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Meager-nowhere dense games (III): Remainder

arXiv:math/9209208v1 [math.LO] 25 Sep 1992Meager-nowhere dense games (III): Remainderstrategies.Marion Scheepers∗Department of Mathematics,Boise State University,Boise, Idaho 83725AbstractPlayer ONE chooses a meager set and player TWO, a nowhere denseset per inning. They play ω many innings.

ONE’s consecutive choicesmust form a (weakly) increasing sequence. TWO wins if the union of thechosen nowhere dense sets covers the union of the chosen meager sets.

Astrategy of TWO which depends on knowing only the uncovered part ofthe most recently chosen meager set is said to be a remainder strategy.TWO has a winning remainder strategy for this game played on the realline with its usual topology.1IntroductionA variety of topological games from the class of meager-nowhere densegames were introduced in the papers [B-J-S], [S1] and [S2]. The existenceof winning strategies which use only the most recent move of either player(so-called coding strategies) and the existence of winning strategies whichuse only a bounded number of moves of the opponent as information (so-called k-tactics) are studied there and in [K] and [S3].

These studies arecontinued here for yet another fairly natural type of strategy, the so-calledremainder strategy.The texts [E-H-M-R] and [W] would be sufficient references for themiscellaneous results from combinatorial set theory which we use.Asfor notation: The symbol JR denotes the ideal of nowhere dense subsetsof the real line (with its usual topology), while the symbol “⊂” is usedexclusively to denote “ is a proper subset of ”. The symbol “⊆” is usedto denote “ is a subset of, possibly equal to”.

Let (S, τ) be a T1-spacewithout isolated points, and let J be its ideal of nowhere-dense subsets.The symbol ⟨J⟩denotes the collection of meager subsets of the space. ForY a subset of S, the symbol J⌈Y denotes the set {T ∈J : T ⊆Y }.∗Supported in part by Idaho State Board of Education grant 91-093.1

The game W MEG(J) (defined in [S2]) proceeds as follows: In the firstinning player ONE chooses a meager set M1, and player TWO respondswith a nowhere dense set N1. In the second inning player ONE chooses ameager set M2, subject to the rule that M1 ⊆M2; TWO responds witha nowhere dense set N2, and so on.

The players play an inning for eachpositive integer, thus constructing a sequence(M1, N1, . .

. , Mk, Nk, .

. .

)which has the properties that Mk ⊆Mk+1 ∈⟨J⟩, and Nk ∈J for each k.Such a sequence is said to be a play of W MEG(J). Player TWO winssuch a play if∞[k=1Mk =∞[k=1Nk.A strategy of player TWO of the form1.

N1 = F(M1) and2. Nk+1 = F(Mk+1\(Skj=1 Nj)) for all kis said to be a remainder strategy.When does TWO have a winningremainder strategy in the game W MEG(J)?It is clear that TWO has a winning remainder strategy in W MEG(J)if J = ⟨J⟩.

The situation when J ⊂⟨J⟩⊆P(S) is not so easy. In Section2 we investigate this question.

We prove among other things Theorem 1,which implies that TWO has a winning remainder strategy in the gameW MEG(JR).The game W MG(J) proceeds just like W MEG(J); only now the win-ning condition on TWO is relaxed so that TWO wins if∞[n=1Mn ⊆∞[n=1Nn.In Section 3 we study remainder strategies for this game; we briefly alsodiscuss the game SMG(J) here. In Section 4 we attend to the versionV SG(J).

The rules of this game turns out to be more advantageous toTWO from the point of view of existence of winning remainder strategies.Some of the theorems in these two sections show that the hypotheses ofTheorem 1 are to some extent necessary.Theorem 12 is due to Winfried Just, while Theorem 16 is due to FredGalvin. I thank Professors Galvin and Just for kindly permitting me topresent their result here and for fruitful conversations and correspondenceconcerning remainder strategies.2The weakly monotonic equal game, WMEG(J).When defining a remainder strategy F for TWO in W MEG(J), we shalltake care that for each A ∈⟨J⟩:1.

F(A) ⊆A, and2

2. F(A) ̸= ∅if (and only if) A ̸= ∅.Otherwise, the strategy F is sure not to be a winning remainder strategyfor TWO in W MEG(J),Theorem 1 If (∀X ∈⟨J⟩\J)(cof(⟨J⟩, ⊂) ≤|J⌈X|), then TWO has awinning remainder strategy in W MEG(J).Theorem 1 follows from the next two lemmas.

In the proof of Lemma2 we use an auxilliary game, denoted REG(J). It is played as follows: Asequence(M1, N1, .

. .

, Mk, Nk, . .

. )is a play of REG(J) if Mk ∈⟨J⟩and Nk ∈J for each k. Player TWOis declared the winner of a play of REG(J) if ∪∞k=1Mk = ∪∞k=1Nk.

TWOhas a winning perfect information strategy in REG(J). (We call REG(J)the “random equal game on J”.

)Lemma 2 If1. cof(⟨J⟩, ⊂) is infinite and2.

(∀X ∈⟨J⟩\J)(cof(⟨J⟩, ⊂) ≤|J⌈X|),then TWO has a winning remainder strategy in W MEG(J).Proof Let A ⊂⟨J⟩\J be a cofinal family of minimal cardinality. Observethat |A| ≤|P(X)| for each X ∈⟨J⟩\J.

Thus, if there is no Y ∈J⌈Xsuch that |A| ≤|P(Y )|, then |Y | < |X| for each Y ∈J⌈X, and wefix a decompositionX =∞[n=1Xnwhere {Xn : n ∈N} is a disjoint collection of sets from ⟨J⟩\J. Foreach such Xn we further fix a representationXn =∞[m=1Xn,mwhere Xn,1 ⊆Xn,2 ⊆.

. .

are from J, and a surjectionΘXn : J⌈Xn→<ωA.For each Y ∈J such that |A| ≤|P(Y )| the set Y is infinite and wealso writeY =∞[n=1Ynwhere {Yn : n ∈N} is a pairwise disjoint collection such that |Yn| =|Y | for each n. Further, choose for each n a surjectionΨYn : P(Yn)\{∅, Yn} →<ωA.3

Let U and V be sets in ⟨J⟩such that we have chosen a decompositionU = ∪∞n=1Un as above. We’ll use the notationU ⊆∗Vto denote that there is an m such that Un ⊆V for each n ≥m; wesay that m witnesses that U ⊆∗V .Fix a well-ordering ≺of ⟨J⟩.

For X ∈⟨J⟩we define:1. Θ(X): the ≺-minimal element A of A such that X ⊆A,2.

Φ(X): the ≺-minimal element Z of ⟨J⟩\J such that Z ⊆∗Xwhenever this is defined, and the empty set otherwise,3. k(X): the minimal natural number which witnesses that Φ(X) ⊆∗X whenever Φ(X) ̸= ∅, and 0 otherwise,4.

Γ(X): the ≺-minimal Y ∈J such that |J⌈X| ≤|P(Y )| andY ⊆∗X whenever this is defined, and the empty set otherwise,and5. m(X): the minimal natural number which witnesses that Γ(X) ⊆∗X whenever Γ(X) ̸= ∅, and 0 otherwise.Let G be a winning perfect information strategy for TWO in thegame REG(J).

We are now ready to define TWO’s remainder strat-egyF : ⟨J⟩→J.Let B ∈⟨J⟩be given.B ∈J: Then we define F(B) = B.B ̸∈J: We distinguish between two cases:Case 1: Γ(B) = ∅.Write X for Φ(B) and n + 1 for k(B). For 1 ≤j ≤n define σj sothatσj =ΘXj (Xj\B)if Xj\B ∈J∅otherwiseLet τ be σ1 ⌢.

. .

⌢σn ⌢⟨Θ(B)⟩, the concatenation of these finitesequences, and choose V ∈J⌈Xn+1 such that ΘXn+1(V ) = τ. ThendefineF(B) = B ∩[X1,n+1 ∪. .

. ∪Xn,n+1 ∪V ∪((∪{G(σ) : σ ⊆τ})\X)].Case 2: Γ(B) ̸= ∅.Write Y for Γ(B) and n + 1 for m(B).

For 1 ≤j ≤n define σj sothatσj =ΨYj (Yj\B)if Yj\B ̸∈{∅, Yj}∅otherwiseLet τ be σ1 ⌢. .

. ⌢σn ⌢⟨Θ(B)⟩, the concatenation of these finitesequences, and choose V ∈P(Yn+1)\{∅, Yn+1} so that ΨYn+1(V ) = τ.Then defineF(B) = B ∩[Y1 ∪.

. .

∪Yn ∪V ∪((∪{G(σ) : σ ⊆τ})\Y )].4

From its definition it is clear that F(B) ⊆B for each B ∈⟨J⟩. Tosee that F is a winning remainder strategy for TWO in W MEG(J),consider a play(M1, N1, .

. .

, Mk, Nk, . .

. )during which TWO adhered to the strategy F.To facilitate theexposition we write:1.

B1 for M1 and Bj+1 for Mj+1\ ∪ji=1 Ni,2. Y j for Γ(Bj),3.

Xj for Φ(Bj),4. Aj for Θ(Bj),5. kj for k(Bj) and6.

mj for m(Bj).We must show that ∪∞j=1Bj ⊆∪∞j=1Nj.We may assume that Bj ̸∈J for each j.Suppose that Y j+1 ̸= ∅for some j. Then Nj+1 is defined by Case2, and as such is of the formBj+1 ∩[Y j+11∪.

. .

∪Y j+1mj+1−1 ∪Vj+1 ∪((∪{G(σ) : σ ⊆τj+1})\Y j+1)]where Vj+1 and τj+1 have the obvious meanings. Now Bj+1\Bj+2 =Nj+1, and thus:• Y j+1 ⊆∗Bj+2 is a candidate for Y j+2, and• Y j+2 ̸= ∅, so that Nj+2 is also defined by Case 2.We conclude that if Y j ̸= ∅for some j, then Y i ̸= ∅for all i ≥j.Moreover, it then also follows that Y i+1 ⪯Y i for each i ≥j.

Since≺is a well-order, there is a fixed k such that Y i = Y k for all i ≥k.Let Y be this common value of Y i, i ≥k. An inductive computationnow shows that (Ak, .

. .

, Aj) ⊆τj for each j ≥k. But thenBj ∩[(G(Ak) ∪.

. .

∪G(Ak, . .

. , Aj))\Y ] ⊆Njfor each j ≥k.It follows that ∪∞j=kBj\Y ⊆∪∞j=kNj.But it isalso clear that Y ∩(∪∞j=1Bj) ⊆∪∞j=kNj.

It then follows from themonotonicity of the sequence of Mj-s that TWO has won this play.The other case to consider is that Y j+1 = ∅for all j. In this case,Xj+1 ̸= ∅for each j, and Nj+1 is defined by Case 1.

In this caseNj+1 is of the form:Bj+1∩[Xj+11,kj+1∪. .

.∪Xj+1kj+1−1,kj+1∪Vj+1∪((∪{G(σ) : σ ⊆τj+1})\Xj+1)],where Vj+1 and τj+1 have the obvious meaning. Now Bj+1\Bj+2 =Nj+1, and so Xj+1 ⊆∗Bj+2, and Xj+1 is a candidate for Xj+2.

Itfollows that Xj+2 ⪯Xj+1 for each j < ω. Since ≺is a well-orderwe once again fix k such that Xj = Xk for all j ≥k.

Let X denoteXk. As before, ⟨Ak, .

. .

, Aj⟩⊆τj for each such j, and it follows thatTWO also won these plays.5

Lemma 3 If ⟨J⟩= P(S), then TWO has a winning remainder strategyin W MEG(J).Proof Let ≺be a well-order of P(S), and write S = ∪∞n=1Sn such thatSn ∈J for each n, and the Sn-s are pairwise disjoint.For eachcountably infinite Y ∈J write Y = ∪∞n=1Yn so that |Yn| = n foreach n, and {Yn : n ∈N} is pairwise disjoint. For X and Y in ⟨J⟩write Y ⊆∗X if Y \X is finite.For each X ∈⟨J⟩\J,• either there is an infinite Y ∈J⌈X,• or else X is countably infinite.In the first of these cases, let Φ(X) be the ≺-first countable elementY of J such that Y ⊆∗X, and let m(X) be the smallest n such thatYm ⊆X for all m ≥n.In the second of these cases, let Φ(X) be the ≺-least element Y of⟨J⟩\J such that Y ⊆∗X, and let m(X) be the minimal n such thatΦ(X) ∩Sm ⊆X for all m ≥n.

Also write Φ(X)j for Φ(X) ∩Sj foreach j, in this case.Then define F(X) so that1. F(X) = X if X ∈J, and2.

F(X) = X∩[(S1∪. .

.∪Sm(X))\Φ(X))∪(Φ(X)1∪. .

.∪Φ(X)m(X)]Then F is a winning remainder strategy for TWO in W MEG(J),for reasons analogous to those in the proof of Lemma 2.Corollary 4 Player TWO has a winning remainder strategy in the gameW MEG(JR).Recall (from [S2]) that G is a coding strategy for TWO if:1. N1 = G(∅, M1) and2.

Nk+1 = G(Nk, Mk+1) for each k.If F is a winning remainder strategy for TWO in W MEG(J), thenthe function G which is defined so that G(W, B) = W ∪F(B\W ) isa winning coding strategy for TWO in W MEG(J). Thus, Corollary 4solves Problem 2 of [S2] positively.We shall later see that the sufficient condition for the existence of awinning coding strategy given in Theorem 1 is to some extent necessary(Theorems 12 and 16).

However, this condition is not absolutely necessary,as we shall now illustrate. First, note that for any decomposition S =∪kj=1Sk, the following statements are equivalent:1.

TWO has a winning remainder strategy in W MEG(J),2. For each j, TWO has a winning remainder strategy in W MEG(J⌈Sj).6

Now let S be the disjoint union of the real line and a countable set S∗.Let X ∈J if X ∩S∗is finite and X ∩R ∈JR. Then S∗∈⟨J⟩, and J⌈S∗isa countable set, while cof(⟨J⟩, ⊂) is uncountable.

According to Corollary4 and Lemma 3, TWO has a winning remainder strategy in W MEG(J).Let λ be an infinite cardinal of countable cofinality. For κ ≥λ, we de-clare a subset of κ to be open if it is either empty, or else has a complementof cardinality less than λ.

With this topology, J = [κ]<λ.Corollary 5 Let λ be a cardinal of countable cofinality, and let κ > λbe a cardinal number. If cof([κ]λ, ⊂) ≤λ<λ, then TWO has a winningremainder strategy in W MEG([κ]<λ).Let A be a subset of ⟨J⟩.The game W MEG(A, J) is played likeW MEG(J), except that ONE is confined to choosing meager sets whichare in A only.

Thus, W MEG(J) is the special case of W MEG(A, J) forwhich A = ⟨J⟩. If there is a cofinal family A ⊂⟨J⟩such that TWO doesnot have a winning remainder strategy in W MEG(A, J), then TWO doesnot have a winning remainder strategy in W MEG(J).Theorem 6 There is a cofinal family A ⊂[ω1]ℵ0 such that TWO doesnot have a winning remainder strategy in W MEG(A, [ω1]<ℵ0).Proof Let F be a remainder strategy for TWO such that F(X) ⊆X forevery countable subset X of ω1.

Put A = {α < ω1 : cof(α) = ω}.We show that there is a sequence⟨(S1, T1), (S2, T2), . .

.⟩such that:1. Sn is a stationary subset of ω1,2.

Tn is a finite subset of ω1,3. Sn+1 ⊆Sn for each n,4.

F(γ) = T1 for each γ ∈S1 and5. F(γ\(∪nj=1Tj)) = Tn+1 for each γ ∈Sn+1 and for each n.To establish the existence of S1 and T1 we argue as follows.For each γ < ω1 which is of countable cofinality we putφ1(γ) = max F(γ)(< γ).By Fodor’s lemma there is a stationary set S0 of countable limitordinals, and an ordinal δ0 < min(S0) such that φ1(γ) = δ0 for eachγ ∈S0.

But then F(γ) is a finite subset of δ0 + 1 for each such γ.Since every partition of a stationary set into countably many setshas at least one of these sets stationary, we find a stationary setS⊂S0 and a finite set T1 ⊂δ0 + 1 such that F(γ) = T1 for eachγ ∈S1.This specifies (S1, T1). Now let 1 ≤n < ω be given and supposethat (S1, T1), .

. .

, (Sn, Tn) with properties 1 through 5 are given.7

For γ ∈Sn we define:φn+1(γ) = max F(γ\(T1 ∪. .

. ∪Tn))(< γ).Once again there is, by Fodor’s Lemma, a stationary set S′ ⊂Snand an ordinal δ′ < ω −1 such that φn+1(γ) = δ′ for each γ ∈S′.As before we then find a stationary set Sn+1 ⊆S′ and a finiteset Tn+1 ⊆δ′ + 1 such that F(γ\(T1 ∪.

. .

∪Tn)) = Tn+1 for eachγ ∈Sn+1.Then (S1, T1), . .

. , (Sn+1, Tn+1) have properties 1 through 5.

It fol-lows that there is an infinite sequence of the required sort.Put δ = sup(∪∞n=1Tn) and choose γn ∈Sn such thatδ < γ1 < γ2 < γ3 < . .

.Then (γ1, T1, γ2, T2, . .

.) is an F-play of W MEG(A, [ω1]<ℵ0), and islost by TWO.Though there may be cofinal families A such that TWO does not havea winning remainder strategy in W MEG(A, J), there may for this verysame J also be cofinal families B ⊂⟨J⟩such that TWO does have awinning remainder strategy in W MEG(B, J).Theorem 7 Let λ be an infinite cardinal number of countable cofinality.If κ > λ is a cardinal for which cof([κ]λ, ⊂) = κ, then there is a cofinalfamily A ⊂[κ]λ such that TWO has a winning remainder strategy inW MEG(A, [κ]<λ).Proof Let (Bα : α < κ) bijectively enumerate a cofinal subfamily of [κ]λ.Writeκ = ∪α<κSαwhere {Sα : α < κ} ⊂[κ]λ is a pairwise disjoint family.Define: Aα = {α} ∪(∪x∈BαSx) for each α < κ, and put A = {Aα :α < κ}.

Then A is a cofinal subset of [κ]λ. Also let Ψ : A →κ besuch that Ψ(Aα) = α for each α ∈κ.Choose a sequence λ1 < λ2 < .

. .

< λn < . .

. of cardinal numbersconverging to λ.For each A ∈A we write A = ∪∞n=1An whereA1 ⊂A2 ⊂.

. .

are such that |An| = λn for each n.For convenience we write, for C and D elements of [κ]≤λ, that C =∗D if |C∆D| < λ. Observe that for A and B elements of A, A ̸= Bif, and only if, |A∆B| = λ.Now define TWO’s remainder strategy F as follows:1.

F(A) = {Ψ(A)} ∪A1 for A ∈A,2. F(A) = {Ψ(B)} ∪(∪({Cm+1 : Ψ(C) ∈Γ(A)}) ∩B) ∪Bm+1if A ̸∈A but A ⊂B and A =∗B for some B ∈A.

Observethat this B is unique. In this definition, Γ(A) = B\A, and m isminimal such that |Γ(A)| ≤λm.8

3. F(A) = ∅in all other cases.Observe that |F(A)| < λ for each A, so that F is a legitimate strategyfor TWO.

To see that F is indeed a winning remainder strategy forTWO, consider a play(M1, N1, . .

. , Mk, Nk, .

. .

)of W MEG(A, [κ]<λ) during which TWO used F as a remainderstrategy.Write Mi = Aαi for each i. By the rules of the game we have:Aα1 ⊆Aα2 ⊆.

. .

.Also, N1 = {α1} ∪A1α1 and n1 minimal is such that |N1| ≤λn1. Aninductive computation shows that Nk+1 = F(Mk+1\(∪kj=1Nj) is theset([{αk+1} ∪(∪{Ank+1γ: γ ∈Nk}) ∪Ank+1αk+1 ) ∩Aα+k+1from which it follows that:1.

N1 ⊆N2 ⊆. .

. ⊆Nk ⊆.

. .,2.

n1 ≤n2 ≤. .

. ≤nk ≤.

. .

goes to infinity,3. αj ∈Nk whenever j ≤k, and thus4.

Apαj ⊆Nk for j ≤k and p ≤nk−1.The result follows from these remarks.Theorem 7 also covers the case when λ = ℵ0.For cofinal familiesA ⊂⟨J⟩which have the special property thatA ̸= B ⇔A∆B ̸∈J(like the one exhibited in the above proof), there is indeed an equivalencebetween the existence of winning coding strategies and winning remainderstrategies in the game W MEG(A, J). Particularly:Proposition 8 Let A ⊂⟨J⟩be a cofinal family such that for A and Belements of A, A ̸= B ⇔A∆B ̸∈J.

Then the following statements areequivalent:1. TWO has a winning coding strategy in W MEG(A, J).2.

TWO has a winning remainder strategy in W MEG(A, J).Proof We must verify that 1 implies 2. Thus, let F be a winning codingstrategy for TWO in the game W MEG(A, J).

We define a remain-der strategy G. Let X be given. If X ∈A we define G(X) = F(∅, X).If X ̸∈A but there is an A ∈A such that X ⊂A and X =∗A,then by the property of A there is a unique such A and we setT = A\X(∈J).

In this case define G(X) = F(T, A). In all othercases we put G(X) = ∅.

Then G is a winning remainder strategy forTWO in W MEG(A, J).9

It is not always the case that there is a cofinal A ⊂⟨J⟩which satisfiesthe hypothesis of Proposition 8. For example, let J ⊂P(ω2) be defined sothat X ∈J if, and only if, X ∩ω is finite and X ∩(ω2\ω) has cardinalityat most ℵ1.

Let {Sα : α < ω2} be a cofinal family. Choose α ̸= β ∈ω2such that:1. ω ⊂(Sα ∩Sβ) and2.

Sα ̸= Sβ.Then Sα∆Sβ ∈J.Coupled with Theorem 7 and an assumption about cardinal arithmetic,the following Lemma (left to the reader) enables us to conclude muchmore.Lemma 9 Let λ be a cardinal of countable cofinality, let µ ≤λ be aregular cardinal number, and let {Sα : α < µ} be a collection of pair-wise disjoint sets such that for each α < µ there is a cofinal familyAα ⊂[Sα]λ for which TWO has a winning remainder strategy in the gameW MEG(Aα, [Sα]<λ). Then there is a cofinal family A ⊂[∪α<µSα]λ suchthat TWO has a winning remainder strategy in W MEG(A, [∪α<µSα]<λ).Corollary 10 Assume the Generalized Continuum Hypothesis.

Let λ be acardinal of countable cofinality. For every infinite set S there is a cofinalfamily A ⊂[S]λ such that TWO has a winning remainder strategy inW MEG(A, [S]<λ).It is clear that if TWO has a winning remainder strategy in the gameW MEG(J), then TWO has a winning remainder strategy in W MG(J).The converse of this assertion is not so clear.Problem 1 Is it true that if TWO has a winning remainder strategyin the game W MG(J), then TWO has a winning remainder strategy inW MEG(J)?3The strongly monotonic game, SMG(J).A sequence (M1, N1, .

. .

, Mk, Nk, . .

.) is a play of the strongly monotonicgame if:1.

Mk ∪Nk ⊆Mk+1 ∈⟨J⟩, and2. Nk ∈J for each k.Player TWO wins such a play if ∪∞j=1Mj = ∪∞j=1Nj.

This game was stud-ied in [B-J-S] and [S1]; from the point of view of TWO this gives TWOa little more control over how ONE’s meager sets increase as the gameprogresses.It is clear that if TWO has a winning remainder strategyin W MG(J), then TWO has a winning remainder strategy in SMG(J).The converse is also true, showing that in the context of remainder strate-gies, the more stringent requirements placed on ONE by the rules of thestrongly monotonic game is not of any additional strategic value for TWO:10

Lemma 11 If TWO has a winning remainder strategy in SMG(J), thenTWO has a winning remainder strategy in W MG(J).Proof Let F be a winning remainder strategy for TWO in SMG(J).We show that it is also a winning remainder strategy for TWO inW MG(J).Let (M1, N1, . .

. , Mk, Nk, .

. .) be a play of W MG(J) during whichTWO used F as a remainder strategy.

Put M ∗1 = M1 and M ∗k+1 =Mk+1 ∪(N1 ∪. .

.∪Nk) for each k. Then (M ∗1 , N1, . .

. , M ∗k, Nk, .

. .) isa play of SMG(J) during which TWO used the winning remainderstrategy F. It follows that ∪∞k=1Mk ⊆∪∞k=1Nk, so that TWO wonthe F-play of W MG(J).We now restrict ourselves to the rules of W MG(J).

As with W MEG(J),a winning remainder strategy for TWO in the game W MG(J) gives riseto the existence of a winning coding strategy for TWO. In general, thestatement that player TWO has a winning remainder strategy in the gameW MG(J) is stronger than the statement that TWO has a winning cod-ing strategy.

To see this, recall that TWO has a winning coding strategyin W MG([ω1]<ℵ0) (see Theorem 2 of [S2]). But according to the nexttheorem, TWO does not have a winning remainder strategy in the gameW MG([ω1]<ℵ0).Theorem 12 (Just) If κ ≥ℵ1, then TWO does not have a winningremainder strategy in the game W MG([κ]<ℵ0).Proof Let F be a remainder strategy for TWO.

For each α < ω1 we putΦ(α) = sup(∪{F(α\T ) : T ∈[α]<ℵ0} ∪α).Then Φ(α) ≥α for each such α. Choose a closed, unbounded setC ⊂ω1 such that:1.

Φ(γ) < α whenever γ < α are elements of C, and2. each element of C is a limit ordinal.Then, by repeated use of Fodor’s pressing down lemma, we induc-tively define a sequence ((φ1, S1, T1), .

. .

, (φn, Sn, Tn), . .

.) such that:1.

C ⊃S1 ⊃. .

. ⊃Sn ⊃.

. .

are stationary subsets of ω1,2. F(α) ∩α = T1 for each α ∈S1, and3.

F(α\(T1 ∪. .

. ∪Tn)) = Tn+1 for each n and each α ∈Sn.Put ξ = sup(∪∞n=1Tn) + ω.

Choose αn ∈Sn so that ξ ≤α1 < α2 <. .

. < αn < .

. .. By the construction above we have:1.

F(α1) ∩ξ = T1 and2. F(αn+1\(T1 ∪.

. .

∪Tn)) ∩ξ = Tn+1 for each n.But then (∪∞n=1Tn) ∩ξ ⊂ξ = (∪∞n=1αn) ∩ξ, so that TWO lost thisplay of W MG([ω1]<ℵ0).11

For a cofinal family A ⊆⟨J⟩, the game W MG(A, J) proceeds just likeW MG(J), except that ONE must now choose meager sets from A only.The proof of Theorem 12 gives a cofinal family A such that TWO doesnot have a winning remainder strategy in the game W MG(A, [ω1]<ℵ0).This should be contrasted with Theorem 7, which implies that there aremany uncountable cardinals κ such that for some cofinal family A ⊂[κ]ℵ0,TWO has a winning remainder strategy in W MG(A, [κ]<ℵ0).Theorem 13 If TWO has a winning coding strategy in W MG(J), andif there is a cofinal family A ⊂⟨J⟩such that A∆B ̸∈J whenever A ̸=B are elements of A, then TWO has a winning remainder strategy inW MG(A, J).Proof Let F be a winning coding strategy for TWO in W MG(J), and letA ⊂⟨J⟩be a cofinal family as in the hypothesis of the theorem. IfB is not in A, but there is an A ∈A such that B ⊂A and A\B ∈J,then this A is unique on account of the properties of A.Define a remainder strategy G for TWO as follows: Let B ∈⟨J⟩begiven.1.

G(B) = F(∅, B) if B ∈A,2. G(B) = F(A\B, A) if B ̸∈A, but B ⊂A and A\B ∈J for anA ∈A, and3.

G(B) = ∅in all other circumstances.Then G is a winning remainder strategy for TWO in W MG(A, J).Corollary 14 Let λ be a cardinal number of countable cofinality.Foreach κ ≥λ, there is a cofinal family A ⊂[κ]λ such that TWO has awinning remainder strategy in W MG(A, J).Proof Write κ = ∪α<κSα where {Sα : α < κ} is a disjoint collection ofsets, each of cardinality λ. For each A ∈[κ]λ, put A∗= ∪α∈ASα.Then A = {A∗: A ∈[κ]λ} is a cofinal subset of [κ]<λ which hasthe properties required in Theorem 13.

The result now follows fromthat theorem and the fact that TWO has a winning coding strategyin W MG([κ]<λ) - see [S4].It is worth noting that Corollary 14 is a result in ordinary set theory,whereas we used the Generalized Continuum Hypothesis in Corollary 10.4The very strong game, VSG(J).Moves by player TWO in the game V SG(J) (introduced in [B-J-S]) consistof pairs of the form (S, T ) ∈⟨J⟩× J, while those of ONE are elements of⟨J⟩. A sequence(O1, (S1, T1), O2, (S2, T2), .

. .

)is a play of V SG(J) if:12

1. On+1 ⊇Sn ∪Tn, and2.

On, Sn ∈⟨J⟩and Tn ∈J for each n.Player TWO wins such a play if∪∞n=1On ⊆∪∞n=1Tn.A strategy F is a remainder strategy for TWO in V SG(J) if(Sn+1, Tn+1) = F(On+1\(∪nj=1Tn))for each n.For X ∈⟨J⟩we write F(X) = (F1(X), F2(X)) when F is a remainderstrategy for TWO in V SG(J). When F is a winning remainder strategyfor TWO, we may assume that it has the following properties:1.

F1(X)∩F2(X) = ∅; for G is a winning remainder strategy if G1(X) =F1(X)\F2(X) and G2(X) = F2(X) for each X.2. X\F2(X) ⊆F1(X); for G is a winning remainder strategy if G1(X) =(X ∪F1(X))\F2(X) and G2(X) = F2(X) for each X.The following Lemma describes a property which every winning remainderstrategy of player TWO for the game V SG(J) must have.Lemma 15 Assume that J ⊂⟨J⟩⊂P(S) and let F be a winning remain-der strategy for TWO in the game V SG(J).

Then the following assertionholds.For each x ∈S there exist a Cx ∈⟨J⟩and a Dx ∈J such that:1. Cx ∩Dx = ∅and2.

x ∈F2(B) for each B ∈⟨J⟩such that Cx ⊆B and Dx ∩B = ∅.Proof Let F be a remainder strategy of TWO, but assume the negation ofthe conclusion of the lemma. We also assume that for each X ∈⟨J⟩,X\F2(X) ⊆F1(X) and F1(X) ∩F2(X) = ∅.Choose an x ∈S witnessing this negation.

Then there is for eachC ∈⟨J⟩and for each D ∈J with x ∈C and C ∩D = ∅a B ∈⟨J⟩such that B ∩D = ∅, C ⊆B and x ̸∈F2(B). We now constructa sequence ⟨(Bk, Ck, Dk, Mk, Sk, Nk) : k ∈N⟩as follows: (we gothrough the first three steps of the construction for clarity, beforestating the general requirements for the sequence)Put C1 = {x} and D1 = ∅.

Choose B1 ∈⟨J⟩such that C1 ⊆B1and x ̸∈F2(B1). Put M1 = B1 and (S1, N1) = F(M1).

This defines(B1, C1, D1, M1, S1, N1).Put C2 = S1 and D2 = N1. Choose B2 ∈⟨J⟩such that C2 ⊆B2,D2 ∩B2 = ∅, and x ̸∈F2(B2).

Put M2 = B2 ∪D2 and (S2, N2) =F(M2\N1). This defines (B2, C2, D2, M2, S2, N2).Put D3 = (N1 ∪N2) and C3 = S2\D3.

Choose B3 ∈⟨J⟩such thatC3 ⊆B3, D3 ∩B3 = ∅, and x ̸∈F2(B3). Put M3 = B3 ∪D3 and(S3, N3) = F(M3\D3).

This defines (B3, C3, D3, M3, S3, N3).13

Let k ≥3 be given, and assume that(B1, C1, D1, M1, S1, N1), . .

. , (Bk, Ck, Dk, Mk, Sk, Nk)have been chosen so that:1.

Dj+1 = (N1 ∪. .

. ∪Nj) ∈J,2.

Cj+1 = Sj\Dj+1 ∈⟨J⟩and x ∈Cj+1 for j < k,3. Cj ⊆Bj, while x ̸∈F2(Bj) and Bj ∩Dj = ∅, and4.

Mj = Bj ∪Dj and5. (Sj, Nj) = F(Mj\Dj) for j ≤k, and6.

(B1, C1, D1, M1, S1, N1) and (B2, C2, D2, M2, S2, N2) are as above.Put Dk+1 = ∪kj=1Nj and Ck+1 = Sk\Dk+1. By hypothesis thereis a Bk+1 ∈⟨J⟩such that Ck+1 ⊆Bk+1, Bk+1 ∩Dk+1 = ∅, andx ̸∈F2(Bk+1).

Put Mk+1 = Bk+1 ∪Dk+1, and put (Sk+1, Nk+1) =F(Mk+1\(N1 ∪. .

. ∪Nk)).Then the sequence(B1, C1, D1, M1, S1, N1), .

. .

, (Bk+1, Ck+1, Dk+1, Mk+1, Sk+1, Nk+1)still satisfies properties 1 through 6 above. Continuing like this weobtain an infinite sequence which satisfies these.

But then(M1, (S1, N1), . .

. , Mk, (Sk, Nk), .

. .

)is a play of V SG(J) during which player TWO used the remainderstrategy F; this play is moreover lost by TWO because TWO nevercovered the point x. This proves the contrapositive of the lemma.Theorem 16 (Galvin) For κ > ℵ1, TWO does not have a winning re-mainder strategy in V SG([κ]<ℵ0).Proof Let F be a remainder strategy for TWO.

If it were winning, choosefor each x ∈κ a Dx ∈[κ]<ℵ0 and a Cx ∈[κ]≤ℵ0 such that:1. Cx ∩Dx = ∅,2.

x ∈Cx and3. x ∈F2(B) for each B ∈[κ]≤ℵ0 such that B ∩Dx = ∅andCx ⊆B.Now (Dx : x ∈κ) is a family of finite sets.

By the ∆-system lemmawe find an S ∈[κ]κ and a finite set R such that (Dx : x ∈S) is a∆-system with root R. For x ∈S define:f(x) = {y ∈S : Dy ∩Cx ̸= ∅}.Then f(x) is a countable set and x ̸∈f(x) for each x ∈S.ByHajnal’s set-mapping theorem we find T ∈[S]κ such that Cx ∩Dy =∅for all x, y ∈T .Let K ∈[T ]ℵ0 be given, and put B = ∪x∈KCx. Then K ⊆F2(B), acontradiction.14

Using similar ideas but with the appropriate cardinality assumptionto ensure that the corresponding versions of the ∆-system lemma and theset-mapping theorems are true, one obtains also:Theorem 17 Let λ be a cardinal of countable cofinality. If κ > 2λ, thenTWO does not have a winning remainder strategy in V SG([κ]<λ).Since for every cardinal λ of countable cofinality, and for each cardinalκ player TWO has a winning coding strategy in W MG([κ]<λ) (see forexample [S4]), Theorems 16 and 17 also show that the existence of awinning remainder strategy for TWO in V SG(J) is a stronger statementthan the existence of a winning coding strategy for TWO.Problem 2 Let λ be an uncountable cardinal of countable cofinality.

Letκ be an infinite cardinal number such that λ<λ < cof([κ]λ, ⊂) ≤2λ. DoesTWO fail to have a winning remainder strategy in any of W MEG([κ]<λ),W MG([κ]<λ) or V SG([κ]<λ)?The following theorem shows that the κ in Theorem 16 cannot bedecreased to ω1.

Thus, the rules of the very strong game are more advan-tageous to TWO than those of the other versions we considered earlier inthis paper.Theorem 18 If cof(⟨J⟩, ⊂) = ℵ1, then TWO has a winning remainderstrategy in V SG(J).Proof We may assume that there is for each X ∈⟨J⟩\J, a Y ∈⟨J⟩\Jsuch that X ∩Y = ∅(else, TWO has an easy winning remainderstrategy even in W MEG(J)). Let ≺be a well-ordering of S, theunderlying set of our topological space.

Choose two ω1-sequences(Cα : α < ω1) and (xα : α < ω1) such that:1. Cα ⊂Cβ ∈⟨J⟩,2.

xα ∈Cβ,3. xβ ̸∈Cβ,4.

xα ≺xβ and5. Cβ\Cα ̸∈J for all α < β < ω1, and6.

{Cα : α < ω1} is cofinal in ⟨J⟩.For each X ∈⟨J⟩we write β(X) for min{α < ω1 : X ⊆Cα}. PutX = {xα : α < ω1}.

Write Ωfor ω1\ω. Let F be a winning perfectinformation strategy for TWO in REG(J), and let G be a winningperfect information strategy for TWO in REG([{xδ : δ ∈Ω}]<ℵ0).We may assume that if σ is a sequence of length r of subsets of Ω,at least one of which is infinite, then |G(σ)| ≥r.

We also define:Kβ = {xγ : γ ≤β} for each β ∈Ω.We define a remainder strategy H for TWO in V SG(J). Thus, letB ∈⟨J⟩be given.1.

If B ∈J: Then put H(B) = (Cβ(B)+ω, {x0, xβ(B)})2. If B ̸∈J:15

(a) If {n < ω : xn ̸∈B} = {0, 1, . .

. , k}:Let T be {xβ(B)} together with the first ≤k + 1 elementsof {xα : α ∈Ω}\B.

PutS = T ∪(∪{G(σ) : σ ∈≤k+2({Kδ : xδ ∈T }),a set in [{xδ : δ ∈Ω}]<ℵ0. Let p be the cardinality of S.Then defineS = {x0, .

. .

, xp}∪S∪((∪{F(σ) : σ ∈≤p({Cα : xα ∈S})})\X).Put H(B) = (Cβ(B)+ω, S). (b) If {n < ω : xn ̸∈B} is not a finite initial segment of ω:Then we put H(B) = (Cβ(B)+ω, {x0, xβ(B).To see that H is a winning remainder strategy for TWO, consider aplay(O1, (S1, T1), .

. .

, On, (Sn, Tn), . .

. )where (S1, T1) = H(O1) and (Sn+1, Tn+1) = H(On+1\(∪nj=1Sj) foreach n.For convenience we put• W0 = T0 = ∅and Wn+1 = Wn ∪Tn+1,• Bn = On\Wn,• βn = β(Bn) and αn = βn + ωfor each nNote that if Bj is such that {n ∈ω : xn ̸∈Bj}(= {0, 1, .

. .

, kj}say) is a finite initial segment of ω, then the same is true for Bj+1.It follows that (Sj, Tj) is defined by Case 2(a) for each j > 1, andthat (kj : j ∈N) is an increasing sequence. It further follows that{xβ1, .

. .

, xβj} ⊂Tj for these j. This in turn implies that:1.

∪∞j=1F(Cβ1, . .

. , Cβj)\X ⊆∪∞n=1Tn,2.

∪∞j=1G(Kβ1, . .

. , Kβj) ⊆∪∞n=1Tn, and3.

{xn : n < ω} ⊆∪∞n=1Tn.But then ∪∞n=1On ⊆∪∞n=1Tn.Corollary 19 TWO has a winning remainder strategy in V SG([ω1]<ℵ0)Using the methods of this paper we can also show that if J ⊂P(S) isa free ideal such that there is an A ∈⟨J⟩such that cof(⟨J⟩, ⊂) ≤|J⌈A|,then TWO has a winning remainder strategy in V SG(J).Corollary 20 For every T1-topology on ω1, without isolated points, TWOhas a winning remainder strategy in V SG(J).16

References[B-J-S]T. Bartoszynski, W. Just and M. Scheepers, Strategies withlimited memory in Covering games and the Banach-Mazurgame: k-tactics, submitted to The Canadian Journal ofMathematics.[E-H-M-R]P. Erd¨os, A. Hajnal, A.

Mate and R. Rado, Combinatorial SetTheory: Partition Relations for Cardinals, North - Holland(1984).[K]P. Koszmider, On coherent families of finite-to-one functions,to appear in Journal of Symbolic Logic.[S1]M.

Scheepers, Meager-nowhere dense games (I): n-tactics,The Rocky Mountain J. of Math. 22 (1992),...[S2]M. Scheepers, Meager-nowhere dense games (II): codingstrategies, Proc.

Amer. Math.

Soc. 112 (1991), 1107-1115.[S3]M.

Scheepers, Concerning n-tactics in the countable-finitegame, The Journal of Symbolic Logic. 56 (1991), 786-794.[S4]M.

Scheepers, Meager-nowhere dense games (IV): n-tacticsand coding strategies, in preparation.[W]N. H. Williams, Combinatorial Set Theory, North - Hol-land (1977).17

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