aa r X i v : . [ m a t h . L O ] J a n FILTERS ON A COUNTABLE VECTOR SPACE
IIAN B. SMYTHE
Abstract.
We study various combinatorial properties, and the im-plications between them, for filters generated by infinite-dimensionalsubspaces of a countable vector space. These properties are analogousto selectivity for ultrafilters on the natural numbers and stability forordered-union ultrafilters on FIN. Introduction
Throughout, we fix a countably infinite-dimensional vector space E overa countable (possibly finite) field F , with distinguished basis ( e n ); one maytake E = L n F and e n the n th unit coordinate vector. We will use theterm “subspace” in reference to E exclusively to mean linear subspace. Ourprimary objects of study here are filters of subsets of E \ { } (we abuseterminology and call these filters on E ), generated by infinite-dimensionalsubspaces. All such filters will be assumed to be proper and contain allsubspaces of finite codimension.We follow the terminology and notation of [20]. A sequence ( x n ) ofnonzero vectors in E is called a block sequence (and its span, a block sub-space ) if for all n , max(supp( x n )) < min(supp( x n +1 )) , where the support of a nonzero vector v , supp( v ), is the finite set of those i ’s such that e i has a nonzero coefficient in the basis expansion of v . Bytaking linear combinations and thinning out, we see that every infinite-dimensional subspace contains an infinite block sequence. Note that supp( v )is an element of FIN, the set of finite nonempty subsets of ω .The set of infinite block sequences in E is denoted by bb ∞ ( E ) and inheritsa Polish topology from E ω , where is E discrete. We denote the set of finiteblock sequences by bb < ∞ ( E ). Block sequences are ordered by their spans:we write X (cid:22) Y if h X i ⊆ h Y i , where h·i denotes the span (with 0 removed),or equivalently, each entry of X is a linear combination of entries from Y .We write X/n (or
X/~x for ~x ∈ bb < ∞ ( E )) for the tail of X consisting of thosevectors with supports entirely above n (or the supports of ~x , respectively),and X (cid:22) ∗ Y if X/n (cid:22) Y for some n . Date : January 19, 2021.2010
Mathematics Subject Classification.
Primary 03E05, Secondary 15A03.The author would like to thank Andreas Blass for many insightful conversations relatedto the material here.
Definition 1.1.
A filter F on E is a block filter if it has a base of sets ofthe form h X i for X ∈ bb ∞ ( E ).From now on, whenever we use the notation h X i , it will be understoodthat X ∈ bb ∞ ( E ).In [20], the author primarily considered families in (bb ∞ ( E ) , (cid:22) ), i.e.,subsets of bb ∞ ( E ) which are upwards closed with respect to (cid:22) ∗ , and fil-ters in (bb ∞ ( E ) , (cid:22) ), those families which are (cid:22) -downwards directed, as animportant special case. As commented there, one can go back and forthbetween filters in (bb ∞ ( E ) , (cid:22) ) and the block filters on E they generate bytaking spans and their inverse images, respectively. Definition 1.2.
Given a block filter F on E :(a) a set D ⊆ E is F -dense if for every h X i ∈ F , there is an infinite-dimensional subspace V ⊆ h X i such that V ⊆ D .(b) F is full if whenever D ⊆ E is F -dense, we have that D ∈ F .Fullness is the analogue in this setting to being an ultrafilter: observethat for filters on ω , U is an ultrafilter if and only if whenever d ⊆ ω hasinfinite intersection with each element of U , d ∈ U .Already, the notion of a full block filter has substantial content: while theycan be constructed using the Continuum Hypothesis ( CH ), Martin’s Axiom( MA ) (Theorem 5.3 in [20]), or by forcing with all infinite-dimensional sub-spaces ordered by containment (Lemma 5.4 in [20]), they project to ordered-union ultrafilters on FIN via supports (Theorem 6.3 in [20]) and thus cannotbe proved to exist in ZFC alone (Corollary 6.5 in [20]). The following addi-tional properties can also be obtained by the same methods:
Definition 1.3.
A block filter F on E is:(a) a ( p ) -filter (or has the ( p ) -property ) if whenever h X n i ∈ F for all n ,there is an h X i ∈ F such that X (cid:22) ∗ X n for all n .(b) spread if whenever I < I < I < · · · is a sequence of intervals in ω ,there is an h X i ∈ F , where X = ( x n ), such that for every n , there is an m such that I < supp( x n ) < I m < supp( x n +1 ).(c) a strong ( p ) -filter (or has the strong ( p ) -property ) if whenever h X ~x i ∈ F for all ~x ∈ bb < ∞ ( E ), there is an h X i ∈ F such that X/~x (cid:22) X ~x for all ~x ⊑ X .In (a) and (c), the X described is called a diagonalization of ( X n ) or ( X ~x ),respectively. If F is both full and a (strong) ( p )-filter, then we refer to it asa (strong) ( p + ) -filter .Much of [20] is devoted to showing that filters with these properties “local-ize” Ramsey-theoretic dichotomies for block sequences in E , due to Rosendal Those readers familiar with [20] should be cautioned that many of the definitions forfamilies therein simplify in the case of filters, and that some of the results in the presentarticle may no longer hold when “filter” is replaced by “family”. This relationship issimilar to that between ultrafilters and the more general class of coideals on ω . ILTERS ON A COUNTABLE VECTOR SPACE 3 [16], and in Banach spaces, due to Gowers [7]. Such dichotomies are phrasedin terms of games:Given X ∈ bb ∞ ( E ), the infinite asymptotic game played below X , F [ X ],is a two player game where the players alternate, with I going first andplaying natural numbers n k , and II responding with nonzero vectors y k forming a block sequence and such that n k < min(supp( y k )). The outcome of a round of the game is the block sequence ( y k ) consisting of II’s moves.Likewise, the Gowers game played below X , G [ X ], is defined with I goingfirst and playing infinite block sequences Y k (cid:22) X , and II responding withnonzero vectors y k forming a block sequence and such that y k ∈ h Y k i . The outcome of a round of G [ X ] is again ( y k ).In both of these games, the notion of a strategy for one of the players isdefined in a natural way. Given a set A ⊆ bb ∞ ( E ), we say that a playerhas a strategy for playing into (or out of ) A if they posses a strategy all ofwhose outcomes line in (or out of) A .The following is the local form of Rosendal’s dichotomy; Rosendal’s orig-inal result can be recovered by simply omitting any mention of F . Theorem 1.4 (Theorem 1.1 in [20]) . Let F be a ( p + ) -filter. If A ⊆ bb ∞ ( E ) is analytic, then there is an h X i ∈ F such that either (i) I has a strategy in F [ X ] for playing out of A , or (ii) II has a strategy in G [ X ] for playing into A . While we won’t deal explicitly with Banach spaces here, the spread con-dition, along with being ( p + ), was used to obtain the local form of Gowersresult for Banach spaces (Theorem 1.4 in [20]). These results are analogousto the way selective ultrafilters on ω and stable ordered-union ultrafilters onFIN localize the respective dichotomies for analytic partitions of [ ω ] ω andFIN [ ∞ ] (see [13] and [2]).One apparent deficiency in Theorem 1.4 is that it is not obvious whethereither conclusion guarantees that F meets A or its complement, in cases (ii)and (i), respectively. In the event that F is a strong ( p + )-filter, this can beensured for Borel sets A (see the discussion following Lemma 4.4 in [20]),but it was unclear whether this is sufficient for analytic sets. This is rectifiedby the following assumption: Definition 1.5.
A block filter F on E is strategic if whenever α is a strategyfor II in G [ X ], where h X i ∈ F , there is an outcome Y of α such that h Y i ∈ F .Under large cardinal assumptions, if F is a strategic ( p + )-filter, thenTheorem 1.4 can be extended to all definable subsets A (Theorem 1.3 in[20]) and moreover, being “strategic ( p + )” exactly characterizes genericity As mentioned in [20], an apparent weakening of the ( p )-property akin to semiselectivity(cf. [5]), namely that a sequence of dense open subsets of F must possess a diagonalizationin F , is all that is used in the proof of this result. It will be a consequence of Theorem4.1 below that this is equivalent to the ( p )-property for full block filters. A proper class of Woodin cardinals suffices.
IIAN B. SMYTHE complete combinatorics K S Theorem 1.2 (cid:11) (cid:19) strategic ( p + )-filter Proposition 4.6 (cid:11) (cid:19) strong ( p + )-filter Lemma 8.13 (cid:15) (cid:15) spread ( p + )-filter (cid:15) (cid:15) ( p + )-filter Theorem 1.1 (cid:15) (cid:15) local Rosendal dichotomy
Proposition 3.6 (cid:15) (cid:15) full block filter . Figure 1.
The implications for block filters proved in [20].of a block filter over the inner-model L ( R ) (Theorem 1.2 in [20]). This latterproperty is known in the literature as having complete combinatorics . Figure 1 shows the implications between these various properties of blockfilters, as proved in [20], with references to the relevant propositions therein(the implication from “spread ( p + )” to “( p + )” is trivial). The unidirectionaldouble arrow = ⇒ indicates a strict implication; it is consistent with ZFC that, when | F | >
2, there is a strong ( p + )-filter which is not strategic, thiswill be proved in the forthcoming [18].The goal of the present article is to investigate the combinatorics of blockfilters having the above properties, as well the possibility of reversing theremaining arrows in Figure 1. We begin by considering the special caseof the finite field of order 2 and its relationship to FIN, where a completeanalysis is possible. In general, we will see that the cardinality of the field F , in so far as it is either 2, finite, or infinite, plays an important role. Those Historically, “complete combinatorics” has been used to describe genericity overHOD( R ) V [ G ] , where G is V -generic for the Le´vy collapse of a Mahlo cardinal [11], a prop-erty proved (implicitly) for selective ultrafilters in [13]. The contemporary usage avoidspassing to a L´evy collapse extension at the expense of stronger large cardinal hypothe-ses. The ZFC content of “complete combinatorics”, in all examples of which the author isaware, is that the filter meets all dense open analytic sets, in the relevant σ -distributiveSuslin partial order. That this characterizes strategic ( p + )-filters in ZFC is implicit in [20].
ILTERS ON A COUNTABLE VECTOR SPACE 5 interested in spoilers may skip ahead to Figure 2 at the end. We also provealternate characterizations of strong ( p + )-filters (Theorem 5.1) and strategic( p + )-filters (Theorem 5.5) using a restricted version of the Gowers game.The results below are inspired by the various equivalent characterizationsof selectivity for ultrafilters, originally proved by Booth and Kunen in [3](see Chapter 11 of [8] for a modern treatment), and of stability for ordered-union ultrafilters, proved by Blass in [2]. Some of our result originate in theauthor’s PhD thesis [19], but have remained otherwise unpublished, whileothers are making their first appearance here.2. FIN and the finite field of order two A sequence ( a n ) in FIN is a block sequence if max( a n ) < min( a n +1 ) forall n . The set of all infinite block sequences in FIN is denoted by FIN [ ∞ ] and inherits a Polish topology from FIN ω . We write FIN [ < ∞ ] for the set offinite block sequences in FIN. Given A, B ∈ FIN [ ∞ ] , we write h A i for theset of all finite unions of entries from A , and A (cid:22) B if h A i ⊆ h B i (likewisefor A/n and A (cid:22) ∗ B ) to agree with our notation for E . Again, we reservethe notation h A i for when A is a block sequence. For A ∈ FIN [ ∞ ] , we write A [ ∞ ] for the set of those B ∈ FIN [ ∞ ] such that B (cid:22) A .The Ramsey theory for FIN is largely a consequence of the finite-unionsform of Hindman’s Theorem [9]: If FIN is partitioned into finitely manypieces, then there is an A ∈ FIN [ ∞ ] such that h A i is contained entirely inone of those pieces.The relevant notions for ultrafilters on FIN were defined by Blass in [2]:An ultrafilter F of subsets of FIN is ordered-union if it has a base of setsof the form h A i . F is stable if whenever h A n i ∈ F for all n ∈ ω , there is a h B i ∈ F such that B (cid:22) ∗ A n for all n . These are connected to block filterson E via the support map: Theorem 2.1 (Theorem 6.3 in [20]) . If F is a full block filter on E , then supp( F ) = { A ⊆ FIN : ∃ X ∈ F ( A ⊇ { supp( v ) : v ∈ X } ) } is an ordered-union ultrafilter on FIN . If, moreover, F is a ( p ) -filter, then supp( F ) is stable. In the case when | F | = 2, nonzero vectors can be identified with theirsupports in FIN, addition of vectors with disjoint supports corresponds totheir union, and scalar multiplication trivializes. Thus, the study of blocksequences of vectors reduces to the study of block sequences in FIN, and F is a full (( p + ), resp.) block filter if and only if it is an (stable, resp.) ordered-union ultrafilter: one direction is Theorem 2.1, while the converse followsfrom the fact that if D ⊆ E is U -dense and U is an ultra filter, then D ∈ U .We will see in Theorem 4.1 below that the second to last implicationin Figure 1 reverses: for block filters, being a ( p + )-filter is equivalent tosatisfying Theorem 4.1, regardless of the field. The | F | = 2 case highlightsa difficulty in understanding the last implication in Figure 1; whether it IIAN B. SMYTHE reverses in this case is equivalent to whether every ordered-union ultrafilteris stable, a long-standing open problem (see, e.g., [10]). We will not attemptto shed any additional light on this question here.While Theorem 1.4 can be rephrased for stable ordered-union ultrafiltersand FIN, a stronger result holds; stable ordered-union ultrafilters localizethe infinite-dimensional form of Hindman’s Theorem [9] due to Milliken andTaylor [15] [21]. This is one of several equivalents to stability proved in [2]:
Theorem 2.2 (Theorem 4.2 in [2]) . Let U be an ordered-union ultrafilteron FIN . The following are equivalent: (i) U is stable. (ii) For any analytic set A ⊆ FIN [ ∞ ] , there is an h A i ∈ U such that either A [ ∞ ] ⊆ A or A [ ∞ ] ∩ A = ∅ . Assuming large cardinal hypotheses, the methods of [5] can be used toextend (ii) to all subsets A in L ( R ) and prove complete combinatorics forstable ordered-union ultrafilters (see also the discussion on p. 121-122 of [2]).Consequently, when | F | = 2, all but possibly the last of the conditions inFigure 1 are equivalent. To see this directly: Corollary 2.3. If U is a stable ordered-union ultrafilter, then U is strategic.Proof. Let h A i ∈ U and α be a strategy for II in G [ A ]. By Lemma 4.7 in[20], there is an analytic set A of outcomes of α which is dense below A , inthe sense of forcing with (FIN [ ∞ ] , (cid:22) ). By Theorem 2.2 applied to A [ ∞ ] , thereis a B (cid:22) A with h B i ∈ U such that either B [ ∞ ] ⊆ A or B [ ∞ ] ∩ A = ∅ . Since A is dense below B , the latter is impossible. In particular, B ∈ A ⊆ [ α ], so U contains an outcome of α . Thus, U is strategic. (cid:3) Another variation on stability appears in the literature [14] [22]: An ultra-filter U on FIN is selective if it is ordered-union and whenever h A a i ∈ U forall a ∈ FIN [ < ∞ ] , there is a h B i ∈ U such that B/a (cid:22) A a for all a (cid:22) B . (Notethe resemblance to our strong ( p )-property.) While this property appearsstronger than stability, it is, again, equivalent: Corollary 2.4. If U is a stable ordered-union ultrafilter on FIN , then U isselective.Proof. Suppose we are given h A a i ∈ U for all a ∈ FIN [ < ∞ ] . Define D = { B ∈ FIN [ ∞ ] : B/a (cid:22) A a for all a (cid:22) B } D = { B ∈ FIN [ ∞ ] : h B i and the h A a i ’s do not generate a filter } . Let D = D ∪ D . It is straightforward to verify that D is analytic and denseopen in (FIN [ ∞ ] , (cid:22) ). By Theorem 2.2, there is a h B i ∈ U such that B ∈ D .Clearly, B / ∈ D , so B ∈ D and thus witnesses selectivity. (cid:3) Both of the previous theorems are instances of complete combinatorics atwork in a
ZFC context.
ILTERS ON A COUNTABLE VECTOR SPACE 7
Returning to the setting of an arbitrary countable field, we have seen thatevery ( p + )-filter produces a stable ordered-union ultrafilter. In Theorem 2.7,we prove a converse. We’ll need some notation: for X = ( x n ) ∈ bb ∞ ( E ), letsupp( X ) = (supp( x n )) ∈ FIN [ ∞ ] . Part (a) of the following lemma impliesthat supp : bb ∞ ( E ) → FIN [ ∞ ] is a projection, in the sense of forcing. Lemma 2.5. (a)
Suppose that X ∈ bb ∞ ( E ) and A ∈ FIN [ ∞ ] are such that A (cid:22) supp( X ) . Then, there is a Y ∈ bb ∞ ( E ) such that Y (cid:22) X and supp( Y ) = A . (b) Suppose that ( X n ) is a (cid:22) ∗ -decreasing sequence in bb ∞ ( E ) and A ∈ FIN [ ∞ ] is such that A (cid:22) ∗ supp( X n ) for all n . Then, there is a Y ∈ bb ∞ ( E ) such that Y (cid:22) ∗ X n for all n and supp( Y ) = A .Proof. Part (a) is easier, so we will prove (b) here instead: Write A =( a k ), with each a k ∈ FIN. For notational convenience, let X − = ( e n ) and m − = −
1. For each n ≥
0, let m n be such that A/m n (cid:22) supp( X n ) and X n /m n (cid:22) X n − . We may assume that each m n = max(supp( a i )) for some i and that m n < m n +1 . For each n ≥ −
1, and each of the finitely many a k ’swith supp( a k ) ⊆ ( m n , m n +1 ], choose y k ∈ h X n i such that supp( a k ) = y k .Let Y = ( y k ). Clearly, supp( Y ) = A . Moreover, for each n and all k withsupp( y k ) ≥ m n , y k ∈ h X n /m n i , and so Y (cid:22) ∗ X n . (cid:3) Lemma 2.6.
Let F be a block filter on E such that supp( F ) is a stableordered-union ultrafilter. If D ⊆ E is F -dense and h Y i ∈ F , then there is a Z (cid:22) Y such that h Z i ⊆ D and supp( Z ) ∈ supp( F ) .Proof. Let D = { A ∈ FIN [ ∞ ] : ∃ Z ∈ bb ∞ ( E )( Z (cid:22) Y ∧ supp( Z ) = A ∧ h Z i ⊆ D ) } , an analytic subset of FIN [ ∞ ] . By Theorem 2.2, there is a h B i ∈ supp( F )with B ∈ FIN [ ∞ ] such that either B [ ∞ ] ⊆ D or B [ ∞ ] ∩ D = ∅ . We claimthe latter cannot happen: As h B i ∈ supp( F ), there is a h Y ′ i ∈ F suchthat supp( Y ′ ) (cid:22) B . Since F is a block filter, we may further assume that Y ′ (cid:22) Y . As D is F -dense, there is a V (cid:22) Y ′ such that h V i ⊆ D , and sosupp( V ) ∈ B [ ∞ ] ∩ D . Thus, B [ ∞ ] ⊆ D , and in particular, B ∈ D . Any Z ∈ bb ∞ ( E ) which witnesses B ∈ D will satisfy the desired conclusion. (cid:3) Theorem 2.7. ( CH ) If U is a stable ordered-union ultrafilter on FIN , thenthere is a ( p + ) -filter F on E such that supp( F ) = U .Proof. Using CH , we can enumerate all subsets of E as D ξ , and all elements A ∈ FIN [ ∞ ] such that h A i ∈ U as A η , for ξ, η < ℵ . We will construct, viatransfinite recursion, a (cid:22) ∗ -decreasing sequence ( X α ) α< ℵ in bb ∞ ( E ) thatwill generate the promised ( p + )-filter F . CH is only used here in so far as it allows us to avoid diagonalizing uncountable-lengthsequences in U . If, instead, MA holds and U was closed under diagonalizations of length < ℵ (such stable ordered-union ultrafilters can be constructed using MA ), then our proofwould go through mutatis mutandis . IIAN B. SMYTHE α = 0: Choose X ′ ∈ bb ∞ ( E ) such that supp( X ′ ) = A . If D is suchthat there is some Y (cid:22) X ′ with h Y i ⊆ D and h supp( Y ) i ∈ U , then choose X to be such a Y . If not, take X = X ′ . α = β + 1: Suppose we have defined X γ for γ ≤ β such that h supp( X γ ) i ∈U . There is some h B i ∈ U such that h B i ⊆ h supp( X β ) i ∩ h A β +1 i . ApplyLemma 2.5(a) to obtain an X ′ β +1 ∈ bb ∞ ( E ) such that X ′ β +1 (cid:22) X β andsupp( X ′ β +1 ) = B . If D β +1 is such that there is some Y (cid:22) X ′ β +1 with h Y i ⊆ D β +1 and h supp( Y ) i ∈ U , then choose X β +1 to be such a Y . If not,take X β +1 = X ′ β +1 . α = β for limit β : Suppose we have defined X γ for γ < β such that h supp( X γ ) i ∈ U . Let ( γ n ) be a strictly increasing cofinal sequence in β . Since U is stable, there is some A ∈ FIN [ ∞ ] such that h A i ∈ U and A (cid:22) ∗ supp( X γ n )for all n . We may, moreover, assume that A (cid:22) A β . By Lemma 2.5(b), thereis an X ′ β ∈ bb ∞ ( E ) such that X ′ β (cid:22) ∗ X n for all n and supp( X ′ β ) = A . If D β is such that there is some Y (cid:22) X ′ β with h Y i ⊆ D β and h supp( Y ) i ∈ U ,then choose X β to be such a Y . If not, take X β = X ′ β . This completes theconstruction.Let F be the block filter on E generated by the X α ’s. Our constructionhas ensured that F has the ( p )-property and that supp( F ) ⊇ U , and hencesupp( F ) = U , since U is an ultrafilter. It remains to verify that F is full.Suppose that D = D ξ is F -dense. By Lemma 2.6 applied to the X ′ ξ (forwhich we’ve ensured h X ′ ξ i ∈ F ) found in stage ξ of the above construction,there must be some Y (cid:22) X ′ ξ such that h Y i ⊆ D and h supp( Y ) i ∈ U , meaningthat X ξ was chosen so that h X ξ i ⊆ D . Thus, F is a ( p + )-filter. (cid:3) It was shown in [2] that if U is an ordered-union ultrafilter, thenmin( U ) = {{ min(supp( a )) : a ∈ A } : A ∈ U } max( U ) = {{ max(supp( a )) : a ∈ A } : A ∈ U } are nonisomorphic selective ultrafilters on ω , and conversely, if V and V are nonisomorphic selective ultrafilters, then (assuming CH ) there is a stableordered-union ultrafilter U on FIN such that min( U ) = V and max( U ) = V . This can now be combined with the previous theorem to get a similarconclusion for ( p + )-filters on E .3. Fullness and maximality
A full block filter F on E is always maximal amongst block filters, and infact is maximal with respect to all filters generated by infinite-dimensionalsubspaces of E . That is, for any infinite-dimensional subspace V of E , if V ∩ X = { } for all X ∈ F , then V ∈ F (to see this, just let D = V inthe definition of “full”). Filters of subspaces with the latter property werestudied by Bergman and Hrushovski in [1], where they were called linearultrafilters ; we will instead call them subspace maximal . ILTERS ON A COUNTABLE VECTOR SPACE 9
Proposition 3.1.
Let F be a filter generated by infinite-dimensional sub-spaces of E . The following are equivalent: (i) F is subspace maximal. (ii) For every subspace V ⊆ E , either V ∈ F or there is some direct com-plement V ′ of V (i.e., V ∩ V ′ = { } and V ⊕ V ′ = E ) such that V ′ ∈ F . (iii) For every linear transformation T on E (to any F -vector space), either ker( T ) ∈ F or there is a subspace X ∈ F such that T ↾ X is injective.Proof. The equivalence if (i) and (ii) is part of Lemma 3 of [1].(i ⇒ iii) Given T , suppose that T is not injective on any subspace Y ∈ F .This means that ker( T ) has nontrivial intersection with every such Y . Hence,by subspace maximality, ker( T ) ∈ F .(iii ⇒ i) Let Y be an infinite-dimensional subspace of E which has non-trivial intersection with every subspace X ∈ F . Let Y ′ be a direct comple-ment to Y in E . Take T : E → E to be the unique linear transformationdetermined by T ( y + y ′ ) = y ′ for y ∈ Y and y ′ ∈ Y ′ . So, ker( T ) = Y . If there was a subspace X ∈ F such that T ↾ X was injective, then by assumption, X ∩ Y is nontrivial and T ↾ X ∩ Y is injective, a contradiction. Thus, Y = ker( T ) ∈ F . (cid:3) We mention here a result from [1] about the relationship between selec-tive ultrafilters on ω and filters of subspaces of E : Proposition 18 in [1] saysthat, given an ultrafilter U on ω , the set {h ( e i ) i ∈ A i : A ∈ U } , together withthe finite-codimensional subspaces of E , generates (via finite intersectionsand supersets) a subspace maximal filter on E if and only if U is selective.However, it is not clear if the resulting filter on E can be a block filter.Moreover, as it is consistent with ZFC that there is a unique (up to isomor-phism) selective ultrafilter, and hence no ordered-union ultrafilters (cf. VI.5in [17] and the comments at the end of the previous section), one cannotobtain a full block filter on E from a selective ultrafilter alone.In contrast to the above forms of maximality, unless | F | = 2, a blockfilter on E is never an ultrafilter (of sub sets ). This is a consequence of theexistence of asymptotic pairs: Definition 3.2. (a) A set A ⊆ E \ { } is asymptotic if for every infinite-dimensional subspace V of E , V ∩ A = ∅ .(b) An asymptotic pair is a pair of disjoint asymptotic sets.A standard construction of an asymptotic pair uses the oscillation of anonzero vector v = P a n e n , defined byosc( v ) = |{ i ∈ supp( v ) : a i = a i +1 }| . It is shown in the proof of Theorem 7 in [12] that if | F | >
2, then on anyinfinite-dimensional subspace of E , the range of osc contains arbitrarily long intervals (i.e., is a thick set), and thus the sets A = { v ∈ E \ { } : osc( v ) is even } A = { v ∈ E \ { } : osc( v ) is odd } form an asymptotic pair. Note that osc, and thus the A i , are invariant undermultiplication by nonzero scalars.Given a block filter F , a set D ⊆ E is F -dense if and only if A = E \ D fails to be asymptotic below every h X i ∈ F . This immediately implies thefollowing alternate characterization of fullness: Proposition 3.3.
A block filter F is full if and only if for every A ⊆ E \{ } ,there is an h X i ∈ F such that either h X i ∩ A = ∅ or A is asymptotic below h X i . (cid:3) The ( p ) -property and its relatives We begin this section by showing that if a block filter witnesses the localform of Rosendal’s dichotomy, then it must have the ( p )-property. Theorem 4.1.
Let F be a block filter. If all clopen subsets of bb ∞ ( E ) satisfy the conclusion of Theorem 4.1, then F has the ( p ) -property.Proof. Let h X n i ∈ F for each n . Define A = { ( x n ) ∈ bb ∞ ( E ) : if m ≤ max(supp( x )), then x ∈ X m } . Clearly, A is a clopen subset of bb ∞ ( E ). By our assumption, applied to A c ,there is an h X i ∈ F such that either (i) I has a strategy in F [ X ] for playinginto A or (ii) II has a strategy in G [ X ] for playing into A c .We claim that (ii) cannot happen. Suppose otherwise, denote II’s strategyby α , and consider the following round of G [ X ]: In the first inning, I plays X and II responds with α ( X ). In the second inning, I plays some Y (cid:22) X such that h Y i ⊆ T m ≤ max( α ( X )) h X m i , which defeats any possible next moveby II, contrary to what we know about α .Thus, (i) holds. Denote by σ the resulting strategy for I. Let Y = X/σ ( ∅ ),so h Y i ∈ F . Let m be given. In the first inning of F [ X ], let I play σ ( ∅ ), andlet II play any y ∈ h Y i such that m ≤ max(supp( y )). In the second inning,I plays σ ( y ), which ensures that for any z ∈ h Y /σ ( y ) i , z ∈ h X m i . In otherwords, Y /σ ( y ) (cid:22) X m . Since m was arbitrary, this shows that Y (cid:22) ∗ X m forall m , verifying the ( p )-property. (cid:3) Next, we show that the ( p )-property implies something which resemblesthe strong ( p )-property, except that the family of elements of the filter whichwe diagonalize is indexed by finite sequences in FIN instead of in E . Note that here, we could take h Y i ∈ F . This shows that block filters witnessingTheorem 5.2 below, while not necessarily full, must still have the ( p )-property. ILTERS ON A COUNTABLE VECTOR SPACE 11
Theorem 4.2.
Let F be a ( p + ) -filter. Then, whenever ( h X ~a i ) ~a ∈ FIN [ < ∞ ] is contained in F , there is an h X i ∈ F such that X/~a (cid:22) X ~a whenever ~a ⊑ supp( X ) .Proof. Let ( h X ~a i ) ~a ∈ FIN [ < ∞ ] be given as described. Since FIN [ < ∞ ] is count-able and F is a ( p )-filter, there is an h X i ∈ F such that X (cid:22) ∗ X ~a for all ~a ∈ FIN [ < ∞ ] . Writing supp( X ) [ < ∞ ] for those finite block sequences in FINcoming from h X i , let B = { ~a a b ∈ supp( X ) [ < ∞ ] : ∀ v ∈ h X i (supp( v ) = b → v ∈ \ {h X ~c i : ~c ⊑ ~a } ) } and B = { A ∈ supp( X ) [ ∞ ] : ∀ n ( A ↾ n ∈ B ) } . Clearly, B is a Borel subset of supp( X ) [ ∞ ] . By Theorem 2.2 applied to thestable ordered-union ultrafilter (by Theorem 2.1) supp( F ), there is h Y i ∈ F with Y = ( y n ) (cid:22) X , such that either supp( Y ) [ ∞ ] ⊆ B or supp( Y ) [ ∞ ] ∩ B = ∅ . Note, however, that the latter is impossible: Since Y (cid:22) ∗ X ~a for all ~a ∈ FIN [ < ∞ ] , we can thin Y = ( y n ) out to a subsequence Y ′ = ( y n k ) suchthat supp( Y ′ ) ∈ B : take y n ∈ h X ∅ i y n ∈ h X ∅ i ∩ h X (supp( y n )) i y n ∈ h X ∅ i ∩ h X (supp( y n )) i ∩ h X (supp( y n ) , supp( y n )) i and so on. Thus, supp( Y ) [ ∞ ] ⊆ B , and in particular, supp( Y ) ∈ B , so Y /~a (cid:22) X ~a whenever ~a ⊑ supp( Y ). (cid:3) Corollary 4.3.
Let F be a ( p + ) -filter. Then, whenever h X n i is in F for all n , there is an h X i ∈ F , with X = ( x n ) , such that X/x n (cid:22) X max(supp( x n )) for all n .Proof. Given h X n i as described, let X ~a = X max( ~a ) for all ~a ∈ FIN [ < ∞ ] andapply Theorem 4.2. (cid:3) Corollary 4.4.
Every ( p + ) -filter is spread.Proof. Let F be a ( p + )-filter and I < I < · · · be an increasing sequence ofnonempty intervals in ω . Let X = ( e n ). For each k ∈ ω , let m k be the leastinteger such that k ≤ max( I m k ) and let X k = X/ max( I m k +1 ). Let Y = ( y n ),with h Y i ∈ F , be as in Corollary 4.3. We may assume Y (cid:22) X/ max( I ).Then, for any n , if k = max(supp( y n )), then Y /k (cid:22) X/ max( I m k +1 ), and so I < supp( y n ) < I m k +1 < supp( y n +1 ). (cid:3) When F is a finite field, we can go one step further: Corollary 4.5.
Assume | F | < ∞ . Every ( p + ) -filter is a strong ( p + ) -filter.Proof. Let F be a ( p + )-filter and ( h X ~x i ) ~x ∈ bb < ∞ ( E ) in F . Note that since | F | < ∞ , for each a ∈ FIN, there are only finitely many vectors v ∈ E having support contained in a . For each ~a ∈ FIN [ < ∞ ] , let h X ~a i ∈ F be suchthat h X ~a i ⊆ \ {h X ~x i : supp( ~x ) ⊑ ~a } . By Theorem 4.2, there is a h X i ∈ F such that X/~a (cid:22) X ~a for all ~a ⊑ supp( X ).So, if ~x ⊑ X , then X/~x = X/ supp( ~x ) (cid:22) X supp( ~x ) (cid:22) X ~x . This verifies the strong ( p )-property. (cid:3) We do not know if this last corollary holds over infinite fields.We note here that the spread condition is analogous to another propertyfor ultrafilters on ω : Recall that an ultrafilter U on ω is a q-point if forevery partition S m I m of ω into finite sets, there exists an x ∈ U suchthat ∀ m ( | x ∩ I m | ≤ U on ω is spread if for every sequence of finite intervals I < I < I < · · · in ω , there exists an x ∈ U such that for every n ,there is an m such that I < x n < I m < x n +1 , where ( x n ) is the increasingenumeration of x . Proposition 4.6.
Let U be an ultrafilter on ω . The following are equivalent: (i) U is a q-point (ii) For every sequence of finite sets I < I < I < · · · in N , there exists a x ∈ U such that ∀ m ( | x ∩ I m | ≤ U is spread.Proof. (i ⇒ ii): This is trivial.(ii ⇒ iii): Let I < I < I < · · · be a sequence of intervals in ω . Let x ∈ U be as in (ii). We may assume that I < x . We partition x = u ∪ v as follows: u n = x n and v n = x n +1 for all n , where ( x n ) is the increasing enumerationof y . For every n , since u n = x n , v n = x n +1 , and u n +1 = x n +2 must becontained in three distinct I k ’s, the middle interval must separate u n and u n +1 , that is, there is an m such that I < u n < I m < u n +1 . Similarly forthe v n . Since U is an ultrafilter, one of u or v must be in U .(iii ⇒ i): Let S m I m be a partition of ω into finite sets. We define aninterval partition ω = S k J k as follows: J = [0 , max I ]. Let J be thesmallest interval immediately above J such that J ∪ J covers I and all I m for which I m ∩ J = ∅ . Continue in this fashion, letting J k +1 be thesmallest interval immediately above J k such that J ∪ · · · ∪ J k ∪ J k +1 covers I k +1 and all I m for which I m ∩ ( J ∪ · · · ∪ J k ) = ∅ . Let x ∈ U be as inthe definition of spread applied to J < J < · · · . Towards a contradiction,suppose that x i < x j are both in some I m . Let n be the least such that I m ⊆ J ∪ · · · ∪ J n . We may assume n > n , I m ∩ ( J ∪ · · · ∪ J n − ) = ∅ . Thus, I m ⊆ J n − ∪ J n . But then, x i and x j fail to be separated by one of the J k ’s, contrary to x witnessingthat U is spread. (cid:3) ILTERS ON A COUNTABLE VECTOR SPACE 13 The restricted Gowers game and strategic filters
Given a block filter F and h X i ∈ F , we define the restricted Gowersgame G F [ X ] below X exactly like G [ X ] except that player I is restrictedto playing Y (cid:22) X such that h Y i ∈ F . Since all subspaces spanned by tailsof X are automatically in F , we may think of G F [ X ] as an intermediatebetween the games F [ X ] and G [ X ]. Throughout this section, we will saythat an outcome Y of one of the games is “in F ” if h Y i is. Our first resulthere relates strategies for I in G F [ X ] to the strong ( p )-property, and is basedon a characterization of selective ultrafilters (Theorem 11.17(b) in [8]). Theorem 5.1.
Let F be a block filter. F has the strong ( p ) -property if andonly if for every X ∈ F and every strategy σ for I in G F [ X ] , there is anoutcome of σ in F .Proof. ( ⇒ ) Towards a contradiction, suppose that σ is a strategy for I in G F [ X ], h X i ∈ F , and no outcome of σ is in F . Define sets A ~x ⊆ F asfollows: A ∅ = {h σ ( ∅ ) i} and in general, A ~x is the set of all h Y i ∈ F suchthat Y is played by I, when I follows σ and ~x = ( x , . . . , x n − ) are the first n moves by II. Some ~x may not be valid moves for II against σ , in which casewe let A ~x = A ~x ′ where ~x ′ is the maximal initial segment of ~x consisting ofvalid moves. Then, for all ~x , A ~x is finite, and A ~x ⊆ A ~y whenever ~x ⊑ ~y .For each ~x , pick h Y ~x i ∈ F such that for all Y ∈ A ~x , Y ~x (cid:22) Y . By the strong( p )-property, there is a h Y i ∈ F , say Y = ( y n ) (cid:22) X , such that Y /~y (cid:22) Y ~y forall ~y ⊑ Y .Consider the play of G F [ X ] wherein I follows σ and II plays y , y , etc.This is a valid play by II by our choice of Y : y ∈ h Y ∅ i ⊆ h σ ( ∅ ) i , y ∈h Y / ( y ) i ⊆ h Y ( y ) i ⊆ h σ ( y ) i , etc. The resulting outcome is Y , and h Y i ∈ F ,a contradiction to our assumption about σ .( ⇐ ) Suppose that F does not have the strong ( p )-property, so there are h X ~x i ∈ F for all ~x ∈ bb < ∞ ( E ) such there for no h X i ∈ F is it the case that X/~x (cid:22) X ~x for all ~x ⊑ X . Take h X i ∈ F arbitrary. We define a strategy σ for I in G F [ X ] as follows: Start by playing Y ∅ (cid:22) X, X ∅ . If II plays y ∈ h Y i ,respond by playing some Y ( y ) (cid:22) Y ∅ , X ( y ) . In general, if II has played( y , . . . , y k ), respond by playing some Y ( y ,...,y k ) (cid:22) Y ( y ,...,y k − ) , X ( y ,...,y k ) .Note that in each move, we can always find such a h Y ~y i ∈ F since F isa block filter. If Y is an outcome of a round of G F [ X ] where I followed σ ,then for every ~y ⊑ Y , Y /~y (cid:22) X ~y . In other words, Y is a diagonalization of h X ~x i ~x ∈ bb < ∞ ( E ) , and thus by assumption, cannot be in F . (cid:3) Since every strategy for I in F [ X ] is also a strategy for I in G F [ X ], itfollows that if F is a strong ( p )-filter, h X i ∈ F , and σ a strategy for I in F [ X ], then there is an outcome of σ in F (this was Theorem 4.3 in [20]).The restricted Gowers game can be used to prove a version of Theorem1.4 for ( p )-filters without the extra assumption of fullness. This result is dueindependently to the author and, in more generality, to No´e de Rancourt: Theorem 5.2 (Theorem 3.11.5 in [19] and Theorem 3.3 in [4]) . Let F bea ( p ) -filter. If A ⊆ bb ∞ ( E ) is analytic, then there is an h X i ∈ F such thateither (i) I has a strategy in F [ X ] for playing out of A , or (ii) II has a strategy in G F [ X ] for playing into A . The following is a version of being strategic for the restricted games.
Definition 5.3.
A block filter F is + -strategic if whenever α is a strategyfor II in G F [ X ], where h X i ∈ F , there is an outcome of α which is in F .What is the relationship between being +-strategic and actually strategic?We will see below that, at least for ( p )-filters, it is captured by fullness.We will need the following notion and a lemma: a tree is a subset T ⊆ bb < ∞ ( E ) which is closed under initial segments. The set [ T ] of infinitebranches through T is a closed subset of bb ∞ ( E ). Lemma 5.4 (cf. Lemma 6.4 in [6]) . Let F be a filter, h X i ∈ F , and α astrategy for II in G F [ X ] . Then, there is a tree T ⊆ bb < ∞ ( E ) such that: (i) [ T ] ⊆ [ α ] , and (ii) whenever ( y , . . . , y n ) ∈ T and h Y i ∈ F , there is a y ∈ h Y i so that ( y , . . . , y n , y ) ∈ T .Proof. We will define a pair of trees T ⊆ bb < ∞ ( E ) and S ⊆ F < ∞ as follows:Put ∅ ∈ T and S . The first level of T consists of all ( y ) ∈ bb < ∞ ( E ) suchthat y is a “first move” by II according to α . That is, there some Y (cid:22) X such that h Y i ∈ F and α ( Y ) = y . For each such y , pick a corresponding Y in its preimage under α ; these comprise the first level of S .We continue inductively. Having put ( y , . . . , y n ) ∈ T and ( Y , . . . , Y n ) ∈ S with α ( Y , . . . , Y i ) = y i for i ≤ n , we put ( y , . . . , y n , y ) if there is some Y (cid:22) X such that h Y i ∈ F and α ( Y , . . . , Y n , Y ) = y . Choose some Y withthis property and put ( Y , . . . , Y n , Y ) into S .Clearly, [ T ] ⊆ [ α ]. To see that [ T ] satisfies (ii), let ( y , . . . , y n ) ∈ T and h Y i ∈ F with Y (cid:22) X . Let ( Y , . . . , Y n ) ∈ S be such that α ( Y , . . . , Y i ) = y i for i ≤ n , and put y n +1 = α ( Y , . . . , Y n , Y ) ∈ h Y i . By construction, there issome Y n +1 with ( Y , . . . , Y n , Y n +1 ) ∈ S and y n +1 = α ( Y , . . . , Y n , Y n +1 ). (cid:3) Theorem 5.5.
Let F be ( p ) -filter. Then, F is + -strategic if and only if F is strategic and full.Proof. ( ⇒ ) First observe that +-strategic implies strategic: Given any h X i ∈F and strategy α for II in G [ X ], let α ′ be the restriction of α to G F [ X ] (inthe obvious sense). Since F is +-strategic, there is an outcome of α ′ , andthus of α , in F .To see that F is full, let D ⊆ E be F -dense and put D = { Y ∈ bb ∞ ( E ) : h Y i ⊆ D } , a closed subset of bb ∞ ( E ). By Theorem 5.2, there is an h X i ∈ F such thateither I has a strategy in F [ X ] for playing into D c , or II has a strategy in ILTERS ON A COUNTABLE VECTOR SPACE 15 G F [ X ] for playing in D . However, the former is impossible: pick Z (cid:22) X in D and let II in F [ X ] always play elements of h Z i . As F is +-strategic, thereis some outcome of II’s strategy in G F [ Y ] in F , verifying fullness.( ⇐ ) Assume that F is strategic and full, that is, F is a strategic ( p + )-filter. We must prove that F is +-strategic. Let h X i ∈ F and α a strategyfor II in G F [ X ]. Let T ⊆ bb < ∞ ( E ) be as in Lemma 5.4. By Theorem1.4, there is a Y (cid:22) X such that h Y i ∈ F and either I has a strategy in F [ Y ] for playing into [ T ] c , or II has a strategy in G [ Y ] for playing into [ T ].The former is impossible as II has a strategy in F [ Y ] for playing into [ T ]:Inductively apply the property in Lemma 5.4(ii) to the tail block sequencesplayed by I in F [ Y ]. Thus, II has a strategy in G [ Y ] for playing into [ T ].As F is strategic, there is some outcome of this strategy, and thus someelement of [ α ], in F . (cid:3) Summary and further questions
Figure 2 shows where the implications between the properties describedin the Introduction stand at the end of this article. The single arrows → indicate that the converse remains open for arbitrary fields (with ZFC as abase theory). In addition to sorting out the remaining implications in thisdiagram, there are a few other questions we wish to highlight for furtherinvestigation (each when | F | > Question.
Does the Continuum Hypothesis (or Martin’s Axiom) imply theexistence of ( p + )-filters which are not strategic? Question.
Is it consistent with
ZFC that there are stable ordered-union ul-trafilters on FIN, but not ( p + )-filters on E ? Question.
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