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MATHEMATICS We note that Theorem 1 implies the main results of refs. dom(s) ⊆ dom(t) and s(i) = t(i) for all i ∈ 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 ∈ 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 = {f ∈ N :(∀i ∈ dom(s)) f (i) = s(i)}. “all sets are measurable” is compatible with the existence of an infinite mad family.
Notation and Background Definitions Definition 2. In this section we summarize the background needed for the 1) For Polish spaces X , Y , and R ⊆ X × Y , we say that R has proof. A good general reference for all of the background full projection on X , if for every x ∈ X there is y ∈ Y such that needed is ref. 12. A comprehensive treatise on modern, infinitary (x, y) ∈ R. Ramsey theory is ref. 13. 2) Let X , Y be Polish spaces, let Z ⊆ X , and let R ⊆ X × Y be a set with full projection. A function ϑ : Z → Y is said to A. Descriptive Set Theory. A topological space X is called Pol- uniformize R on Z if for all x ∈ Z we have (x, ϑ(x)) ∈ R. ish if it is separable and admits a complete metric that 3) The Ramsey uniformization principle, abbreviated R-Unif, is the ∞ induces the topology. In this paper we will be working with following statement: For all Y Polish and all R ⊆ [N] × Y 2N = {0, 1}N N with full projection, there is an infinite set A ⊆ N and a function the Polish space and N (with the product topol- ∞ ∞ ogy, taking {0, 1} and N discrete) and subspaces of these ϑ :[A] → Y which uniformizes R on [A] . spaces. Recall the following key notion from descriptive set That the Ramsey uniformization principle holds in Solovay’s theory: model follows by the argument given in ref. 14, section 1.12, p. 46 with Random forcing replaced everywhere by Mathias forcing. Definition 1. A subset A ⊆ X of a Polish space X is analytic if there is a continuous f : Y → X from a Polish space Y to X such that D. Invariance under Finite Changes. The last ingredient for the A = ran(f ). proof is the notion of “E0 invariance.” Since NN maps continuously onto any Polish space, we have that A ⊆ X is analytic iff there is a continuous f : NN → X Definition 3. such that ran(f ) = A. We will use this characterization as our ∞ 1) E0 is the equivalence relation defined on [ ] by definition of the analytic set below. N For the proof of Theorem 1 we need the following combi- n xE0y ⇐⇒|x4y| < ∞. natorial description of the topology on NN. We denote by N 2 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1906183116 Schrittesser and Tornquist¨ Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 all 3. Subclaim 65. p. of top T that scniuu on continuous is some find can every we for Then claim. the fies all are there on if only all for z ucam2. Subclaim 1. Subclaim that and 12 ref. of sense the in map identifying By xacniuu function continuous a fix iuu.Then tinuous. replacing all for hr is there | anClaim. Main theorem: the prove will oethat Note (a nraigfnto uhthat such function increasing o nifiiesbe fa lmn of element an of subset infinite an now crtesradT and Schrittesser infi- Dedekind are sets that infinite assume all may that We implies nite). DC that use + we DC + ZF that show assumptions: property.” Ramsey following the have the sets “all + under R-Unif work We 1 Theorem of Proof suefrsmlct htif that simplicity for assume replacing possibly by that Moreover, shows imal. diagonalization easy an otherwise since ˆ ˜ z ˜ (k n rmnwo efix we on now From Proof: Proof: Proof: For ups h li sfle hnb asyuniformization Ramsey by Then false. is claim the Suppose ealta when that Recall Let Let ∩ o all for y y [W 0 ) 0 0 y n z + [N] ∈N ∈ ∈ | x 0 7→ z i (a z ⊆ A < | i 0 ] a z ˜ f f ∈ W ∞ ∞ ˆ = ) ∈ ∈ \ z ˆ ∈ n xdaoe en o each for define above, fixed 0 (N (N T ∞. ( s z ∩ tflosthat follows It N. W ) . Suppose [W By A [N] ups no Suppose k T → N | 0 n z 0 ∈ z [N] ˜ ∈ ups hr are there Suppose h function The t s z f hr is There , ∈N +i z 0 0 u sntmaximal. not is scntn on constant is 0 s ∩ (x m [N] hr is There ∞ 2 rpsto 2 Proposition [W iha nnt ustof subset infinite an with ) ) 0 (k ] ornquist = ¨ 1 ∞ ) ∞ N B P a and and and (ˆ let , 0 ea netv euneo lmnsin elements of sequence injective an be )| MATHEMATICS Remarks: in V [x] as in the proof of Theorem 1, is almost disjoint from every 00 ∞ z ∈ A. Otherwise, we can choose a Mathias condition (s, A) with 1) In the proof above, a crucial point was obtaining W0 ∈ [W0] 00 ∞ 0 0 s ⊆ x ⊆ A and a name y˙ such that p y˙ ∩ x˜˙G is infinite and Ψ(y ˙ ) such that (∀z ∈ [W0 ] )z ˜ ⊆ an or z˜ ∩ an = ∅. Note that an 00 (where x˙G is a name for the Mathias real). By a well-known prop- alternative and quick way to obtain such W0 is to appeal to W 00 erty of Mathias forcing (so-called continuous reading of names) Ramsey’s theorem for pairs and take 0 to be a homoge- ∞ {n, j } we can assume that there is a continuous function ϑ:[N] → neous set for the 2-coloring of unordered pairs , where ∞ assuming n < j we let [N] with code in V such that p forces that y˙ = ϑ(x ˙G ). It is easy to see that ϑ(y) ∈ A for any y ∈ [x]∞ (here we use the definabil- ( 0 ity of A). But then ran(ϑ) would be an analytic almost disjoint 1 if an (j ) ∈ an , c(n, j ) = c family such that any element of {y˜ : y ∈ [x]∞} has infinite inter- 0 otherwise. section with some element of ran(ϑ), which is impossible by the proof of Theorem 1. 2) Finally, the following obvious question must be addressed: Surprisingly, the connection between the Ramsey property in Can a similar result be obtained without assuming R-Unif? ∞ [N] and mad families relative to the ideal of finite sets extends We think it is unlikely and that the assumptions of Theo- to much more complicated Borel ideals. We construct a family rem 1 are the minimal natural assumptions needed to obtain of ideals using the familiar Fubini sum: Given, for each n ∈ , a positive solution to Mathias’ problem; but we do not know. N an ideal Jn on a countable set Sn we obtain an ideal J on S = F S Corollaries and Further Results n n as follows: Corollary 3 (Tornquist¨ (5)). There are no mad families in Solovay’s M ∞ model. J = Jn = {X ⊆ S :(∀ n) X ∩ Sn ∈ Jn } Proof: By ref. 14, Solovay’s model is a model of ZF + DC. n That the Ramsey property holds in this model follows from where (∀∞n) means “for all but finitely many n.” The Fubini ref. 1. Finally, the Ramsey uniformization principle holds by our sum L FIN (where FIN denotes the ideal of finite sets on remarks after Definition 2. n FIN × FIN FIN2 We point out that the proof of Theorem 1 above localizes as N) is also known as or ; iterating Fubini FINα α < ω follows. sums into the transfinite we obtain , 1. This fam- ily of ideals of lies cofinally in the Borel hierarchy in terms of Corollary 4. If Γ, Γ0 are reasonable pointclasses such that every complexity. relation R ∈ Γ can be uniformized on a Ramsey positive set by a The notion of the mad family can be extended to arbitrary Γ0-measurable function and all sets in Γ0 are Ramsey, then there ideals on a countable set: If J is such an ideal, a J -almost dis- joint family is a subfamily A of P(S) \J such that for any 2 are no infinite mad families in Γ. 0 0 Note that in particular, this gives an additional proof of distinct A, A ∈ A, A ∩ A ∈ J .A J -mad family is of course a Mathias’ classical result that there are no analytic infinite mad J -almost disjoint family which is maximal under ⊆ among such families: Apply the corollary taking the pointclass of analytic sub- families. ∞ In ref. 16 we show the following: sets of [N] for Γ, and the σ algebra generated by the analytic sets for Γ0, and use the Jankov–von Neumann uniformization Theorem 7. (ZF + DC + R-Unif) Let α < ω1. If all sets theorem (ref. 12, theorem 18.1). α Our first corollary also allows us to draw consequences have the Ramsey property, then there are no infinite FIN -mad regarding the axiom of projective determinacy (in short, PD). families. As for classical mad families, we immediately obtain corollar- Corollary 5 (7, 8). PD implies there are no projective infinite mad ies regarding the axiom of projective determinacy and Solovay”s families. model. The first corollary was already shown in ref. 8 using Proof: The hypotheses of the previous theorem hold with forcing over inner models. Γ0 = Γ equal to the class of projective sets: PD implies that this pointclass has the uniformization property, and by ref. 15, all Corollary 8. (ZF + PD) For each α < ω1 there are no infinite α projective sets are completely Ramsey under PD. projective FIN -mad families. Another consequence of our proof is that Mathias forcing α destroys mad families from the ground model: Corollary 9. For each α < ω1 there are no infinite FIN -mad families in Solovay’s model. Theorem 6. In the Mathias extension, there is no infinite mad family Σ ACKNOWLEDGMENTS. This paper is dedicated to the memory of Professor which is definable by a 1 formula in the language of set theory with Richard Timoney (School of Mathematics, Trinity College Dublin), who sadly parameters in the ground model. In particular, no infinite almost passed away on 1 January 2019. We thank Professor Christian Rosendal and disjoint family from the ground model is maximal in the Mathias the anonymous reviewers for their useful comments and suggestions. D.S. extension. thanks the Austrian Science Fund for support through Grant P29999. A.T. Proof: Suppose x is Mathias over V and that in V [x], A = thanks the Danish Council for Independent Research for generous support ∞ through Grant 7014-00145B. A.T. also gratefully acknowledges his associa- {z ∈ [N] : Ψ(z)} is an infinite almost disjoint family, where Ψ(z) tion with the Center for Symmetry and Deformation, funded by the Danish is Σ1 (with parameter in V ). We show that x˜, where x˜ is defined National Research Foundation (DNRF92). 1. A. R. D. Mathias, Happy families. Ann. Math. Logic 12, 59–111 (1977). 8. K. Bakke Haga, D. Schrittesser, A. Tornquist,¨ Maximal almost disjoint families, 2. F. Galvin, K. Prikry, Borel sets and Ramsey’s theorem. J. Symb. Log. 38, 193–198 (1973). determinacy, and forcing. arXiv:1810.03016 (6 October 2018). 3. J. Silver, Every analytic set is Ramsey. J. Symb. Log. 35, 60–64 (1970). 9. H. Horowitz, S. Shelah, A Borel maximal eventually different family. arXiv:1605.07123 4. E. Ellentuck, A new proof that analytic sets are Ramsey. J. Symb. Log. 39, 163–165 (23 May 2016). (1974). 10. D. Schrittesser, Compactness of maximal eventually different families. Bull. Lond. 5. A. Tornquist,¨ Definability and almost disjoint families. Adv. Math. 330, 61–73 (2018). Math. Soc. 50, 340–348 (2018). 6. H. Horowitz, S. Shelah, Can you take Tornquist’s¨ inaccessible away? arXiv:1605.02419 11. H. Horowitz, S. Shelah, Madness and regularity properties. arXiv:1704.08327 (26 April (9 May 2019). 2017). 7. I. Neeman, Z. Norwood, Happy and mad families in L(R). J. Symb. Log. 83, 572–597 12. A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics (2018). (Springer, New York, NY, 1995), vol. 156. 4 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1906183116 Schrittesser and Tornquist¨ Downloaded by guest on October 1, 2021 Downloaded by guest on October 1, 2021 4 .M ooa,Amdlo e-hoyi hc vr e frasi Lebesgue is reals of set every which in set-theory of model A Solovay, M. R. 14. Todorcevic, S. 13. crtesradT and Schrittesser measurable. Studies) PictnUiest rs,Pictn J 00,vl 174. vol. 2010), NJ, Princeton, Press, University (Princeton n.Math. Ann. nrdcint asySae A-7)(naso Mathematics of (Annals (AM-174) Spaces Ramsey to Introduction ornquist ¨ –6(1970). 1–56 92, 6 .Shitse,A T A. ordinals. Schrittesser, on games D. of determinacy 16. the On Kechris, S. A. Harrington, A. L. 15. aiis ri:94084(5Arl2019). April (15 arXiv:1904.05824 families. Logic 0–5 (1981). 109–154 20, rqit h asypoet n ihrdmninlmad dimensional higher and property Ramsey The ornquist, ¨ NSLts Articles Latest PNAS n.Math. Ann. | f5 of 5 MATHEMATICS