Partition relations for Hurewicz-type selection hypotheses

P AR TITION RELA TIONS F OR H UR EWICZ-TYPE SELECTION H Y POTHESES NAD A V SAMET, MARION SCHEEPERS, AND BO AZ TSABAN Abstract. W e give a general metho d to reduce Hurewicz-type selection hypotheses in to standard ones. The method cov ers the known results of this kind and g ives some ne w ones. Building on that, we sho w how to deriv e Ramsey theoretic c har- acterizations for these selection h yp otheses. 1. Introduction In [7], Menger introduced a hypothesis whic h generalizes σ -compact- ness of top o logical spaces. Hurewicz [4] prov ed that Menger’s prop ert y is equiv alen t to a pro p ert y of the follow ing type. S fin ( A , B ) : F or each sequence {U n } n ∈ N of mem b ers of A , there exist finite subsets F n ⊆ U n , n ∈ N , suc h t ha t S n F n ∈ B . Indeed, Hurewicz observ ed t ha t X has Menger’s prop erty if, and only if, X s atisfies S fin ( O , O ), where O is the collection of a ll op en co v ers of X . Motiv ated by a conjecture of Menger, Hurewicz [4] introduced a h yp othesis of the follow ing type. U fin ( A , B ) : F or eac h sequence { U n } n ∈ N of elemen ts o f A w hic h do not contain a finite sub cov er, there exist finite (p ossibly empty ) subsets F n ⊆ U n , n ∈ N , suc h t ha t { S F n : n ∈ N } ∈ B . Hurewicz w as in terested in U fin ( O , Γ), where Γ is the collection of all op en γ -cov ers o f X . ( U is a γ -c over of X if it is infinite, and each x ∈ X is an elemen t in all but finitely man y mem b ers o f U .) While the Hurewicz-t ype selection hypotheses U fin ( A , B ) are stan- dard notio ns in the field of selection principles, they are less standard in the more general field of infinitary com binatorics. The reason f or that is that the finite subse ts are “ glued” b efore considering the resulting ob ject. Moreov er, the definition of U fin ( A , B ) is somewhat less elegan t than that of S fin ( A , B ), and consequen tly is less conv enien t to w ork with. 2000 Mathematics Su bje ct Classific ation. 05C55, 05D10, 5 4D20 . Key wor ds and phr ases. Ramsey theor y of op en covers, selection principles, Hurewicz c overing prop e rty , τ -c ov ers. 1 2 NADA V SA MET, MA RION SCHEEPERS, AND BOAZ TSA BAN U is an ω -c ove r of X if X / ∈ U , but fo r each finite F ⊆ X , there is U ∈ U such that F ⊆ U . One of the main results of [6] is the result that U fin ( O , Γ) is equiv alen t to S fin (Ω , B ) for an appropriate mo dification B of Γ . A similar resu lt w as established in [1] for U fin ( O , Ω). In these pap ers, these reductions were used to o btain Ramsey theoretic c haracterizations of U fin ( O , Γ) and of U fin ( O , Ω), resp ectiv ely . W e generalize these r esults. As applications, w e repro duce the main results of [6], strengthen t he main results of [1], and obt a in standard and Ra msey theoretic equiv alen ts f o r a prop ert y in tro duced in [11 ]. As each cov er of a t ype considered in our Ramsey theoretic results is infinite, eac h of our Ramsey theoretic results implies Ramsey’s clas- sical theorem and can therefore b e view ed as a structural extension of Ramsey’s Theorem. 2. Reduction of sele ction hyp otheses Convention. T o simplify t he presen tation, b y c over of X we a lw a ys mean a c ountable collection U of op en subse ts of X , such that S U = X and X / ∈ U . Also, A and B a lwa ys denote families of (suc h) co v ers of the underlying space X . Definition 1. A is R amseyan if for eac h U ∈ A and eac h partitio n of U in to finitely many (equiv alently , t w o) pieces, one of these pieces b elongs to A . Lemma 2 ( [8 ]) . Assume that A is R amseyan. Then A ⊆ Ω . Pr o of. Assume that A is Ramsey an, and A 6⊆ Ω . Fix U ∈ A \ Ω, and a finite subset F ⊆ X suc h that F is no t contained in an y U ∈ U . Since U is a cov er of X , | F | ≥ 2. F or eac h C ( F , let U C = { U ∈ U : U ∩ F = C } . Then U = S C ∈ P ( F ) \{ F } U C is a partition of U in to finitely man y pieces. As A is Ramsey an, there is C ( F suc h that U C ∈ A . But then the elemen ts of F \ C are not co v ered by any mem b er o f U C . A Con- tradiction.  Examples of Ramsey an collections of cov ers are Ω and Γ, defined in the in tro duction. W e will giv e one more example in Section 5. A cov er U of X is multifinite [12] if there exists a partition of U in to infinitely man y finite cov ers of X . Definition 3 (The Gimel op er ator on families of co v ers) . Let A b e a family of cov ers of X . ג ( A ) is the family of a ll cov ers U of X suc h that: Either U is m ultifinite, or there exists a partition P of U in t o finite sets suc h that { S F : F ∈ P } \ { X } ∈ A . P AR TITION RELA TIONS FOR HUREWICZ SELECTIONS 3 R emark 4 . F or eac h A , A ⊆ ג ( A ). An elemen t of ג ( A ) will b e called A -glue a ble . This explains our c hoice of the Hebrew letter Gimel ( ג ). Definition 5. A cov er V is a fini te-to-on e der efinement of a co v er U , if there exists a finite-to- one surjection f : U → V suc h that for eac h U ∈ U , U ⊆ f ( U ). A is finite-to-one der efinable if for each U ∈ A and each finite-to-one derefinemen t V ∈ O o f U , V ∈ A . A useful to ol in the study of selec tion principles and their relatio n to Ramsey theory is the game G fin ( A , B ). This game is pla y ed b y t w o pla y ers, ONE and TW O, with a n inning p er each natural n um b er n . A t the n th inning O NE c ho oses a cov er U n ∈ A and TW O chooses a finite subset F n of U n . TW O wins if S n ∈ N F n ∈ B . Otherwise, ONE wins. Our goal in this section is proving the f o llo wing. Theorem 6. L et B b e R amseyan and finite-to-one der efinable. The fol lowing ar e e quivalent: (1) U fin ( O , B ) . (2) S fin ( O , O ) and Λ = ג ( B ) . (3) ONE has no winning str ate gy in G fin (Ω , ג ( B )) . (4) S fin (Ω , ג ( B )) . Mor e over, in (3) and (4), Ω c an b e r eplac e d by any o f Λ or Γ . W e pro v e this theorem in a sequence of lemmas. As it ma y b e of indep enden t in terest, some of these lemmas use w eak er (or no) require- men ts on B than those p osed in Theorem 6. U is a lar ge c over o f X if eac h p oint x ∈ X belong s to infinitely man y U ∈ U . Let Λ b e the collection of all coun table large co v ers of X . The followin g pro of is similar to that of [6, Lemma 8]. Lemma 7. Assume that B is R am seyan an d fini te-to-o n e der efinable. Then U fin ( O , B ) implie s Λ = ג ( B ) . Pr o of. By Lemma 2, B ⊆ Ω. Th us, each U ∈ ג ( B ) is large. Let U b e a lar g e cov er of X . If U is m ultifinite, then U ∈ ג ( B ), and we are done. W e no w treat the remaining tw o cases. Case 1. U has no finite sub co v er. Let { U n : n ∈ N } bijectiv ely en u- merate a large co v er U of X . W e may assume that no finite subset of 4 NADA V SA MET, MA RION SCHEEPERS, AND BOAZ TSA BAN U cov ers X : If U con tains infinitely man y disjoin t finite sub cov ers then it is multifinite. F or m, n ∈ N , w e use the con v enien t no t a tion U [ m,n ) = [ m ≤ i m 1 , m 2 , . . . , m k n ; (2) There is i suc h that k n ≤ i < k n +1 and U [ i,m i ) 6 = ∅ ; and (3) U [ k n − 1 ,k n +1 ) / ∈ { U [ k i − 1 ,k i +1 ) : i < n } . (3) is p ossible since U [1 ,k n ) 6 = X . F or eac h n , let V n = { U [ i,m i ) : k n ≤ i < k n +1 } . As [ n V 2 n − 1 ∪ [ n V 2 n = { U [ i,m i ) : i ∈ N } ∈ B and B is Ramsey an, there is j ∈ { 0 , 1 } suc h tha t S n V 2 n − j ∈ B . W e consider the case j = 0 (the other case can b e treat ed similarly). F or eac h n , eac h elemen t of V 2 n has the form U [ i,m i ) with k 2 n ≤ i < k 2 n +1 . By (1), U [ i,m i ) ⊆ U [ k 2 n ,k 2 n +2 ) . Th us, { U [ k 2 n ,k 2 n +2 ) : n ∈ N } is a finite-to-one derefinemen t of S n V 2 n , and is therefore a mem b er of B . As B is finite-to-o ne derefinable, { U [1 ,k 4 ) } ∪ { U [ k 2 n ,k 2 n +2 ) : n > 1 } ∈ B , either, and this witnesses that the part it ion of U into the pieces { U i : 1 ≤ i < k 4 } and { U i : k 2 n ≤ i < k 2 n +2 } , n > 1, is as required in the statemen t U ∈ ג ( B ). Case 2 . U has only finitely man y disjoin t finite subcov ers. Let F b e the f amily of all elemen ts in these finite sub co v ers. U \ F is a large co v er of X not con taining a n y finite sub cov er. By what w e ha v e j ust pro v ed, U \ F ∈ ג ( B ) . As U \ F is not m ultifinite, there is a partition P of U \ F in to finite pieces, suc h that { S V : V ∈ P } ∈ B . Fix V 0 ∈ P . P ′ = {V 0 ∪ F } ∪ P \ {V 0 } is a partition of U in to finite pieces. Define f : { S V : V ∈ P } → { S V : V ∈ P ′ } by f ( S V ) = S ( V ∪ F ) if S V = S V 0 , and f ( S V ) = S V otherwise. As f is finite-to-one and B is finite-to-o ne derefinable, { S V : V ∈ P ′ } ∈ B , and thu s U ∈ ג ( B ).  P AR TITION RELA TIONS FOR HUREWICZ SELECTIONS 5 Note that fo r each family of co v ers B , U fin ( O , B ) implies U fin ( O , O ). Clearly , U fin ( O , O ) = S fin ( O , O ). Th us, Lemma 7 sho ws that the impli- cation (1) ⇒ (2) of Theorem 6 ho lds for eac h Ramsey an and bijectiv ely derefinable family of co v ers B . The followin g will b e used oft en. Lemma 8 ( [9 ]) . S fin ( O , O ) = S fin (Λ , Λ) = S fin (Ω , Λ) = S fin (Γ , Λ) . Lemma 9. The fol low i n g ar e e quivalent: (1) S fin ( O , O ) . (2) ONE has no winning str ate gy in G fin (Λ , Λ) . (3) ONE has no winning str ate gy in G fin (Ω , Λ) . (4) ONE has no winning str ate gy in G fin (Γ , Λ) . Pr o of. Recall that S fin ( O , O ) = S fin (Λ , Λ). S fin (Λ , Λ) is equiv alen t to (2) [10, Theorem 5]. As Γ ⊆ Ω ⊆ Λ, (2 ) ⇒ (3 ) ⇒ (4). But (4) implies S fin (Γ , Λ), whic h is the same as (1) [9].  Corollary 10. The c onjunction of S fin ( O , O ) and Λ = ג ( B ) implies that ONE h as no winning str ate gy in any of the games G fin (Λ , ג ( B )) , G fin (Ω , ג ( B )) , or G fin (Γ , ג ( B )) . Pr o of. Lemma 9 a nd the assumption Λ = ג ( B ).  This giv es (2) ⇒ (3 ) of Theorem 6. (3 ) ⇒ (4 ) in that theorem is clear. It remains to sho w that (4) ⇒ (1). As Γ ⊆ Ω ⊆ Λ, it suffices to pro v e t he f ollo wing. Lemma 11 . Assume that B is finite-to-one der efinable. Then S fin (Γ , ג ( B )) implies U fin ( O , B ) . Pr o of. Assume that X satisfies S fin (Γ , ג ( B )). As U fin ( O , B ) = U fin (Γ , B ) [9], it suffices t o pro v e that X satisfies U fin (Γ , B ). Let U n , n ∈ N , b e disjoin t o p en γ -cov ers of X whic h do not con tain a finite sub co v er. Enum erate each U n bijectiv ely as { U n k : k ∈ N } . F or eac h n , let V n = { U 1 m ∩ U 2 m ∩ · · · ∩ U n m : m ∈ N } . F or eac h n , V n is an op en γ -cov er of X . Apply S fin (Γ , ג ( B )) to obtain for each n a finite subset F n ⊆ V n suc h that S n F n ∈ ג ( B ). Eac h U ∈ S n F n is a subset of some elemen t o f U 1 , hence for eac h finite subset F ⊆ S n F n , S F 6 = X . Therefore S n F n is not multifinite. Let {X m : m ∈ N } b e a partition of S n F n in to finite pieces suc h that { S X m : m ∈ N } ∈ B . Let f ( m ) = min { k : X m ∩ F k 6 = ∅} , 6 NADA V SA MET, MA RION SCHEEPERS, AND BOAZ TSA BAN and put Y n = [ m ∈ f − 1 ( n ) X m The sets { f − 1 ( n ) : n ∈ N } form a partition of N . Since eac h F k is finite and the X m ’s are disjoin t, f − 1 ( n ) is finite fo r all n . It follo ws that each Y n is a finite set. Eac h member o f Y n b elong t o some F k ⊆ V k for some k ≥ n . F or eac h n , c ho ose ψ ( n ) ∈ N suc h that: (1) If U n k ∈ U n app ear as term in the sets of Y n then ψ ( n ) ≥ k . (2) The sets S k ≤ ψ ( n ) U n k are distinct for differen t v alues o f n . This is p ossible since { S k ≤ m U n k : m ∈ N } , n ∈ N , are γ - co v ers. Define Z n = { U n k : k ≤ ψ ( n ) } . The sets Z n are finite and disjoin t. F or eac h n ∈ N , S Y n ⊆ S Z n 6 = X . Hence { S Z n : n ∈ N } is a finite derefinem en t of { S X n : n ∈ N } . Therefore { S Z n : n ∈ N } ∈ B and the sequence { Z n } n ∈ N witnesses that X has the pro p ert y U fin (Γ , B ).  This completes the pro of of Theorem 6 . 3. P ar tition re la tions for g lueable covers The sym b o l [ A ] n denotes the set of n -elemen t subsets of A . F or a p ositiv e inte ger k , the Baumgartner-T aylor p artition r elation [2] A → ⌈ B ⌉ 2 k denotes t he following statemen t: F o r eac h A in A and eac h f : [ A ] 2 → { 1 , . . . , k } , there a re (1) B ⊆ A suc h that B ∈ B ; (2) A partition of B into finite pieces B = S n ∈ N B n ; and (3) j ∈ { 1 , . . . , k } , suc h that f ( { U, V } ) = j for all U, V ∈ B whic h do not b elong to the same B n . The Baumgar t ner-T a ylor partition relation is one of the most imp or- tan t partition relations in the studies of op en cov ers and their com bi- natorial prop erties – see [5] fo r a surv ey of t his field. Lemma 12 ([8]) . If e ach mem b er of B is in fi nite and A → ⌈ B ⌉ 2 2 holds, then A ⊆ Ω . T ogether with Theorem 6, the fo llowing give s a Ramsey theoretic c haracterization of prop erties o f the fo rm U fin ( O , B ). Theorem 13. Assume that B is R amse yan and finite-to-on e der efin- able. Th e fol lowing ar e e quivalent: P AR TITION RELA TIONS FOR HUREWICZ SELECTIONS 7 (1) S fin (Ω , ג ( B )) . (2) F or e ach k , Ω → ⌈ ג ( B ) ⌉ 2 k holds. (3) Ω → ⌈ ג ( B ) ⌉ 2 2 . Pr o of. (1 ⇒ 2) This follows fr om Theorem 6 and the follow ing. Lemma 14 ([6]) . Assume that A is R amseyan . If ONE has no win- ning str ate gy in the game G fin ( A , B ) , then for e ach k , A → ⌈ B ⌉ 2 k holds. (3 ⇒ 1) Assume that X satisfies Ω → ⌈ ג ( B ) ⌉ 2 2 . By Theorem 6, it suffices to sho w that X satisfies U fin (Γ , B ). Let U n , n ∈ N , b e o p en γ -cov ers of X whic h do not contain a finite sub co v er. En umerate eac h U n bijectiv ely as { U n k : k ∈ N } . F or eac h n , define V n = { U 1 k ∩ U 2 k ∩ · · · ∩ U n k : k ∈ N } , and let V = S n ∈ N V n . Then, V is an ω -co v er o f X . F or each elemen t o f V fix a represen tation o f the form U 1 k ∩ U 2 k ∩ · · · ∩ U n k . Define a function f : [ V ] 2 → { 1 , 2 } by f ( { V 1 , V 2 } ) = ( 1 if V 1 and V 2 are from the same V n , 2 otherwise. Cho ose W ⊆ V suc h that W ∈ ג ( B ), a partition W = S k W k in to finite pieces, and a color j ∈ { 1 , 2 } , suc h tha t for A and B fro m distinct W k ’s, f ( { A, B } ) = j . Consider the p ossible v alues of j . j = 1: Then there is an n suc h that for all A ∈ W w e hav e A ⊆ U 1 n 6 = X . Hence W is not a co v er. Contradiction. j = 2: Let F n = W ∩ V n . Then each F n is finite. F rom this p o in t, the pro of con tin ues as in t he pr o of o f Lemma 11.  4. Se lecting one element fr om each cover W e no w consider the following selection principle. S 1 ( A , B ) : F or eac h sequence {U n } n ∈ N of elemen ts of A , t here exist U n ∈ U n , n ∈ N , suc h that { U n : n ∈ N } ∈ B . The corresp onding game G 1 ( A , B ), is defined as follows: A t the n th inning ONE c ho oses a co v er U n ∈ A and TW O c ho oses U n ∈ U n . TW O wins if { U n : n ∈ N } ∈ B . Otherwise, O NE wins. The corresp onding part ition relation, called the ordinary partition relation, is defined as follow s. F or p ositiv e in tegers n a nd k , A → ( B ) n k means: F or each A ∈ A and eac h f : [ A ] n → { 1 , . . . , k } , there is B ⊆ A suc h that B ∈ B , a nd f | [ B ] n is constan t. 8 NADA V SA MET, MA RION SCHEEPERS, AND BOAZ TSA BAN The following t heorem was prov ed in [6 ] for B = Γ, and in [1] for B = Ω. Theorem 15. L e t B b e R amseyan an d finite-to-one der efinable. The fol lowing ar e e quivalent. (1) S 1 ( O , O ) and U fin ( O , B ) . (2) S 1 (Λ , ג ( B )) . (3) S 1 (Ω , ג ( B )) . (4) ONE has no winning str ate gy in the game G 1 (Ω , ג ( B )) . (5) Ω → ( ג ( B )) 2 2 . (6) Ω → ( ג ( B )) 2 k for al l k . Pr o of. (1 ⇒ 2) S 1 ( O , O ) = S 1 (Λ , Λ). By Lemma 7, Λ = ג ( B ) for X . Th us, X satisfies S 1 (Λ , ג ( B )). (2 ⇒ 3) Ω ⊆ Λ. (3 ⇒ 1) As S 1 (Ω , ג ( B )) implies S fin (Ω , ג ( B )), w e ha v e by Lemma 11 that U fin ( O , B ) holds, and that ג ( B ) = Λ. Th us, X satisfies S 1 (Ω , Λ), whic h is the same as S 1 ( O , O ) [9 ]. (1 ⇒ 4) By [1 0 , Theorem 3], S 1 ( O , O ) implies that ONE do es not ha v e a strategy in G 1 (Λ , Λ), and in particular in G 1 (Ω , Λ). Again, use Theorem 6 to get that Λ = ג ( B ). (4 ⇒ 6) F ollow s from [6, Theorem 1]. (6 ⇒ 5) is immediate. (5 ⇒ 3) As B is Ramsey an, B ⊆ Ω, and therefore ג ( B ) ⊆ Λ. Th us, (5) implies Ω → (Λ) 2 k . Using the metho ds of [6], one can prov e that Ω → (Λ) 2 k implies S 1 (Ω , Λ) [8]. Clearly , (5 ) also implies Ω → ⌈ ג ( B ) ⌉ 2 2 , and by Theorem 13, w e g et Λ = ג ( B ).  5. Applica tions 5.1. γ -cov er s. As ev ery infinite subset of a γ - co v er is again a γ -co v er of the same space, Γ is R amsey a n. Lemma 16. Γ is finite-to-one der efinable. Pr o of. Assume that U ∈ Γ and f : U → V is finite-t o -one and surjec- tiv e. As f is finite-to- one and U is infinite, V is infinite. Assume that x ∈ X and W = { V ∈ V : x / ∈ V } is infinite. F o r eac h V ∈ W and eac h U ∈ f − 1 ( V ), U ⊆ V a nd thus x / ∈ U . As f is surjectiv e, S V ∈W f − 1 ( V ) is infinite. A con tradiction.  Th us, we can directly a pply Theorems 6, 1 3, and 15, and obtain the follo wing. Theorem 17 ( [6 ]) . The fol low ing ar e e quivalent: P AR TITION RELA TIONS FOR HUREWICZ SELECTIONS 9 (1) U fin ( O , Γ) . (2) S fin ( O , O ) and Λ = ג (Γ) . (3) ONE has no winning str ate gy in G fin (Ω , ג (Γ)) . (4) S fin (Ω , ג (Γ)) . (5) F or e ach k , Ω → ⌈ ג (Γ) ⌉ 2 k holds. (6) Ω → ⌈ ג (Γ) ⌉ 2 2 .  Theorem 18 ( [6 ]) . The fol low ing ar e e quivalent. (1) S 1 ( O , O ) and U fin ( O , Γ) . (2) S 1 (Λ , ג (Γ)) . (3) S 1 (Ω , ג (Γ)) . (4) ONE has no winning str ate gy in the game G 1 (Ω , ג (Γ)) . (5) Ω → ( ג (Γ)) 2 2 . (6) Ω → ( ג (Γ)) 2 k for al l k .  5.2. ω -co v ers. Definition 19. A cov er V is a der efine m ent of a co v er U if U refines V . A is der efinable if for eac h U ∈ A and eac h derefinemen t V ∈ O of U , V ∈ A . Ω is derefinable, and in particular finite-to-one derefinable. Lemma 20 ( f olklore) . Ω is R amseyan. Pr o of. Assume that U ∈ Ω and U = U 1 ∪ . . . U n and no U i ∈ Ω. F or eac h i , c ho ose a finite subse t F i of X witnessing U i / ∈ Ω. Then F = F 1 ∪ · · · ∪ F n is not co v ered b y an y elemen t of U . A contradiction.  In the forthcoming Theorem 2 1 , w e repro duce the statemen ts of The- orems 2 and 3 of [1]. One direction in the pro of of Theorem 3 in [1] uses Theorem 4 o f [6], whic h in turn requires that ג (Ω) is derefinable. Unfortunately , b y Theorem 2 of [1], spaces dealt with in this theorem only hav e Λ = ג (Ω). But Λ is not derefinable: F ix distinct a, b, x n ∈ X , n ∈ N . Then the large co v er { X \ { a, x n } , X \ { b, x n } : n ∈ N } refines { X \ { a } , X \ { b }} . Our r esults give a corrected pro of of this direction. Theorem 21. The fol low i n g ar e e quivalent: (1) U fin ( O , Ω) . (2) S fin ( O , O ) and Λ = ג (Ω) . (3) ONE has no winning str ate gy in G fin (Ω , ג (Ω)) . (4) S fin (Ω , ג (Ω)) . (5) F or e ach k , Ω → ⌈ ג (Ω) ⌉ 2 k holds. (6) Ω → ⌈ ג (Ω) ⌉ 2 2 . Pr o of. Ω is derefinable and Ramsey an (Lemma 20). Apply Theorems 6 a nd 13.  10 NADA V SA MET, MA RION SCHEEPERS, AND BOAZ TSA BAN As in the previous theorem, the follow ing Theorem 22 repro duces Theorem 5 of [1] a nd fixes a problem similar to the ab ov e- mentioned one in the o riginal pro o f of the implication (5) ⇒ (3) b elo w. Theorem 22. The fol low i n g ar e e quivalent. (1) S 1 ( O , O ) and U fin ( O , Ω) . (2) S 1 (Λ , ג (Ω)) . (3) S 1 (Ω , ג (Ω)) . (4) ONE has no winning str ate gy in the game G 1 (Ω , ג (Ω)) . (5) Ω → ( ג (Ω)) 2 2 . (6) Ω → ( ג (Ω)) 2 k for al l k . Pr o of. Apply Theorem 15.  5.3. τ ∗ -co v ers. Let [ N ] ℵ 0 = { A ⊆ N : | A | = ℵ 0 } . F or A, B ∈ [ N ] ℵ 0 , A ⊆ ∗ B means tha t A \ B is finite. A family Y ⊆ [ N ] ℵ 0 is lin e arly r efinab l e if for eac h y ∈ Y there exists a n infinite subset ˆ y ⊆ y suc h that the family ˆ Y = { ˆ y : y ∈ Y } is linearly ordered by ⊆ ∗ . A countable cov er U = { U n : n ∈ N } of X is a τ ∗ -c over of X if {{ n : x ∈ U n } : x ∈ X } is linearly refinable. T ∗ is the collection of all τ ∗ -co v ers. Γ ⊆ T ∗ ⊆ Ω. T ∗ is derefinable [11]. In particular, T ∗ is finite-to-one derefinable. (This lat ter assertion is easier to see.) Prop osition 23. T ∗ is R amseyan. Pr o of. Let { U n : n ∈ N } b e a bijectiv e en umeration of a τ ∗ -co v er U of X . F or eac h x ∈ X , let x U = { n : x ∈ U n } , and let ˆ x U b e an infinite subset of x U suc h that the sets ˆ x U are linearly ordered b y ⊆ ∗ . Consider a partition U = V ∪ ( U \ V ). Define A = { n : U n ∈ V } . W e ma y assume that b oth A and its complemen t are infinite. F or eac h x ∈ X , define ˆ x V = ˆ x U ∩ A and ˆ x U \V = ˆ x U ∩ A c . If { ˆ x V : x ∈ X } ⊆ [ N ] ℵ 0 or { ˆ x U \V : x ∈ X } ⊆ [ N ] ℵ 0 then we ar e done. If this is not the case, then there are some x, y ∈ X suc h that ˆ x V and ˆ y U \V are finite. Without loss of generality , assume that ˆ y U ⊆ ∗ ˆ x U . Th us, ˆ y V = ˆ y U ∩ A ⊆ ∗ ˆ x U ∩ A = ˆ x V but ˆ y V is infinite and ˆ x V is finite. A con tradiction.  By Theorems 6 and 13, w e hav e t he f ollo wing. Theorem 24. The fol low i n g ar e e quivalent: (1) U fin ( O , T ∗ ) . (2) S fin ( O , O ) and Λ = ג (T ∗ ) . (3) ONE has no winning str ate gy in G fin (Ω , ג (T ∗ )) . P AR TITION RELA TIONS FOR HUREWICZ SELECTIONS 11 (4) S fin (Ω , ג (T ∗ )) . (5) F or e ach k , Ω → ⌈ ג (T ∗ ) ⌉ 2 k holds. (6) Ω → ⌈ ג (T ∗ ) ⌉ 2 2 .  By Theorem 15, w e hav e the follo wing. Theorem 25. The fol low i n g ar e e quivalent. (1) S 1 ( O , O ) and U fin ( O , T ∗ ) . (2) S 1 (Λ , ג (T ∗ )) . (3) S 1 (Ω , ג (T ∗ )) . (4) ONE has no winning str ate gy in the game G 1 (Ω , ג (T ∗ )) . (5) Ω → ( ג (T ∗ )) 2 2 . (6) Ω → ( ג (T ∗ )) 2 k for al l k .  Ac kno wledgmen t. W e thank the r eferee for the useful comments on this pap er. Reference s [1] L. Babinkostov a, Lj. Koˇ cinac, a nd M. Scheepe rs, Combinatorics of op en c overs (VIII) , T op ology a nd its Applications 140 (2004 ), 15 –32. [2] J. Baumga rtner a nd A. 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Tsaban, Str ong γ -s ets and other singular sp ac es , T op ology and its Applica- tions 153 (2005), 620–6 39. 12 NADA V SA MET, MA RION SCHEEPERS, AND BOAZ TSA BAN (Nadav Samet) Dep ar tment of Ma thema tics, Weizmann Institute of Science, Rehov ot 7610 0, Israel Curr ent addr ess : Go ogle Ireland Ltd., Go rdon House, Barr ow Str e et, Dublin 4, Ireland E-mail addr ess : thesam et@gm ail.com (Marion Scheeper s) Dep ar tment of Ma thema tics, Boise St a te Univer- sity, Boise, ID 83725, USA E-mail addr ess : marion @diam ond.boisestate.edu (Boaz Tsa ban) Dep ar tment of Ma thema tics, Bar-Ilan U niversity, Rama t- Gan 529 00, I srael; and Dep ar tment of Ma thema tics, Weizmann Insti- tute of S cience, Rehov ot 76100, Israel E-mail addr ess : tsaban @math .biu.ac.il