Superfilters, Ramsey theory, and van der Waerdens Theorem

SUPERFIL TERS, RAMSEY THEOR Y, AND V AN DER W AERDE N’S THEOREM NAD A V SAMET AND BOAZ TSABAN Abstract. Superfilter s are generaliz ations of ultrafilters, and cap- ture the underlying concept in Ramsey theoretic theorems such as v a n der W a erden’s Theorem. W e establis h several proper ties of super filters, which generalize b oth Ramsey ’s Theorem a nd its v arian ts fo r ultrafilters on the natural n um ber s. W e use them to confirm a conjecture of Koˇ cinac a nd Di Maio, whic h is a gene ral- ization of a Ramsey theor etic result of Sch eep ers, concerning se- lections from op en covers. F ollowing Ber gelson and Hindman’s 1989 Theorem, we pr esent a new simultaneous gener alization of the theorems of Ramsey , v an der W aerden, Sch ur, F olkman-Ra do- Sanders, Rado, and o thers, wher e the colo red sets can be muc h smaller than the full s et of natura l num b ers . 1. A unified Ramsey Theore m It is a simple observ ation that when each elemen t of an infinite set is color ed b y one of finitely many colors, the set m ust con tain an in- finite mono c hromatic subset. When replacing infinite by c ontaining arithmetic pr o gr essions of arbitr ary leng th , w e obtain v an der W aer- den’s Theorem [29]. Some of the b est references for many b eautiful theorems of this kind, together with applications, are the classical [11], the mon umen tal [12], the elegan t Protasov [19], and the more recen t [17]. These resu lts lead naturally to the concept of sup erfilter. Definition 1.1. F or a set S , [ S ] n = { F ⊆ S : | F | = n } , a nd [ S ] ∞ is the family of infinite subsets of S . A nonempt y fa mily S ⊆ [ N ] ∞ is a sup e rfilter if for all A, B ⊆ N : (1) If A ∈ S and B ⊇ A , then B ∈ S . (2) If A ∪ B ∈ S , then A ∈ S or B ∈ S . Sup erfilters we re iden t ified at least as early as in Berge’s 19 59 mono- graph [3] (under the name gril le ). 1 They w ere also considered under 1 (Added after publicatio n) F re deric Mynar d po in ts o ut that the notion of sup er- filter (under the name of grille) go es back at least to: Gustave Cho quet, Sur les notions de filtr e et de gri l le , C. R. Acad. Sci. Paris 224 (1 947), 171– 173. 1 2 NADA V SAMET AND BOAZ TSABAN the name c oide al (e.g., [8]). Sup erfilters are la rge t yp es of Banakh a nd Zdomskyy’s semifilters and unsplit semifilte rs [1 ]. Recall that a nonprinc ip al ultr afi lter is a family a s in Definition 1.1 whic h is also closed und er finite in tersections. 2 F or brev it y , b y ult r afilter w e alw ays me an a nonprincipal one. Example 1.2 . (1) Ev ery ultrafilter is a s up erfilter. (2) Ev ery union of a f amily of ultra filters is a sup erfilter. (3) [ N ] ∞ is a sup erfilter whic h is not an ultrafilter. In fact, o ne can sho w that ev ery sup erfilter is a union of a family of ultrafilters, but w e will not use this here. Definition 1.3. A P is t he family of all subsets of N con taining arbi- trarily long arithmetic pro gressions. Clearly , A P is not an ultrafilter. The finitary v ersion of v an der W aerden’s Theorem implies the follo wing. Theorem 1.4 (v an der W aerden) . A P is a sup erfi lter. Definition 1.5. S → ( S ) n k is the s tatemen t: F or eac h A ∈ S a nd eac h coloring c : [ A ] n → { 1 , 2 , . . . , k } , t here is M ⊆ A suc h that M ∈ S and c is constant on [ M ] n . The set M is called mono chr omatic for the coloring c . Th us for up w ards-closed S ⊆ [ N ] ∞ , the follo wing are equiv alen t: (1) S is a superfilter. (2) S → ( S ) 1 2 . (3) S → ( S ) 1 k for all k . The asse rtion S → ( S ) n k b ecomes stronger when n or k is increased . Definition 1.6. A s up erfilter S is: (1) R am s e y if S → ( S ) n k holds for all n and k . (2) Str ongl y R amsey if for a ll pairwise disjoint A 1 , A 2 , . . . with S n ≥ m A n ∈ S for all m , there is A ⊆ S n A n suc h that A ∈ S and | A ∩ A n | ≤ 1 for all n . (3) We akly R amsey if for all pairwise disjoint A 1 , A 2 , . . . / ∈ S with S n A n ∈ S , there is A ⊆ S n A n suc h tha t A ∈ S and | A ∩ A n | ≤ 1 for a ll n . 2 Definition 1 .1 do es not change if w e assume that A, B are disjoint in (2). But if, in addition, we replace there or by exclu s ive or , we obtain a characterization o f ultrafilter. That is, the assumption ab out intersections need not be stated explicitly . SUPERFIL TERS AND RAMSEY THEOR Y 3 Clearly , strongly Ramsey sup erfilters are w eakly Ramsey . W e will so on show that Ramsey is sandwic hed b etw een stro ngly Ra msey and w eakly Ramsey . Before doing so, w e giv e examples sho wing that con- v erse implications cannot be pro v ed. Example 1.7 . Fix a partition N = S n I n with eac h I n infinite. Let S b e the up w a rds closure of S n [ I n ] ∞ . It is easy to see that S is a sup erfilter. S is Ra msey: Let A ∈ S , and c : [ A ] n → { 1 , . . . , k } b e a coloring of A . Pick m suc h that A ∩ I m is infinite, and use Ramsey’s Theorem 1.14 for the coloring c : [ A ∩ I m ] 2 → { 1 , . . . , k } to obtain an infinite M ⊆ A ∩ I m whic h is mono c hromatic for c . S is not strongly Ramsey: F or eac h m , S n ≥ m I n ∈ S , but if | A ∩ I n | ≤ 1 for a ll n , then A / ∈ S . Example 1.8 . F ollo wing is an example of a w eakly Ramsey sup erfilter whic h is not Ramsey . Essen tia lly the same e xample w as, indep enden tly , found b y Filip´ ow, Mro ˙ zek, Rec la w, and Szuc a [9]. Let N ∗ b e the set of all finite sequences of natural n um b ers. F o r σ , ρ ∈ N ∗ , write σ ⊇ ρ if the sequence ρ is a prefix of σ . As N ∗ is coun table, w e ma y use it instead of N to define our superfilter. Sa y that a set D ⊆ N ∗ is somewher e dense if there is ρ ∈ N ∗ suc h that f or eac h σ ∈ N ∗ with σ ⊇ ρ , there is η ⊇ σ suc h that η ∈ D . Let S b e the family of all somewhere dense subsets of N ∗ . It is not difficult to see that S is a sup erfilter, and that it is w eakly Ramsey . T o see that it is not Ramsey , define a colo ring c : [ N ∗ ] 2 → { 1 , 2 } b y c ( σ, η ) = 1 if one o f σ, η is a prefix of the o ther, and 2 otherwise. If M ⊆ N ∗ is mono c hromatic of color 1, then M is a branc h in N ∗ , and t h us M / ∈ S . On the other hand, if M is somewhere dense, then it m ust contain a t least tw o ele men ts, o ne of whic h a prefix of the other. Th us, M is not mono c hromatic of color 2 , e ither. Examples 1.7 and 1.8 sho w that some hy p othesis is required to make the Ramsey an not ions coincide . W e suggest a rather mild one. Definition 1.9. A s up erfilter S is sh rin kable if , for all pairwise disjoin t A 1 , A 2 , . . . with S n ≥ m A n ∈ S for all m , there are B n ⊆ A n suc h that B n / ∈ S and S n B n ∈ S . R emark 1.10 (Th uemmel) . A s up erfilter S is shrink able if, and only if, for eac h sequence S 1 ⊇ S 2 ⊇ . . . of elemen t of S , there is S ∈ S suc h that for eac h n , S \ S n / ∈ S . T o see this, iden tify S m with S n ≥ m A n for eac h m ∈ N , and S with S n B n . All ultrafilters are s hrink able, for a trivial reas on: If a disjoin t union S n A n is in the ultrafilter, a nd some A m is in the ultrafilter, then S n>m A n is not in the ultrafilter. 4 NADA V SAMET AND BOAZ TSA BAN The sup erfilters in Examples 1.7 and 1.8 are not shrink able. F or shrink able sup erfilters, we ha v e a complete c haracterization o f b eing Ramsey . Theorem 1.11. F or sup e rfilters S , the fol lo wing ar e e quivalent: (1) S is str ongly R amse y. (2) S is R amse y and sh rinkable. (3) S → ( S ) 2 2 , and S is shrinkable. (4) S is we ak l y R am s ey and s hrinkable. Pr o of . (1 ⇒ 2) As singletons do not b elong to sup erfilters, strongly Ramsey implies shrink able. It therefore suffices to pro v e the follo wing. Lemma 1.12. Every str on gly R am sey sup erfilter is R amsey. Pr o of . Let S b e a strongly Ramsey sup erfilter, A ∈ S , and c : [ A ] d → { 1 , . . . , k } . The pro of is b y induction on d , with d = 1 following from S b eing a sup erfilter. Induction step: W e r ep eatedly apply the f ollo wing f act. F or eac h A ∈ S and each n ∈ A , there is M ⊆ A \ { n } suc h that M ∈ S , and a color i ∈ { 1 , . . . , k } , suc h t hat for eac h F ∈ [ M ] d − 1 , c ( { n } ∪ F ) = i . Indeed, w e can define a coloring c n : [ A \ { n } ] d − 1 → { 1 , . . . , k } by c n ( F ) = c ( { n } ∪ F ) and use the induction h yp othesis. En umerate A = { a n : n ∈ N } . Cho o se A a 1 ⊆ A \ { a 1 } and a color i a 1 suc h t hat A a 1 ∈ S and for eac h F ∈ [ A a 1 ] d − 1 , c ( { a 1 } ∪ F ) = i a 1 . In a similar manner, choose inductiv ely for eac h n > 1 A a n ⊆ A a n − 1 \ { a n } and a color i a n suc h that A a n ∈ S and for eac h F ∈ [ A a n ] d − 1 , c ( { a n } ∪ F ) = i a n . As a n / ∈ A a n for a ll n , T n A a n = ∅ . Le t B 0 = A \ A a 1 and for eac h n > 0 , let B n = A a n \ A a n +1 . The sets B n are pairwise disjoin t, S n B n = A , and S n ≥ m B n = A a m ∈ S for all m . As S is strongly Ramsey , there is B ⊆ A suc h that B ∈ S and | B ∩ B n | ≤ 1 for all n . Fix a c olor i suc h that C = { n ∈ B : i n = i } ∈ S . Let c 1 = min C . Inductiv ely , for each n > 1 choose c n ∈ C suc h that c n > c n − 1 and C \ [1 , c n ) ⊆ A c n − 1 . 3 F or e ac h n , C ∩ [ c n , c n +1 ) is finite and th us not a mem b er o f S . As S n ( C ∩ [ c n , c n +1 )) = C ∈ S and S is w eakly Ramsey , there is D ∈ S suc h that D ⊆ C and | D ∩ [ c n , c n +1 ) | ≤ 1 for all n . As D = D ∩ [ n ∈ N [ c 2 n , c 2 n +1 ) ! ∪ D ∩ [ n ∈ N [ c 2 n − 1 , c 2 n ) ! , 3 E.g., let k = | C \ A c n − 1 | + 1 and let c n be the k - th element o f C . SUPERFIL TERS AND RAMSEY THEOR Y 5 there is l ∈ { 0 , 1 } suc h that M = D ∩ S n [ c 2 n − l , c 2 n +1 − l ) ∈ S . Let m 1 < m 2 < · · · < m d b e mem b ers of M . Let n b e minimal suc h that m 1 < c n . Then m 2 , . . . , m d ∈ C \ [1 , c n +1 ) ⊆ A c n ⊆ A m 1 , and th us c ( { m 1 , . . . , m d } ) = c ( { m 1 } ∪ { m 2 , . . . , m d } ) = i m 1 = i .  (2 ⇒ 3 ) T rivial. (3 ⇒ 4 ) In fact, the follow ing holds. Lemma 1.13. If S → ( S ) 2 2 , then S is we akly R amsey. Pr o of . Let A 1 , A 2 , . . . b e as in the definition of w eakly Ra msey . Let D = S n A n , and define a coloring c : [ D ] 2 → { 1 , 2 } by c ( m, k ) = ( 1 ( ∃ n ) m, k ∈ A n 2 otherwise As S is Ra msey , there is a mono chromatic A ⊆ D with A ∈ S . If all elemen ts of [ A ] 2 ha v e color 1, then A ⊆ A n for some n , and thus A n ∈ S , a contradiction. Th us, all elemen ts of [ A ] 2 ha v e color 2, whic h means that | A ∩ A n | ≤ 1 for all n .  (4 ⇒ 1) Let A 1 , A 2 , . . . b e a s in the definition o f strongly Ramsey . As S is shrink able, there a re B n ⊆ A n suc h that B n / ∈ S and B = S n B n ∈ S . As S is we akly Ramsey , there is a subset A o f B suc h that A ∈ S and | A ∩ B n | ≤ 1 for all n . As B n ⊆ A n for all n and the sets A n are pairwise disjoin t, | A ∩ A n | ≤ 1 for all n . This completes the pro of of Theorem 1.11.  Corollary 1.14 (Ramsey [21]) . [ N ] ∞ → ([ N ] ∞ ) n k for al l n and k . Pr o of . Clearly , [ N ] ∞ is strongly Ramsey .  Corollary 1.15 (Bo oth-Kunen [5]) . An ultr afi lter is we akly R amsey if, and on ly if, it is R amsey. Pr o of . Ultrafilters are s hrink able.  The follo wing definition and s ubsequen t result will b e useful later. Definition 1.16 (Sche ep ers [23]) . S 1 ( S , S ) is the statemen t: Whenev er S 1 , S 2 , · · · ∈ S , there are s n ∈ S n , n ∈ N , suc h that { s n : n ∈ N } ∈ S . Theorem 1.17. F or sup e rfilters S : (1) If S is str ongly R amsey, then S 1 ( S , S ) h olds. (2) S 1 ( S , S ) i m plies that S is sh rin kable. 6 NADA V SAMET AND BOAZ TSA BAN Pr o of . (1) W e first observ e that , in the definition of strongly Ramsey , there is no need for the se ts A n to be pairwise disjoin t. Lemma 1.18. If a sup erfi lter S is str ong ly R amsey, then for a l l non e mpty A 1 , A 2 , . . . w i th S n ≥ m A n ∈ S f o r al l m , ther e ar e a n ∈ A n , n ∈ N , such that A = { a n : n ∈ N } ∈ S . Pr o of . Assume that S n ≥ m A n ∈ S for all m . Let L = \ m ∈ N [ n ≥ m A n . If L ∈ S , enume rate L = { l n : n ∈ N } . Pic k m 1 suc h t hat a m 1 := l 1 ∈ A m 1 . F or each n > 1, there is m n > m n − 1 suc h that a m n := l n ∈ A m n . F or m / ∈ { m n } n ∈ N , pic k an y a m ∈ A m . Then w e obtain a sequence as required. Th us, assume tha t L / ∈ S . T akin g B n = A n \ L for all n , we hav e that [ n ≥ m B n = ( [ n ≥ m A n ) \ L ∈ S for all m . No w, T m S n ≥ m B n = ∅ , that is, eac h k ∈ S n B n b elongs to only finitely man y B n . F or eac h n , let C n = B n \ [ m>n B m . The s ets C n are pairwis e disjoin t, and for eac h m , S n ≥ m C n = S n ≥ m B n ∈ S . As S is strongly Ramsey , w e obtain A ⊆ S n C n suc h that A ∈ S and | A ∩ C n | ≤ 1 for all n . F or each n , let a n ∈ A ∩ C n if | A ∩ C n | = 1 , and an arbitrary elemen t of A n otherwise. Then the seq uence { a n } n ∈ N is as required.  Th us, assume that A 1 , A 2 , · · · ∈ S . Clearly , they are all nonempty , and S n ≥ m A n ∈ S for all m . By Lemma 1.18, there ar e a n ∈ A n , n ∈ N , suc h that { a n : n ∈ N } ∈ S . (2) Apply S 1 ( S , S ) to the sequence S n ≥ m A n , m ∈ N , and recall that finite sets do not b elong to superfilters.  As Ramsey does not imply strongly Ramsey (Example 1.7), but do es for shrink able superfilters (Theorem 1.11 (4)), w e ha ve that the con v erse of Theorem 1.17(2) is false. Unfor tunately , w e do not hav e a concrete example for the fo llo wing. Conjecture 1.19. T h er e is a sup erfilter S such that S 1 ( S , S ) holds, but S is not str ongly (e quivalently, by The or em 1 . 1 7(2), we akly) R amsey. SUPERFIL TERS AND RAMSEY THEOR Y 7 2. An ap plica tion to topological selection p rinciples Our initial motiv ation for studying sup erfilters came fr om an attempt to pro vide a ( mainly) com binatorial pro of of a ma jor Ramsey-theoretic result of Sc heep ers, conce rning selections from op en cov ers. The general theory has connections and applicatio ns fa r b ey ond Ramsey theory , and the in terested reader is referred to the s urv ey pap ers [24, 14, 28]. The Ramsey-theoretic asp ect of this t heory is surv ey ed in [1 5]. Here, w e presen t only the conce pts whic h are neces sary fo r the presen t pap er. Fix a t op ological space X . A family U of subsets of X is a c o ver of X if X / ∈ U but X = S U . A cov er U of X is an ω -c over if for eac h finite F ⊆ X , there is U ∈ U such that F ⊆ U . Let Ω = Ω( X ) denote the family of all op en ω -co v ers of X . According to Definition 1.5, the statemen t Ω → (Ω) 2 2 mak es sense, and it is natural to ask what is required f rom X for this statemen t to b e true. Sa y that X is Ω - Lindel¨ of if eac h elemen t of Ω con tains a coun table elemen t o f Ω. The follo wing result is essen tially prov ed in [23], using an a uxiliary result from [13]. In the general form stated here , it is pro ved in [16]. Theorem 2.1 (Sc heep ers [23, 13, 16]) . F or Ω -Lindel¨ of sp ac es, the fol- lowing ar e e quiva lent: (1) S 1 (Ω , Ω) . (2) Ω → (Ω) 2 2 . (3) Ω → (Ω) n k for al l n, k . W e pro ceed in a general manner that will pro v e, in a ddition to Sc heepers’s Theorem, a conjecture of D i Maio, Koˇ cinac, and Mecca- riello from [6 ], and a subsequen t one of D i Maio and Koˇ cinac from [7]. Let C ( X ) denote the space of contin uous real-v a lued functions of X . ω - co v ers arise when considering the closure op erator in C ( X ), with the top ology of p oin t wise conv ergence [10]. When considering the c ompact-op en top olo gy , k -c overs arise, whic h are co v ers suc h that eac h compact set is con tained in a mem ber of the co v er (e.g., [6] and references therein). In [6] it is conjectured t hat Sc heep ers’s Theorem also holds whe n ω - co v ers a re replaced by k -co v ers. A natural generalization of these top ologies on C ( X ) giv es rise to the follo wing notion. An abstr act b ounde dness is a fa mily B of no nempt y closed subsets of X whic h is closed under taking finite u nions a nd closed subsets, and con t ains all singletons [7]. A cov er U is a B -c over if eac h B ∈ B is con tained in some mem b er of U . In [7] it is conjectured that Sc heepers’s Theorem holds in general, when ω - co v ers are r eplaced b y B -co v ers for an y abstract b oundedness notion B . 8 NADA V SAMET AND BOAZ TSA BAN Closing an abstract b oundedness notion B do wn w a rds will not c hange the notion o f B -co v ers. Th us, fo r simplicit y w e use a more familiar no- tion. A nonempt y family I of subs ets of X is an ide a l on X if X / ∈ I , { x } ∈ I for all x ∈ X , and f or all A, B ∈ I , A ∪ B ∈ I . Definition 2.2. Fix an ideal I on X . U is an I -cov er of X if X / ∈ U , and for eac h B ∈ I there is U ∈ U such that B ⊆ U . O I is the family of all op en I -cov ers of X . Lemma 2.3. (1) If U 1 ∪ U 2 ∈ O I , then U 1 ∈ O I or U 2 ∈ O I . (2) Each U ∈ O I is infi nite. Pr o of . (1) Assume that B 1 , B 2 ∈ I witness that U 1 , U 2 / ∈ O I , respec- tiv ely . Then no ele men t of U 1 ∪ U 2 con tains B 1 ∪ B 2 . (2) O I ⊆ Ω.  Let U ∈ O I . If U is countable, w e ma y use it as an index set instead of N , and consider sup erfilters on U . Definition 2.4. U I = {V ⊆ U : V ∈ O I } = P ( U ) ∩ O I . Lemma 2.3 implies the fo llo wing. Corollary 2.5. F or e ach c ountable U ∈ O I , U I is a sup erfilter.  U I cannot be assumed to b e an ultrafilter when proving Sche ep ers’s Theorem 2.1: If S 1 (Ω , Ω) ho lds, then each U ∈ Ω can be split in to t w o disjoin t elemen ts o f Ω [23]. W e are no w ready t o pro v e the general statemen t. Say that X is O I -Lindel¨ of if eac h elemen t o f O I con tains a countable elemen t of O I . Theorem 2.6. L et I b e an ide al on X . F or O I -Lindel¨ of sp ac es, the fol lowing ar e e quivalent: (1) S 1 ( O I , O I ) . (2) F or al l disjoi n t U 1 , U 2 , . . . / ∈ O I with S n U n ∈ O I , ther e is V ⊆ S n U n such that V ∈ O I and |V ∩ U n | ≤ 1 for al l n . (3) O I → ( O I ) 2 2 . (4) O I → ( O I ) n k for al l n, k . Pr o of . Using O I -Lindel¨ ofness, w e may restrict attention to coun table I - co v ers in all of our argumen ts. More precisely , w e prov e the stro nger assertion, where O I is replaced with the family o f c ountable op en I - co v ers, and no assumption is p osed on the space X . (4 ⇒ 3 ) T rivial. (3 ⇒ 2) Let U 1 , U 2 , . . . b e as in (2 ). Set U = S n U n . Then U ∈ O I , and by Corollar y 2.5, U I is a sup erfilter. By (3), we ha v e in particular SUPERFIL TERS AND RAMSEY THEOR Y 9 U I → ( U I ) 2 2 . By Theorem 1.13, U I is w eakly Ramsey . As U 1 , U 2 , . . . / ∈ U I and S n U n = U ∈ U I , there is V ∈ U I ⊆ O I as required. (2 ⇒ 1) Assume that U 1 , U 2 , · · · ∈ O I . Fix { U n : n ∈ N } ∈ O I . F or eac h n , le t V n = { U n ∩ U : U ∈ U n } . Then U = [ n ∈ N V n ∈ O I . By Corolla ry 2.5, U I is a sup erfilter. By (2), U I is weak ly Ramsey . No w, S n V n = U ∈ U I , and for each n , V n / ∈ U I . By thinning out the sets V n if necessary , we ma y assume tha t they are disjoint. Th us, there is V ⊆ U suc h that V ∈ U I and |V ∩ V n | ≤ 1 for all n . F or eac h n , if |V ∩ V n | = 1, tak e the U ∈ U n suc h t hat U n ∩ U ∈ V , and ot herwise ta k e a n arbitra ry U ∈ U n . W e obta in a I -co v er of X with one ele men t from eac h U n . (1 ⇒ 4) L et U ∈ O I . Let V b e the closure o f U under finite in ter- sections. V is countable, and U ∈ V I ⊆ O I . Consider the superfilter V I . By S 1 ( O I , O I ), w e ha v e S 1 ( V I , V I ). By Theorem 1.17, V I is shrink a ble. By Theorem 1 .11, it remains to prov e that V I is w eakly R amsey . Let V 1 , V 2 , . . . / ∈ V I b e pairwise disjoint with S n ≥ m V n ∈ V I for a ll m . F or eac h n , let U n = ( \ m ∈ I V m : I ⊆ N , | I | = n, ( ∀ m ∈ I ) V m ∈ V m ) . Claim 2.7. U n ∈ V I . Pr o of . As V is closed under finite interse ctions, U n ⊆ V . Assume that there is B ∈ I not con tained in an y mem b er of U n . Let I = { m : ( ∃ U ∈ V m ) B ⊆ U } . Then | I | < n . F or each m ∈ I c ho ose B m ∈ I witnessing that V m / ∈ O I . Then B ∪ S m ∈ I B m is not co v ered by an y U ∈ S n V n , a con tradiction.  Apply S 1 ( V I , V I ) to the se quence U n , n ∈ N , t o obtain eleme n ts U n ∈ U n with { U n : n ∈ N } ∈ V I . Let m 1 b e suc h that V m 1 := U 1 ∈ V m 1 . Inductiv ely , for eac h n > 1, U n is an in tersection of elemen ts from n man y V m -s, and th us there are m n distinct from m 1 , . . . , m n − 1 , and an elemen t V m n ∈ V m n , suc h tha t U n ⊆ V m n . Then A = { V m n : n ∈ N } ∈ V I . A ⊆ S n V n , and | A ∩ V n | ≤ 1 for all n .  A t the price of a sligh tly less combinatorial pro of, we can w eak en the restriction of O I -Lindel¨ ofness substan tially . 10 NADA V SAMET AND BOAZ TSA BAN Theorem 2.8. Assume that X has a c ountable op en I -c over. Then the four items of The or em 2.6 ar e e quivalent. Pr o of . The pro of is the same a s that o f Theorem 1.13, but w e argue directly in some of its steps. W e do this briefly . (1 ⇒ 4 ) By (1), X is O I -Lindel¨ of, and the argumen t in the proof of Theorem 2.6 applies . (3 ⇒ 2) Let U 1 , U 2 , . . . / ∈ O I b e disjoin t with S n U n ∈ O I . Set U = S n U n . Define a c oloring c : [ U ] 2 → { 1 , 2 } b y c ( U, V ) = ( 1 ( ∃ n ) U, V ∈ U n 2 otherwise By (3 ), there is a mono c hromatic V ⊆ U with V ∈ O I . It is easy to see that V is as req uired in (2). (2 ⇒ 1) Use the premised { U n : n ∈ N } ∈ O I : Assume that U 1 , U 2 , · · · ∈ O I . F or eac h n , let V n = { U n ∩ U : U ∈ U n } . No w, S n V n = U ∈ U I , and for each n , V n / ∈ U I . By thinning out the sets V n if necessary , w e ma y assume that they a re disjoint. By (2), there is V ⊆ U suc h that V ∈ O I and |V ∩ V n | ≤ 1 for all n .  F or T 1 top ological spaces, the a ssumption that X has a countable op en I -co v er can b e simplified. Lemma 2.9. L et I b e an ide al on a T 1 sp ac e X . Ther e is a c ountable I -c over of X if, a n d only if, ther e is a c ountable D ⊆ X such that D / ∈ I . Pr o of . ( ⇒ ) Let U b e a countable I -cov er of X . F or each U ∈ U , pic k x U ∈ X \ U . T ak e D = { x U : U ∈ U } . ( ⇐ ) U = { X \ { x } : x ∈ D } is a countable I -co v er of X .  In particular, Sc heepers’s Theorem 2.1 is true for al l T 1 sp ac es : It is trivially true for finite spaces, and in the remaining case there is a coun tably infinite subs et. In the case of k -co v ers, it suffices to assume that X has a countable subset with noncompact closure. 3. Ba ck to v an der W aerden’s Theorem W e reconsider v an der W aerden’s sup erfilter A P of all s ets con taining arbitrarily long arithmetic progress ions. SUPERFIL TERS AND RAMSEY THEOR Y 11 Example 3.1 . F ur sten b erg and W e iss (unpublished) prov ed that A P 9 (A P) 2 2 . Using Theorem 1 .13, w e can repro duce their observ ation b y sho wing that A P is not ev en we akly R amsey : Let A 1 = { 1 } , and for eac h n > 1, let m n = 2 max A n − 1 , and A n = { m n + 1 , m n + 2 , ..., m n + n } . F or eac h n , A n / ∈ A P, and S n A n ∈ A P. But there is no arithmetic progression of length 3 with at most one elemen t in eac h A n . Example 3.1 motiv ates us to lo ok for a prop erty whic h is weak er than b eing Ramsey but still implies Ramsey’s Theorem, and whic h is satisfied b y A P. A natural candidate is a v ailable in the literature. Definition 3.2 (Baumgartner-T a ylor [2]) . S → ⌈S ⌉ n k is the statemen t: F or eac h A ∈ S and each coloring c : [ A ] n → { 1 , 2 , . . . , k } , there is M ⊆ A s uc h that M ∈ S , and a partition of M in to finite pieces, suc h that c is cons tant on elemen ts of [ M ] n con taining at most one elemen t from eac h piece. An y pro v a ble ass ertion of the form S → ⌈S ⌉ n k with ∅ 6 = S ⊆ [ N ] ∞ and n, k ≥ 2 is an impro v emen t of Ramsey’s Theorem: G iv en a coloring of N , t ak e M ∈ S and a par tition of M in to finite sets as promised b y S → ⌈S ⌉ n k . Then any c hoice of one elemen t from each piece giv es an infinite mono c hromatic set. S → ⌈S ⌉ n k also implies that S is a sup erfilter. Lemma 3.3. F or e ach upwar ds-close d ∅ 6 = S ⊆ [ N ] ∞ : (1) If S → ⌈S ⌉ n k , l ≤ n , and m ≤ k , then S → ⌈S ⌉ l m . (2) F or e ach k , S → ⌈S ⌉ 1 k is e quiva le nt to S → ( S ) 1 k . Pr o of . (1) Giv en c : [ A ] l → { 1 , . . . , m } , define f : [ A ] n → { 1 , . . . , k } b y letting f ( F ) b e the c -color of the l smallest elemen ts of F . Us e S → ⌈S ⌉ n k to obtain M ⊆ A suc h that M ∈ S , and a partition of M into finite sets, suc h that sets with elemen ts coming from distinct pieces of M all ha v e the same f -color i . F or eac h F ∈ [ A ] l with elemen ts coming from distinct pieces of M , tak e arbitr ary n − l elemen ts from o ther pieces of M , whic h are g reater than all elemen ts of F (t his can b e done since M is infinite, and the pieces are finite). Add these elemen ts to F , to obta in F ′ . Then c ( F ) = f ( F ′ ) = i . (2) Immediate from the definition.  Definition 3.4. A sup erfilter S is a P -p o i n t if for a ll mem b ers A 1 ⊇ A 2 ⊇ . . . o f S , there is A ∈ S suc h that A \ A n is finite f or all n . Definition 3.5 (Sc heep ers [23]) . S fin ( S , S ) is the statemen t: Whenev er S 1 , S 2 , · · · ∈ S , there are finite F n ⊆ S n , n ∈ N , suc h that S n F n ∈ S . 12 NADA V SAMET AND BOAZ TSA BAN Theorem 3.6. The fol lowing ar e e quivalent for s up erfilters S : (1) S is a P -p oint. (2) S fin ( S , S ) . (3) F or al l disjoint A 1 , A 2 , . . . with S n ≥ m A n ∈ S for al l m , ther e is A ⊆ S n A n such that A ∈ S and A ∩ A n is finite for a l l n . (4) F or e ach p artition N = S n A n with S n ≥ m A n ∈ S for al l m , ther e is A ∈ S s uch that A ∩ A n is fin i te for al l n . (5) S → ⌈S ⌉ 2 2 and S is shrinkable . (6) S → ⌈S ⌉ n k for al l n, k , and S is shrinkable . Pr o of . (1 ⇒ 2 ) Assume that S 1 , S 2 , · · · ∈ S . F or eac h n , let A n = S m ≥ n S m . By (1), there is A ∈ S suc h that A \ A n is finite for all n . F or eac h n , let F n = ( A ∩ S n ) \ A n +1 . Let B = A ∩ T n A n . F or eac h n , add at most finitely man y elemen ts of B to F n , in a w a y that F n remains finite, F n ⊆ S n , and S n F n ⊇ B . Then A \ S n F n is finite, and th us S n F n ∈ S . (2 ⇒ 3 ) apply S fin ( S , S ) to the sequence S n ≥ m A n , m ∈ N . (3 ⇒ 4 ) T rivial. (4 ⇒ 1) Assume that B 1 ⊇ B 2 ⊇ . . . are mem b ers of S . W e ma y assume that B 1 = N . Let A 0 = T n B n . If A 0 ∈ S we a re do ne, so assume that A 0 / ∈ S . F or eac h n , let A n = B n \ B n +1 . N = A 0 ∪ S n A n is a partition of N as required in (3): S n A n ∈ S a s A 0 / ∈ S . F or eac h n , S m ≥ n A m = B n \ A 0 ∈ S , since B n ∈ S . T ak e A ∈ S suc h that A ∩ A n is finite for all n . Then A \ B n is finite for all n . (5 ⇒ 3 ) Consider disjoin t A 1 , A 2 , . . . with S n ≥ m A n ∈ S f or a ll m . As S is shrink able, w e may assume that A n / ∈ S for all n . Let D = S n A n , and define a coloring c : [ D ] 2 → { 1 , 2 } b y c ( m, k ) = ( 1 ( ∃ n ) m, k ∈ A n 2 otherwise By S → ⌈S ⌉ 2 2 , there is a part ition M = S n F n ⊆ D into finite sets, suc h that M ∈ S and c is constant on pairs of elemen ts coming from differen t F n -s. Assume that these pairs ha v e color 1. Fix k ∈ F 1 , and n suc h that k ∈ A n . F or eac h m 6 = 1 and eac h i ∈ F m , c ( k , i ) = 1 and th us i ∈ A n , to o. But then each l ∈ F 1 has c ( i, l ) = 1, and th us l ∈ A n , to o. Thus , M ⊆ A n . As M ∈ S , w e hav e that A n ∈ S ; a con tradiction. Thus , a ll pairs coming from different F n -s, m ust come from differen t A n -s. T ak e A = S n F n . SUPERFIL TERS AND RAMSEY THEOR Y 13 (1 , 3 ⇒ 6 ) Clearly , (3) implies that S is shrink able. W e prov e that S → ⌈S ⌉ d k for all d , k , by induction on d . Let S b e a P -point sup erfilter, A ∈ S , and c : [ A ] d → { 1 , . . . , k } . The case d = 1 follo ws from S b eing a sup erfilter. Induction step: En umerate A = { a n : n ∈ N } . Cho ose A a 1 ⊆ A \ { a 1 } and a color i a 1 suc h that A a 1 ∈ S , and a partition of A a 1 in to finite sets, suc h that for eac h F ∈ [ A a 1 ] d − 1 with a t most one elemen t in each piece, c ( { a 1 } ∪ F ) = i a 1 . In a similar manner, c ho ose inductive ly for eac h n > 1 A a n ⊆ A a n − 1 \ { a n } and a color i a n suc h that A a n ∈ S , and a partition o f A a n in to finite sets, suc h that for each F ∈ [ A a n ] d − 1 with at most one elemen t in eac h piece , c ( { a n } ∪ F ) = i a n . As S is a P - p oin t, there is B ∈ S suc h tha t B \ A a n is finite f or all n . F ix a color i suc h that C = { n ∈ B : i n = i } ∈ S . Let c 1 = min C . Inductiv ely , for eac h n > 1 c ho ose c n ∈ C suc h that: (1) c n > c n − 1 ; (2) F or eac h piece from the partitions o f A a 1 , . . . , A a n whic h inter- sects [1 , c n − 1 ), c n is greater than all e lemen ts of that piece; and (3) C \ [1 , c n ) ⊆ A c n − 1 . As C = C ∩ [ n ∈ N [ c 2 n , c 2 n +1 ) ! ∪ C ∩ [ n ∈ N [ c 2 n − 1 , c 2 n ) ! , there is l ∈ { 0 , 1 } suc h that M = C ∩ S n [ c 2 n − l , c 2 n +1 − l ) ∈ S . Let m 1 < m 2 < · · · < m d b e mem b ers of M coming from distinct in terv als [ c 2 n − l , c 2 n +1 − l ). Let n b e minimal w ith m 1 < c n . Then m 2 , . . . , m d ∈ C \ [1 , c n +1 ) ⊆ A c n ⊆ A m 1 , and m 2 , . . . , m d come from distinct pie ces of the partition of A m 1 . Th us, c ( { m 1 , . . . , m d } ) = c ( { m 1 } ∪ { m 2 , . . . , m d } ) = i m 1 = i . (6 ⇒ 5 ) T rivial.  The equiv alence of (1) and (3) in the follow ing corollary can b e sho wn, using a well known ar gumen t, to b e the same as the equiv a- lence of (i) and (iii) in The orem 2.3 of Baumgartner and T a ylor [2]. Corollary 3.7. F or ultr afi lters U , the fol lowi n g ar e e quivalent: (1) U is a P -p o int. (2) S fin ( U , U ) . (3) U → ⌈U ⌉ 2 2 . (4) U → ⌈U ⌉ n k for al l n, k . Pr o of . Recall that ultrafilt ers are shrink a ble.  14 NADA V SAMET AND BOAZ TSA BAN Definition 3.8. A fa mily F of subsets of N gener ates an upw ards- closed family S if F ⊆ S and each elemen t of S con tains an elemen t of F . An up w ards-closed family S ⊆ [ N ] ∞ is c om p actly ge n er ate d if there are upw ards-closed families F 1 , F 2 , . . . ⊆ P ( N ), eac h generated b y finite subsets of N , suc h that S = T n F n . Example 3.9 . [ N ] ∞ is compactly generated: T ak e F n = [ N ] ≥ n , n ∈ N . A P is compactly generated: Let F n b e the f amily of all sets containing arithmetic progressions of length n . Similarly , the F olkma n-R ado-San d ers superfilter [2 2] o f sets contain- ing arbitrarily large finite subsets together with all of their sub set sums is compactly generated. Sc h ur’s Theorem [26] states that if the natural n um b ers are colored in finitely many colors, then there is a mono chromatic solution to the equation x + y = z . Rado’s Theorem [20] extends Sc h ur’s Theorem to arbitrary regular homogeneous systems of equations. A homoge- neous system of equations Ax = 0 with in teger co efficien ts is r e gular if the columns of A can b e partitioned in to sets P 1 , . . . , P k suc h that P v ∈ P 1 v = 0, and for eac h i > 1, eac h elemen t of P i is a linear com bi- nation of elemen ts of P 1 ∪ · · · ∪ P i − 1 . The family of all sets containing a solution t o a regular homoge- neous system of equations is not a superfilter. This problem can b e solv ed by using t he follow ing op eration o n up w ards-closed families (see Prop osition 3.12 below ). Definition 3.10. F or an up wards-closed family F of s ubsets of N and k ∈ N , P ar k ( F ) is the family of all A ⊆ N suc h that for e ac h partition of A in t o k pieces , one of the pieces belongs to F . P ar( F ) = T k P ar k ( F ). F or upw ards-closed families F , P ar( F ) ⊆ F , and F is a sup erfilter if, and only if, P ar( F ) = F . Lemma 3.11. Assume that F ⊆ P ( N ) is upwar ds-close d and gener ate d by finite subsets of N . Then the sam e is true for P ar k ( F ) , for al l k . Pr o of . This is a reformulation of the compactness theorem for parti- tions, see Theorem 2.5 in [19].  Note that N ∈ P ar( F ) if, and only if, P ar( F ) is nonempty . Prop osition 3.12. L et F b e a n upwar ds-close d family of subsets of N . Assume that F do es not c ontain any singleton, and N ∈ P ar( F ) . Then: (1) P ar( F ) is the maximal sup erfilter c ontaine d in F . (2) If F is c omp actly gener a te d, then so is P ar( F ) . SUPERFIL TERS AND RAMSEY THEOR Y 15 Pr o of . (1) It is easy to s ee that P ar( F ) is closed upw ards. Assume that A ∪ B ∈ P ar( F ), and A ∩ B = ∅ . If A, B / ∈ P ar( F ), then there are a partition of A in to n pieces and a partition of B in to m pieces, suc h tha t none of the pieces b elong to F . But this yields a partition o f A ∪ B in to n + m pieces, none o f whic h from F , that is, A ∪ B / ∈ P ar n + m ( F ). A contradiction. As P a r( F ) ⊆ F , t here are no singletons in P a r( F ), and consequen tly no finite sets. If S is an y superfilter contained in F , then S = Par( S ) ⊆ P ar( F ). (2) Assume that F = T n F n , with eac h F n up w ards-closed a nd g en- erated by finite subsets. Replacing eac h F n b y T m ≤ n F m , w e ma y assume that F 1 ⊇ F 2 ⊇ . . . . It follo ws t hat for eac h k , P ar k ( T n F n ) = T n P ar k ( F n ), and th us P ar( F ) = \ k ∈ N P ar k ( F ) = \ k ∈ N P ar k ( \ n ∈ N F n ) = \ k ,n ∈ N P ar k ( F n ) . By Lemma 3.11, each P ar k ( F n ) is upw ards-closed a nd generated b y finite sets.  Example 3.13 . Let F b e the family of all subsets of N containing a solution to the equation x + y = z . Let P ar( x + y = z ) = Par( F ). Sc h ur’s Theorem tells that N ∈ Par( x + y = z ). By Prop osition 3.12, P ar( x + y = z ) is a compactly-generated sup erfilter. W e can define similarly P ar( Ax = 0) for an arbitrary regular system of homo- geneous equations, and by Rado’s Theorem ha v e that P ar( Ax = 0) is a compactly-generated superfilter. W e no w s tate the main application of Theorem 3.6. Theorem π . Assume that S is a c omp actly-gener ate d sup erfil ter. Then S → ⌈S ⌉ n k for al l n, k . Pr o of . By Theorem 3.6, it suffices to sho w tha t S fin ( S , S ) holds. Let F 1 , F 2 , . . . ⊆ P ( N ) be up w a rds-closed and generated by finite sets, suc h that S = T n F n . Assume that A 1 , A 2 , · · · ∈ S . F or eac h n , pick a finite F n ∈ F n suc h that F n ⊆ A n . Then S n F n ∈ S .  Theorem π is a sim ultaneous impro v emen t of the theorems of Ram- sey , v an der W aerden, Sc h ur, Rado, F o lkman-Rado-Sanders, and man y more. In particular, w e hav e the follo wing. Corollary 3.15. A P → ⌈ A P ⌉ n k , for al l n, k .  Theorem π can be restated as follo ws. 16 NADA V SAMET AND BOAZ TSA BAN Corollary 3.16. Assume that S is a sup erfilter c omp actly gener ate d by F 1 , F 2 , . . . . T h en for al l r , k , A ∈ S , c : [ A ] r → { 1 , . . . , k } , a n d m 1 < m 2 < . . . , ther e ar e dis j o int F n ∈ F m n , n ∈ N , such that S n F n ∈ S , and c is c o nstant on s e ts with at mos t one el e ment fr om e ach F n . Pr o of . W e ma y assume that F 1 ⊇ F 2 ⊇ . . . . Assume that A ∈ S a nd c : [ A ] n → { 1 , 2 , . . . , k } . Using Theorem π , tak e M ⊆ A suc h t hat M ∈ S , and a partitio n of M in to finite pieces, suc h that c is constan t on sets containing at most o ne elemen t fro m each piece. M contains some finite elemen t of F m 1 . Let F 1 b e the union of as man y pieces o f M as required so that F 1 con tains this elemen t of F m 1 . M \ F 1 ∈ S , and is partitioned b y the remaining pieces, thus we can t ak e a union of finitely man y of the remaining pieces, F 2 , containing some elemen t of F m 2 , etc. S n F n con tains an elemen t o f eac h F n , and t h us b elongs to S .  Example 3.17 . Consider Corollary 3.16 w ith S = A P . Fix an a rbitrarily quic kly increasing sequence m n , and a ssume tha t w e colo r an arbitrar ily sparse A ∈ A P. Then eac h F n con tains, and t h us ma y b e assumed to b e, an arithmetic progress ion of length m n . The special case A = N is the main corolla ry in Be rgelson and Hindman’s 1989 paper [4]. Bergelson and Hindman’s pro of in [4] sho ws that it suffices to as- sume that the colored set A is an elemen t of a combin atorially large ultrafilter (see [4]). Elemen ts of A P need not lie in a com binatorially large ultrafilter, and w e do not kno w a simple w a y to deduce Corollary 3.15 (or 3.16) from Bergelson and Hindman’s Corollary , and not ev en from their m uc h stronger Theorem 2.5 of [4]. 4. An additional applica tion to topological selection principles Using Theorem 3 .6 and arguments similar to those in the pro of of Theorem 2.6 , w e also o btain the follow ing Theorem 4.1. In the case that I is the ideal of finite sets ( O I = Ω), the equiv a lence of (2) and (4) w as prov ed by Just, Miller, Sche ep ers, and Szept yc ki in [13]. In the case that I is the ideal of subsets of compact sets, the equiv alence of (2) and (4) was prov ed by Di Maio, Ko ˇ cinac, and Meccariello in [6]. Theorem 4.1. L et I b e an ide al on X . If X is O I -Lindel¨ of, then the fol lowing ar e e quivalent: (1) F or al l U 1 ⊇ U 2 ⊇ . . . fr om O I , ther e is U ∈ O I such that U \ U n is finite for a l l n . (2) S fin ( O I , O I ) . SUPERFIL TERS AND RAMSEY THEOR Y 17 (3) F or al l disjoint U 1 , U 2 , . . . with S n ≥ m U n ∈ O I for al l m , ther e is U ⊆ S n U n such that U ∈ O I and U ∩ U n is finite for a l l n . (4) O I → ⌈O I ⌉ 2 2 . (5) O I → ⌈O I ⌉ n k for al l n, k .  Here to o, by using direc t argumen ts as in the pro o f of Theorem 2.8, “ X is O I -Lindel¨ of ” can b e we ak ened t o “ X has a coun table op en I - co v er”, or equiv alently for T 1 spaces, to “there is a coun table D ⊆ X suc h that D / ∈ I ”. 5. Final comments Mathias defines in [18] h a ppy famil i e s , certain types of sup erfilters whic h w ere later named sele ctive by F arah [8]. F arah po in ts out in [8] that ev ery selectiv e sup erfilter is Ramsey . It is immediate that ev ery selectiv e sup erfilter is strongly Ramsey , and arguments similar to those in the pro of of L emma 1.12 sho w tha t ev ery strongly Ramse y s up erfilter is selectiv e. Give n F arah’s obse rv ation, one can obtain a simpler pro of of Lemma 1 .12. Rec la w has informed us o f his indep enden t w ork with Filip´ o w, Mro ˙ zek, and Szuca [9], whic h contains related results, mainly o f a descriptiv e set theoretic flav or. In the topolo gical res ults, considering fro m the start only coun table co v ers remo v es any restriction from the considered top ological spaces. F or example, our resu lts immediately apply to the corresp onding fam- ilies of coun table Bor el cov ers, since the Borel sets form a base for a top ology o n X . A general study of coun table Borel co v ers in the con text of se lection principles is a v ailable in [25]. Theorem π and its Corollary 3.16 should b e view ed as a simp le w a y to lift one-dimens ional Ramsey theoretic results to higher dimens ions. It do es not generalize the Bergelson-Hindman Theorem from [4], but it extends it to co v er additional classes of sup erfilters, and assumes less on the color ed se t. Ac kno wledgmen t s. W e thank Adi Jarden, Jan Stary , Egb ert Th uem- mel, and Lyub omy r Z domskyy for their useful commen ts and sugges- tions, Andreas Blass for p oin ting out Berge’s reference to us, and Vitaly Bergelson and Neil Hindman fo r their encouraging reaction to these re- sults, when presen ted in the conference Ultr a m ath 2 008 (Pisa, Italy , June 20 08). There is no doubt tha t these results ow e m uc h to the pioneering works of Sc heep ers and his follo w ers, some of whic h a re men tioned in the bibliograph y , and in the reference s therein. 18 NADA V SAMET AND BOAZ TSA BAN Reference s [1] T. B anakh and L. Zdomskyy , Co herence of Semi filters , www.franko .lviv.ua/fac ulty/mechmat/Departments/Topology/booksite.html [2] J. Baumgartner and A T aylor, Partition the or ems and ultr afilters , T ra nsactions of the American Mathematical So cie t y 241 (1978), 283– 309. [3] C. 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(Nadav Samet) Dep ar tment of Ma thema tics, Weizmann Institute of Science, Rehov ot 76100, Israel Curr ent addr ess : Go ogle Ir eland Ltd., Gordon Ho use, Bar row Street, Dublin 4 , Ireland E-mail addr ess : thesam et@gma il.com (Boaz Tsaba n) Dep ar tment o f Ma thema tics, Bar-Ilan University, Rama t- Gan 52900, Israel; and Dep ar tment of Ma thema tics, Weizmann I nsti- tute of Science, Rehov ot 76100, Israel E-mail addr ess : tsaban @math. biu.ac.il