THE FILTER DICHOTOMY PRINCIPLE DOES NOT IMPLY THE SEMIFILTER TRICHOTOMY PRINCIPLE 3 HEIKEMILDENBERGER 1 0 2 Abstract. We answer Blass’ question from 1989 of whether the in- equality u < g is strictly stronger than the filter dichotomy principle n a [4, page 36] affirmatively. We show that there is a forcing extension in J whicheverynon-meagrefilteronωisultrabyfinite-to-oneandthesemi- 3 filter trichotomy does not hold. This trichotomy says: every semifilter is either meagre or comeagre or ultra by finite-to-one. The trichotomy ] is equivalent to the inequality u < g by work of Blass and Laflamme. O Combinatorics of block sequences is used to establish forcing notions L that preservesuitable properties of block sequences. . h t a m 1. Introduction [ We separate two useful combinatorial principles: We show the filter di- 1 chotomy principle is strictly weaker than the semifilter trichotomy princi- v 6 ple. Consequencesofthelatterandequivalentstatementstothelatterinthe 9 realm of measure, category, rarefication ordersareinvestigated in [21,17,5]. 3 PaulLarsonprovesin[22]along-standingquestionaboutmediallimits: The 0 . filter dichotomy implies that there are none. Our result on the combinator- 1 ical side thus separates some very powerful principles in analysis. 0 3 We first recall the definitions: For B ⊆ ω and f: ω → ω, we let f′′B = 1 {f(b) : b ∈ B} and f−1′′B = {n : f(n) ∈ B}. By a filter we mean a : v proper filter on ω. We call a filter non-principal if it contains all cofinite Xi sets. Let F be a non-principal filter on ω and let f: ω → ω be finite- to-one (that means that the preimage of each natural number is finite). r a Then also f(F) = {X : f−1′′X ∈ F} is a non-principal filter. It is the filter generated by {f′′X : X ∈ F}. From now on we consider only non- principal filters. Two filters F and G are nearly coherent, if there is some finite-to-one f: ω → ω suchthat f(F)∪f(G)generates afilter. We also say to this situation that f(F) and f(G) are coherent. The set of all infinite subsets of ω is denoted by [ω]ω. A semifilter S is a subset of [ω]ω that contains ω as an element and that is closed under almost supersets, i.e., (∀X ∈ S)(∀Y ∈ [ω]ω)(X rY finite → Y ∈ S). In particular, [ω]ω is a semifilter. Date: Dec. 23, 2012. 2010 Mathematics Subject Classification. 03E05, 03E17, 03E35, 28A20. Key words and phrases. Iterated proper forcing, combinatorics with block-sequences, filters, semifilters, preservation theorems for forcing. 1 2 HEIKEMILDENBERGER The filter dichotomy principle, abbreviated FD, says that for every filter there is a finite-to-one function f such that f(F) is either the filter of cofinite sets (also called the Fr´echet filter) or an ultrafilter. In the latter case we call F ultra by finite-to-one or nearly ultra. A semifilter S is called meagre/comeagre if the set of the characteristic functions of the members of S is a meagre/comeagre subset of the space 2ω. We recall a connection between meagreness and unboundedness in the eventual domination order. Let ωω denote the set of functions from ω to ω. For f,g ∈ ωω we say g eventually dominates f and write f ≤∗ g iff (∃n ∈ ω)(∀n≥ n )(f(n) ≤ g(n)). Talagrand [34] showed 0 0 Lemma 1.1. For every semifilter S the following are equivalent (1) There is a finite-to-one function such that {X : (∃S ∈ S)(f′′S ⊆ X)} is the Fr´echet filter. (2) S is meagre. (3) The set of enumerating functions of members of S is ≤∗-bounded. The semifilter trichotomy principle, abbreviated SFT, says that for every semifilter S either S is meagre or f(S) is ultra or f(S) = [ω]ω for some finite-to-one f. Thelatter is equivalent toS beingcomeagre, foran explicit proof see [22, Th. 4.1]. The semifilter trichotomy can also formulated in terms of two cardinal characteristics: Let F be a filter on ω. B ⊆ F is a base for F if for every X ∈ F there is some Y ∈ B such that Y ⊆ X. The character of F, χ(F), is the smallest cardinality of a base of F. The cardinal u is the smallest character of a non-principal ultrafilter. We denote by g be the groupwise density number, that is the smallest number of groupwise dense sets whose intersection is empty. A set G ⊆ [ω]ω is called groupwise dense if it is closed under almost subsets and for every strictly increasing function f: ω → ω there is an infinite A such that [f(n),f(n+1)) ∈ G. Laflamme [21, n∈A Theorem 8] showed that u < g imSplies SFT, and Blass [5] showed that SFT implies u< g. The purpose of this paper is to show the following: Main Theorem. “FD and the negation of SFT” is consistent relative to ZFC. A groupwise dense family that is closed under finite unions is called a groupwise dense ideal. The groupwise density number for filters, g , is the f smallest number of groupwise dense ideals with empty intersection. From [8] and Blass [5], just read for groupwise dense ideals, it follows that u< g f is equivalent to FD. Moreover, FD implies b = u < g = d = c [5]. Hence f FD and and not SFT is equivalent to g ≤ u < g . Brendle [12] constructed f a c.c.c. extension with κ = g < g = b = κ+, and asked whether b = g < g f f is consistent. By Shelah’s g ≤ b+ in ZFC [33], the only constellation for f b ≤ g < g is b = g < g = b+. Since in any model of the dichotomy f f and u ≥ g we have the cardinal constellation b = u = g < g = c, the f FD DOES NOT IMPLY SFT 3 main theorem also answers a question by Brendle [12, Question 10] about separating g and g above b in the ℵ -ℵ -scenario. f 1 2 For S,X ∈ [ω]ω we say S splits X iff X ∩S and X rS are both infinite. A set SP ⊆ [ω]ω is called splitting or a splitting family iff for every X ∈ [ω]ω there is some S ∈ SP splitting X. The smallest cardinal of a splitting family is called the splitting number and denoted by s. Necessarily the splitting number s mustbe boundedby u for FD and u≥ g, because by [24, Cor. 4.4.], FD together with s> u implies u< g. The same argument shows: Proposition 1.2. g ≤ s implies g = g . f f Proof. Assume that we have groupwise dense families G , α < κ for some α κ< g . Thenthere is adiagonalisation D of thegenerated ideals, thatis for f every α < κ there are A ∈ G , i ≤ n such that D ⊆ A ∪···∪A . α,i α α α,0 α,nα Then if κ < s these A ∩ D are not a splitting family on [D]ω and α,i hence there is some infinite D′ ⊆ D and there are i , α < κ, such that α (∀α< κ)(D′ ⊆ A ). So D′ ∈ G and g > κ. (cid:3) α,iα Tα<κ α P-points and some cardinal characteristics are involved in our forcing construction. We recall some definitions: We say “A is almost a subset of B” and write A ⊆∗ B iff ArB is finite. Similarly, the symbol =∗ denotes equality up to finitely many exceptions in [ω]ω or in ωω, the set of functions from ω to ω. An ultrafilter U is called a P-point if for every γ < ω , for every A ∈ U, 1 i i < γ, there is some A ∈ U such that for all i < γ, A ⊆∗ A ; such an i A is called a pseudo-intersection or a diagonalisation of the A , i < γ. i Let P be a notion of forcing. We say that P preserves an ultrafilter U if (cid:13) “(∀X ∈ [ω]ω)(∃Y ∈ U)(Y ⊆ X∨Y ⊆ ωrX))” and in the contrary case P we say “P destroys U”. If P is proper and preserves U and U is a P-point, then U stays a P-point [9, Lemma 3.2]. The only models of FD that have been known so far are also models of u< g (and hence SFT). A ground model with CH is extended by an iterated forcing hP ,Q : β ≤ ω ,α < ω i that is built in the usual way: The β α 2 2 iterand Q i˜s a P -name and P = P ∗Q , and at limits we build P α α α+1 α α α with cou˜ntable supports. The iterands are˜proper forcings that preserve at least one, indeed any P-point, and thus keep u small. The principle of near coherence of filters says that for any two filters (recall: they contain the Fr´echet filter) are nearly coherent. An equivalent formulation is that any two non-principal ultrafilters are nearly coherent. The filter dichotomy implies NCF by [8]. If u< d, which follows from NCF, then by Ketonen [19] every filter witnessing u is a P-point, so we do not have to worry whether there is some non-P ultrafilter with a base of a smaller size. Let us write V for VPα, an arbitrary extension by a P -generic filter G . Although u is α α α kept small, at least at stationarily many limit steps α < ω of cofinality ω 2 1 4 HEIKEMILDENBERGER the next iterand adds a real that has supersets in all groupwise dense sets in VPα = V and thus g = ℵ [8]. α 2 Some types of such models of u < g are known: an iteration of length ω with countable support of Blass–Shelah forcing over a ground model of 2 CH [9] gives ℵ = u < s = g = 2ℵ0 = ℵ and an iteration of length ω with 1 2 2 countable support of Miller forcing over a ground model of CH [10] gives ℵ = u = s < g = 2ℵ0 = ℵ . A third type of model of u < g is given by a 1 2 countable support iteration of Matet forcing [4]. Other proper tree forcings that preserve P-points can be interwoven into the iteration and, as long as atstationarily manysteps of cofinality ω areal isaddedthat hasasuperset 1 in each groupwise dense family in the intermediate model, the outcome is u< g. The proof of our main theorem works with a construction that uses some techniques developed in Mildenberger and Shelah’s construction of a model of NCF and not FD in [26]. As there, we use Hindman’s theorem and Eis- worth’s theorem on Matet forcing with stable ordered-unionultrafilters. We combine this with new work on Matet forcing with stable ordered union ul- trafilters (also known as Milliken–Taylor ultrafilters). The main difficulty was to find the properties (I4) and (I5) of the iteration. We give an overview over the construction: We let S2 = {α ∈ ω : i 2 cf(α) = ω }, i = 0,1. A ground model with CH and ♦(S2) is extended i 1 by a countable support iteration hP ,Q : β ≤ ω ,α < ω i, with iterands β α 2 2 of the form Qα = MUα. Alternatively˜, instead of assuming a diamond in the ground model we can force in a first forcing step with approximations hP ,Q : β ≤ γ,α < γi, γ < ω . For such a representation of a forcing β α 2 see [2˜7, Section 2]. The tricky part of our proof is to find suitable stable ordered-union ultrafilters U . In contrast to in [26] this time also the steps α α 6∈ S2 have a more subtle influence and they must be gentler than the 1 Blass–Shelah forcing, that increases the splitting number. The iterands are proper, indeed σ-centred iterands can be chosen. Their definition is based on ultrafilters and other names that are defined by induction. The essential properties of the iteration are: First: As in [26], we preserve only one arbitrary P-point E ∈ V that will 0 be fixed forever, and we destroy many others. Eisworth [14] is the tool for preserving E. Second: We get FD with the aid of a diamond. A diamond sequence on S2 is a sequence hD : α ∈ S2i such that for any X ⊆ ω the set 1 α 1 2 {α ∈ S2 : X ∩α = D˜} is stationary. If the diamond D guesses a non- 1 α α meagre filter D [G] =˜F, we add with Q a (generically˜flat) finite-to-one α α function f , su˜ch that f (F) = f (E) and E stays an ultrafilter. Thus, by α α α a reflection argument for countable support iterations of proper iterands of size ≤ ℵ , FD holds in the final model. 1 FD DOES NOT IMPLY SFT 5 Third: During our construction, a semifilter S witnessing the failure of the semifilter trichotomy will be constructed as (1.1) S ={x ∈ [ω]ω : (∃α∈ ω rS2)(x ⊇∗ s )} 2 1 α where the s is simply the Q -generic real over VPα = V for α ∈ ω . The α α α 2 α ∈ ω rS2 contribute to the name of S, and for the α ∈ S2, s has an 2 1 1 α even more important role: To ensure that the iteration can be continued from P to P . For this construction we use for α ∈ S2 Matet forcing M α α+1 1 thinned out with a Milliken–Taylor ultrafilter Uα ∈ VPα, called MUα, such that U guarantees that our inductive construction can be carried on. Let α R∗ stand for the set of finite-to-finite relations (explained in Section 4). The aim is to build an iteration of length ω with the following property: 2 (I5) For all γ ∈ ω rS2: 2 1 s is ≤∗-unboundedover V and(∀β ∈ γrS2)(∀R ∈ R∗∩V ) P (cid:13) γ γ 1 β (cid:0) γ+1 (s 6⊆∗ Rs ) . γ β (cid:1) For this we will have to work quite a bit. An impatient knowledgeable reader might first want to read Lemma 5.1 in which we prove why (I5) to- gether with other well-known properties of Matet forcing and of Matet forc- ing with suitable Milliken–Taylor ultrafilters ensures that VPω2 is a model of FD and not SFT. A side remark: (I5) means that the counterexample S is a semifilter consisting of somewhat solid sets s , α ∈ ω r S2. Matet reals are more α 2 1 “solid” than Blass–Shelah reals: Matet blocks muststay or are droppedas a whole,whereasinBlass–Shelah forcingwecanthinouteach blockaccording to a logarithmic measure. Both have very large complements, which is the reason that both diagonalise many families of groupwise dense sets. Matet forcing will be defined and explained in Section 2. Blass–Shelah forcing will not appear any more in this work. By Proposition 1.2 it is ruled out as a candidate for iterands. Observation 1.3. From the short proof of Laflamme’s [21] theorem u < g → SFT in [7, Lemma 9.15, Theorem 9.22] we read off the groupwise dense families that witness g ≤ u in our forcing extensions. In the ground model, fix a basis {E : ε < ℵ } for the P-point E. Then we let ε 1 (1.2) G = {Z ∈ [ω]ω : (∃S ∈ S)(∀m,n ∈ Z) ε ([m,n)∩S 6= ∅ → [m,n)∩E 6= ∅)∧ ε (∃T)(ωrT ∈[ω]ω rS)(∀m,n ∈ Z) ([m,n)∩T 6= ∅→ [m,n)∩E 6= ∅)}. ε Since S is not meagre and not comeagre, the sets G are groupwise dense, ε and G = ∅ since S is not equal to E by finite-to-one. Tε<ω1 ε The paper is organised as follows: In Section 2 we explain Matet forcing with centred systems and show how to preserve the non-meagreness and the density of a given set. In Section 3 we recall Matet forcing with stable 6 HEIKEMILDENBERGER ordered-union ultrafilters and Eisworth’s work. Section 4 introduces some notionsforhandlingblocksequencesandfinite-to-onefunctions. InSection5 we define the iterated forcing orders that establish our consistency results. Undefined notation on cardinal characteristics can be found in [2, 7]. Undefined notation about forcing can be found in [20, 32]. In the forcing, we follow the Israeli style that the stronger condition is the larger one. A very good background in proper forcing is assumed. 2. Preserving a non-meagre set and a dense set We define a variant of Matet forcing. For this purpose, we first introduce some notation about block-sequences. Our nomenclature follows Blass [3] and Eisworth [14]. WeletFbethecollection ofallfinitenon-emptysubsetsofω. Fora,b ∈ F we write a < b if (∀n ∈ a)(∀m ∈ b)(n < m). A filter on F is a subset of P(F)that isclosed underintersections andsupersets. Asequencea¯ = ha : n n ∈ ωi of members of F is called unmeshed if for all n, a < a . The set n n+1 (F)ω denotes the collection of all infinite unmeshed sequences in F. If X is a subset of F, we write FU(X) for the set of all finite unions of members of X. We write FU(a¯) instead of FU({a : n ∈ ω}). n Definition 2.1. Given a¯ and ¯b in (F)ω, we say that ¯b is a condensation of a¯ and we write ¯b ⊑ a¯ if ¯b ⊆ FU(a¯). We say ¯b is almost a condensation of a¯ and we write ¯b ⊑∗ a¯ iff there is an n such that hb : t ≥ ni is a condensation t of a¯. We also call ¯b ⊑∗ a¯ a strengthening a¯. We use the verb “to strengthen a¯” as an abbreviation for “to replace a¯ by an appropriate strengthening and call that strengthening again a¯”. Definition 2.2. A set C ⊆ (F)ω is called centred, if for any finite C ⊆ C there is a¯ ∈ C that is almost a condensation of any c¯∈ C. Definition 2.3. In the Matet forcing, M, the conditions are pairs (a,c¯) such that a ∈ F and c¯∈ (F)ω and a < c . The forcing order is (b,d¯) ≥ (a,c¯) 0 (recall the stronger condition is the larger one) iff a ⊆ b and bra is a union of finitely many of the c and d¯is a condensation of c¯. n Definition 2.4. Given a centred system C ⊆ (F)ω, the notion of forcing M(C) consists of all pairs (s,a¯′), such that s ∈ F and there is a¯ ∈ C such that a¯′ is an end-segment of a¯, i.e., a¯′ = ha : n ≥ ki. The forcing n order is the same as in the Matet forcing. In the special case that C is the set of members of a ⊑∗-descending sequence a¯α, α < β, we also write M(a¯α : α < β) for M(C). In this section we use M(a¯α : α < β) for ⊑∗-descending sequences of length 1, of length < κ and of length κ where κ = (2ω)V is assumed to be regular. The forcing M(a¯α : α < β) diagonalises (“shoots a real through”) {aα : n < ω}, α < β. We write set(a¯) for {a : n < ω}. n n S S FD DOES NOT IMPLY SFT 7 That means that in the generic extension VM(a¯α:α<β) the real s = {w : ∃a¯(w,a) ∈ G} is an almost subset of any set(a¯α). So, in V[G] theSsemi- filter generated by {set(a¯α) : α < β} is meagre even if it was not meagre before. This means that our forcings are very specific: They destroy the non-meagreness of some sets, but may preserve the non-meagreness of oth- ers. We now work on this and a related preservation property. We let f: ω → ω be strictly increasing with f(0) = 0 and let x: ω → 2. The meagre set coded by f and x is (2.1) M = {y ∈ω2 : (∀∞n)(y ↾ [f(n),f(n+1))6= x ↾ [f(n),f(n+1)))} (f,x) see [2, 2.2.4]. Every meagre subset of ω2 is a subset of a meagre set of the form M , which is F . (f,x) σ We write names for reals and for meagre sets in c.c.c. forcings P in a standardised form. Let g: ω → H(ω), H(ω) is the set of hereditarily finite sets. A standardised name for g is g = Name(k¯,p¯) = {h(n,k ),p i : n,m ∈ ω}, n,m n,m ˜ such that {p : m ∈ω} is predensein P and p (cid:13) g ↾ n = kˇ , k ∈ n,m n,m P n,m n,m H(ω), and such that kn′,m′ ↾ n = kn,m if pn′,m′ and pn,m˜are compatible and n′ ≥ n. In the case of meagre sets we let g code a name (f,x) for a pair (f,x)asinEquation(2.1). Thisisdoneasfo˜llows: Inorderto˜easenotation, we let k consist of f : n+2 → ω and x : f (n+1) → 2 for a strictly n,m n n n increasing f ∈ n+2ω and x ∈ f(n+1)2. Thus the k are possibilities for n n n,m f ↾ n+2, x ↾ f(n+1). If g codes (f,x) we call the name given by entering (f,x) into Equa- tion ˜(2.1) simpl˜y M . The code is evaluated in various Z˜FC models and g always gives a meagr˜e set. We write P⊆ P′ iff P ⊆ P′ and for any p,q ∈ P, if p and q are incompat- ic ible in P then they are also incompatible in P′. If P ⊆ P′ then not every ic standardised P-name for a real is also P′-name for a real. This happens, however, if any maximal antichain {p : m ∈ ω} in P stays maximal in n,m P′. This explains our adding the clauses “is a M(¯bβ)-name of a meagre set that can be construed as an M(c¯β)-name” in the inductive construction. In the end we want to evaluate only names in the final order and each such name appears at some stage of countable cofinality (see Lemma 2.5) and then can be construed also as a name of a forcing order of any later stage. We let P⋖Q denote that P is a complete suborder of Q. If C ⊆ C′ are centred systems, then M(C) ⊆ M(C′). The constructions in the section ic are based on the following fact: If g is a M(C)-name and an M(C′)-name, C ⊆ C′ and p ∈ M(C) and k ∈ H˜(ω), then: Iff p (cid:13)M(C) g(nˇ) = kˇ then p (cid:13)M(C′) g(nˇ) = kˇ. In general M(C) is not a complete subo˜rder of M(C′), and in ge˜neral the Cohen forcing, which is equivalent to M(a¯), is not a complete suborder of M(C). For example, there are M(C) that preserve an ultrafilter from the ground model [14, Theorem 2.5]. 8 HEIKEMILDENBERGER The following, until the end of this section, is used as a crucial step. We construct a σ-centred forcing Q = M(C) as the union of an ⊆ -increasing ic chain M(C ), α < κ. Q will serve as an iterand in an iteration of length κ+ α with finite or with countable supports. If there is d¯∈ C such that ∀c¯∈ C, d¯⊑∗ c¯, then, M(C) is equivalent to M(d¯) and this is in turn equivalent to Cohen forcing. Lemma 2.5. Let {c¯ε : ε < δ}, be a ⊑∗-centred set. Assume Q = M({c¯ε : ε < δ}) and cf(δ) > ω and g is a Q-name for a member of ωH(ω) or of a 0 meagre set. Then we can fin˜d an ε < δ such that for every ε∈ [ε ,δ) there 0 0 are p ∈ M(c¯ε) and k ∈ ωH(ω) such that {p : m < ω} is predense n,m n,m n,m in Q and p (cid:13) g(n) = k . n,m Q n,m ˜ Proof. We assume that g = {h(n,h ),q i : m,n < ω}. Since cf(δ) > ω, n,m n,m there is some ε < δ suc˜h that all q are in M({c¯β : β ≤ ε }). Now, given 0 n,m 0 ε∈ [ε ,δ), we take 0 I = {q ∈M(c¯ε) : (∃m)(q ≥ q )}. n Q n,m Then I is predense in Q. Now let p , m < ω, list I and choose k n n,m n n,m such that p (cid:13) g(n)= k . Then k¯, p¯describe g as desired. (cid:3) n,m Q n,m ˜ ˜ In this section we deal with the simple case that C = {c¯ε : ε < δ} is the range of a ⊑∗-descending sequence hc¯ε : ε < δi. The following lemma is the key to the preservation properties used in our construction. Therefore we prove it very much in detail. Recall MA (σ-centred) is Martin’s axiom <κ for σ-centred posets and < κ dense sets. A poset P is called centred if for any finite F ⊆ P there is q stronger than any of the p ∈ F. P is σ-centred if it is the union of countably many centred sub-posets. Let Γ be a class of forcings. MA (Γ) says: For any P ∈ Γ for any γ < κ for any collection <κ {D : α < γ} there is a filter G⊆ P such that (∀α < γ)(D ∩G 6= ∅). α α In the next lemma we use a technique called “sealing antichains” or “pro- cessing names” that was used in the set theory of the reals [1, 11, 13, 15, 25] and possibly elsewhere and is a very important technique in constructing forcings under the assumption of large cardinals. “Sealing the antichains” is a crucial step in the constructions of the iterands Q = M(hc¯ : ε < ω i) α ε 1 in the proofs of Lemma 5.3 and of Lemma 5.4. It allows us to preserve the non-meagreness of up to (2ω)V sets and to preserve the density in the ∗ ⊑ -order. We state and prove the following lemma for an arbitrary regular κ, though we use it for κ= ℵ in the proof of the main theorem. 1 Lemma 2.6. Assume that κ is a regular uncountable cardinal, 2ω = κ, MA (σ-centred), and that X = {x : α < κ} is not meagre. Then there <κ α is a ⊑∗-descending sequence hc¯ε : ε < κi such that of size κ such that Q = M(c¯ε : ε <κ) fulfils (cid:13) “X is not meagre ” Q FD DOES NOT IMPLY SFT 9 Proof. Let h¯bα,gα : ε < κi list the pairs (¯b,g) such that ¯b ∈ (F)ω and g = {h(n,k ),˜p i : m,n ∈ ω} is an M(¯b)-˜name for a meagre set. We n,m n,m ˜assume that each pair (¯b,g) appears in the list κ many times. Let gδ = {h(n,kδ ),pδ i : m,n ∈˜ω}. ˜ n,m n,m We choose by induction on ε< κ a sequence c¯ε ∈(F)ω with the following properties: (a) If δ < ε then c¯ε ⊑∗ c¯δ. (b) If ε = δ +1 and for some γ ≤ δ, ¯bδ = c¯γ and gδ is a M(¯bδ)-name of a meagresetthatcan beconstruedas anM(c¯δ)-n˜ame, thenc¯ε guarantees that for some ζ < κ, ε (cid:13) x 6∈ M . Q ζε gδ ˜ For ε = 0 we let c¯0 = h{n} : n < ωi. Let ε < κ bea limit ordinal. We apply MA (σ-centred) to the σ-centred <κ forcingnotion{(c¯,n,F) : c¯isafiniteunmeshedsequenceofsubsetsofnand F is a finite subsetof ε}, (Israeli) ordered by (¯b,n′,F′) ≥ (c¯,n,F) iff n′ ≥ n, F′ ⊇ F, ¯b =⊑ c¯, b = c for i < n and (∀γ ∈ F)(∀k)(b ⊆ [n,n′) → b ∈ i i k k FU(c¯γ)),andthedensesetsI = {(c¯,m,F) : c¯rn 6= ∅∧δ ∈ F∧m ≥ n}, δ,n δ < ε, n < ω, and thus we get a filter G intersSecting all the I and set δ,n c¯ε = {c¯ : (∃n,F)((c¯,n,F) ∈ G)}. Then c¯ε is as desired. S Step ε = δ+1, and c¯δ is chosen. We assume that for some γ ≤ δ, ¯bδ = c¯γ and gδ is a M(¯bδ)-name of a meagre set that has an equivalent M(c¯δ)-name. Othe˜rwise we can take c¯δ+1 = c¯δ. Recall that by our coding gδ = Name(k¯δ,p¯δ) ={hhn,kδ i,pδ i : n,m ∈ ω} n,m n,m ˜ and kδ = (fδ ,xδ ), fδ : n+2 → ω, xδ : fδ (n+1) → 2 n,m n,m n,m n,m n,m n,m By induction on r ∈ ω we first choose cε+ ∈ F and b(r) ∈ ω, u ∈ F such r r thatcε+ = {cδ : n ∈ u }. We let cε+ = cδ, b(0) = max(cε+)+1, u = {0}. r S n r 0 0 0 0 Suppose that cε+, b(r) are chosen. Let {w : i < 2b(r)} enumerate all r r,i subsets of b(r). Now by subinduction on i < 2b(r) we choose nδ(r,i) = nδ(r,w ), mδ(r,i) = mδ(r,w ) such that r,i r,i (1) pδ ≥ (w ,c¯δ ↾ [max(u )+1,ω)). nδ(r,0),mδ(r,0) r,0 r (2) For1≤ i ≤ 2b(r)−1,pδ = (w ∪c(pδ ),c¯((pδ )). nδ(r,i),mδ(r,i) r,i nδ(r,i),mδ(r,i) nδ(r,i),mδ(r,i) (3) For 0≤ i ≤ 2b(r) −1, c¯((pδ ) is an end segment of c¯δ. nδ(r,i),mδ(r,i) (4) For 0≤ i < 2b(r) −1, (c(pδ ),c¯(pδ )) nδ(r,i),mδ(r,i) nδ(r,i),mδ(r,i) ≤ (c(pδ ),c¯(pδ )). nδ(r,i+1),mδ(r,i+1) nδ(r,i+1),mδ(r,i+1) 10 HEIKEMILDENBERGER (5) For 0 ≤ i ≤ 2b(r) −1, pδ determines g ↾ nδ(r,i)+2, and in nδ(r,i),mδ(r,i) ˜ particular it determines xδ : fδ (nδ(r,i)+1) → 2. nδ(r,i),mδ(r,i) nδ(r,i),mδ(r,i) (6) For 0≤ i < 2b(r) −1, nδ(r,w ) ≥ r and r,i b(r)≤ fδ (nδ(r,i)) < fδ (nδ(r,i)+1) ≤ nδ(r,i),mδ(r,i) nδ(r,i),mδ(r,i) fδ (nδ(r,i+1)). nδ(r,i+1),mδ(r,i+1) (7) B(r)= {[fδ (nδ(r,i)),fδ (nδ(r,i)+1)) : i < 2b(r)}. S nδ(r,i),mδ(r,i) nδ(r,i),mδ(r,i) Once the subinduction is performed, we choose a finite set cε+ = {c(pδ ) : i < 2b(r)} r+1 [ nδ(r,i),mδ(r,i) This also determines u . We let r+1 b(r+1) = max(cε+ )+1. r+1 Thus the induction step is finished. We have (2.2) c¯(pδ )= c¯δ ↾ [max(u )+1,∞). nδ(r,2b(r)−1),mδ(r,2b(r)−1) r+1 Since X is not meagre there is some ζ < κ such that there is an infinite ε set Y = {r : i < ω} ⊆ω such that i (m1) hr : i < ωi is an increasing enumeration of Y, and i (m2) for every j ∈ ω, x ↾ B(r ) = x ↾ ζε j [ (cid:16) nδ(rj,i),mδ(rj,i) i<2b(rj) fδ (nδ(r ,i)),fδ (nδ(r ,i)+1) . (cid:2) nδ(rj,i),mδ(rj,i) j nδ(rj,i),mδ(rj,i) j (cid:1)(cid:17) We let (2.3) c¯ε = hcε+ : j < ωi. rj+1 Now we show that Q = M(c¯ε : ε < κ) is as desired. It is σ-centred, because for every w ∈ F, Q = {(w,c¯δ ↾ [ℓ,ω)) : ℓ ∈ ω,w < cδ,δ ∈ κ} is w ℓ centred. We show that (cid:13) X is not meagre. Q Assume towards a contradiction that there is a Q-name g for a meagre set and there is p ∈ Q such that p (cid:13) “X ⊆ M ”. Since˜cf(κ) > ω, by Q g Lemma 2.5 there is some γ < κ such that g is an M˜(c¯γ)-name. Since in the enumeration every name appears cofinally˜often, for some δ ≥ γ we have (¯bδ,gδ) = (c¯γ,g). So at stage ε = δ +1 in our construction we take care of g’s e˜quivalent˜M(c¯δ)-name Name(k¯δ,p¯δ). Let ζ and c¯ε be as in this step. ε ˜ Assume that there are some q ≥ p and some n(∗) such that q (cid:13) (∀n ≥ Q n(∗))(x ↾ [f (n),f (n+1)) 6= x ↾ [f (n),f (n+1))). By the form of Q, ˜g ˜g ˜g ζε ˜g ˜g