of Baire measurability in the Ellentuck mathematics since the late 1960s. topology on [ ]1. N Author contributions: D.S. and A.T. performed research and wrote the paper.y Concurrent with these developments in Ramsey theory, Cohen’s introduction of the method of forcing for indepen- The authors declare no conflict of interest.y dence proofs in set theory in the early 1960s set off an explosion This article is a PNAS Direct Submission.y of independence results, among the most famous of which is Published under the PNAS license.y Solovay’s model of Zermelo–Fraenkel set theory (ZF) in which 1 D.S. and A.T. contributed equally to this work.y only a fragment of the axiom of choice–namely dependent 2 To whom correspondence may be addressed. Email: [email protected] choice (DC)—holds and in which all subsets of R are Lebesgue Published online August 29, 2019. www.pnas.org/cgi/doi/10.1073/pnas.1906183116 PNAS j September 17, 2019 j vol. 116 j no. 38 j 18883–18887 Downloaded by guest on October 4, 2021 We note that Theorem 1 implies the main results of refs. dom(s) ⊆ dom(t) and s(i) = t(i) for all i 2 dom(s); we will write 5 and 7. s ? t (“s and t are incompatible”) if s 6⊆ t and t 6⊆ s. < Theorem 1 may seem all the more surprising given another For each s 2 N N, let recent result of Horowitz and Shelah (11), who show that for a N variety of measurability notions including the Lebesgue measure, Ns = ff 2 N :(8i 2 dom(s)) f (i) = s(i)g: “all sets are measurable” is compatible with the existence of an infinite mad family. <N The family fNs : s 2 N g is easily seen to form a basis for the Let us briefly comment on the proof of Theorem 1 and the topology on NN. difficulties that have to be overcome. For this discussion, sup- <N 1 Note that <N is countable, and so 2N is a Polish space pose A ⊆ [N] is an infinite mad family, and assume “all sets are N Ramsey” and “Ramsey uniformization” (Definition 2). (isomorphic to 2N) in the product topology, taking 2 = f0, 1g dis- 1 The first difficulty encountered is that the set of x 2 [N] crete. This view will be important later in the proof of Theorem which meet exactly 1 element of A in an infinite set is clearly 1 where we will describe the properties of a certain continuous z Ramsey conull when A is a mad family. The key idea is to over- function f defined on NN in terms of a “derived” function z 7! T 1 < come this difficulty by associating to each z 2 [N] a carefully from N to 2N N . 1 N chosen very, very sparse set z~ 2 [N] , which is constructed using a fixed, infinite sequence (an )n2N chosen from A (it is here that B. The Ramsey Property. For any set X we define we use the principle of DC). A basic property of the map z 7! z~ is that it is equivariant under finite differences; that is, if z4z 0 is [X ]1 = fA ⊆ X : A is infiniteg: finite, then z~4z~0 is finite. 1 1 Because we assumed that A is maximal, for each z 2 [N] Recall from the Introduction that a set S ⊆ [N] is Ramsey (or 1 1 there is some yz 2 A such that z~ \ y is infinite, and so R-Unif has the Ramsey property) if there is B 2 [N] such that [B] ⊆ 1 1 gives us a function f :[N] !A such that f (z) \ z~ is infinite S or S \ [B] = ;.
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