SPM Bulletin 25

S P M BULLETIN ISSUE NUMBER 25: August 2008 CE Contents 1. Editor’s note 2 2. Researc h announcemen ts 2 2.1. Com binatorial and mo del-theoretical principles related to regularit y o f ultrafilters and compactness of top olog ical sp aces, I 2 2.2. F r ´ ec het-Urysohn fans in free top olo gical groups 2 2.3. P ac king index of subsets in P olish gr o ups 2 2.4. Symmetric mono c hromatic subsets in colorings of the Lobac hevsky plane 3 2.5. Structural R a msey theory of metric spaces a nd top ological dynamics of isometry groups 3 2.6. Distinguishing Num b er of Coun table Homogeneous Relational St r uctures 3 2.7. Indestructible colourings and ra in b o w Ramsey theorems 3 2.8. Pro ducts of Borel subgroups 4 2.9. Selection theorems and treeabilit y 4 2.10. Com binatorial and mo del-theoretical principles related to regularity of ultrafilters and compactness of top olog ical spaces, IV 4 2.11. A prop ert y of C p [0 , 1] 4 2.12. A Dedekind Finite Borel Set 4 2.13. Aronsza jn Compacta 5 2.14. A strong an tidiamond principle compatible with CH 5 2.15. On the strength of Hausdorff ’s gap condition 5 2.16. Nonhomogeneous analytic fa milies of trees 5 2.17. Reasonable non-Radon- Nik o dym ideals 5 2.18. σ -contin uit y and related for cings 6 2.19. An exact Ramsey principle for blo ck sequences 6 2.20. Baire reflection 6 2.21. T uk ey classes of ultrafilters on ω 6 2.22. Coun tably determined compact ab elian groups 7 2.23. A top ological reflection principle equiv alent to Shelah’s St r o ng Hyp othesis 7 2.24. Sup erfilters, Ramsey theory , and v a n der W aerden’s Theorem 7 3. Unsolv ed pro blems from earlier issues 8 1 2 S P M BULLETIN 25 (AUGUST 2008) 1. Editor ’s note The slides of t he talks giv en at the conference Ultr am ath (Pisa, Italy , June 200 8) are a v ailable at http://www. dm.unipi.it/~ultramath/abstracts.html Enjo y . Bo az Tsab an , t saba n@ma t h.biu.ac.il http://www. cs.biu.ac.il/~tsaban 2. Res ear ch announcements 2.1. Com binatorial and mo del-theoretical principles related to regularit y of ultr afilters and compactness of top ological spaces, I. W e begin the study of the consequences of the existence of certain infinite matrices. Our presen t application is to compactness of pro ducts of top ological spaces. http://arxi v.org/abs/0803.3498 Paolo Lip p arini 2.2. F r´ ec het-Urysohn fans in free top ological groups. In this pap er w e answ er the question of T. Banakh and M. Zarichn yi constructing a cop y of the F r ´ ec het- Urysohn fan S ω in a to p o lo gical group G admitting a functorial em b edding [0 , 1] ⊂ G . The latter means tha t eac h autoho meomorphism of [0 , 1] extends to a contin uous ho- momorphism of G . This implies t ha t many natura l free t op ological group construc- tions (e.g. the constructions of the Marko v fr ee to p o lo gical gro up, free ab elian top o- logical group, free t o tally b ounded group, free compact group) applied to a T yc honov space X con taining a to p o lo gical cop y of the space Q of rationals give top ological groups con taining S ω . http://arxi v.org/abs/0803.4117 T ar as Bana kh, Du ˇ san R ep ov ˇ s, and Lyub o m yr Zdomskyy 2.3. P ac king index of subsets in P olish groups. F or a subset A o f a P olish group G , w e study the (almost) pack ing index ind P ( A ) (resp. Ind P ( A )) of A , equal to the suprem um of car dina lit ies | S | of subsets S ⊂ G such that the family of shifts { xA } x ∈ S is (a lmost) disjoint (in the sense t ha t | xA ∩ y A | < | A | for any distinct p oints x, y ∈ S ). Subsets A ⊂ G with small (almost) pac king index are small in a g eometric sense. W e show that ind P ( A ) ∈ N ∪ {ℵ 0 , c } for a n y σ - compact subset A of a Polish group. If A ⊂ G is Borel, then the pack ing indices ind P ( A ) and Ind P ( A ) cannot take v a lues in the half-interv al [ sq (Π 1 1 ) , c ) where sq (Π 1 1 ) is a certain uncountable cardinal that is smaller than c in some mo dels of ZF C. In each non-discrete P olish Ab elian group G w e construct t w o closed subsets A, B ⊂ G with ind P ( A ) = ind P ( B ) = c and Ind P ( A ∪ B ) = 1 a nd then a pply this result to show that G con tains a nowh ere dense Haar null subset C ⊂ G with ind P ( C ) = Ind P ( C ) = κ for an y giv en cardinal n um b er κ ∈ [4 , c ]. S P M BULLETIN 25 (August 2008) 3 http://arxi v.org/abs/0804.1333 T ar as Bana kh, Nadya Lyaskovska, and D u ˇ san R ep ov ˇ s 2.4. Symmetric mono c hromatic subsets in colorings of t he Lobac hevsky plane. W e pro ve t ha t for eac h partitio n of the Lobachev sky plane in to finitely many Borel pieces one of the cells of the partition con tains an un b ounded cen trally sym- metric subset. http://arxi v.org/abs/0804.1335 T. B a nakh, A. Dudko and D. R ep ov ˇ s 2.5. Structural Ramsey theory of metric spaces and t opological dynamics of isometry groups. In 2003 , Kec hris, P esto v and T o dorcevic show ed that the structure o f certain separable metric spaces - called ultra homogeneous - is closely related to the com binatorial b eha vior of the class of t heir finite metric spaces. The purp ose of the presen t pap er is to explore the differen t asp ects of this connection. http://arxi v.org/abs/0804.1593 L. Nguyen V a n Th´ e 2.6. Distinguishing N um b er of Coun table Homogeneous Relational St ruc- tures. The distinguishing n umber of a graph G is the smallest p ositive inte ger r such that G has a lab eling of its v ertices with r lab els fo r which there is no non-trivial automorphism of G preserving t hese lab els. Alb ertson and Collins computed the distinguishing n um b er for v arious finite graphs, and Imrich, Kla v ˇ zar and T rofimo v computed the distinguishing n um b er of some infinite graphs, sho wing in part icular that the Random G raph has distinguishing num ber 2. W e compute the distinguish- ing nu mber of v arious other finite a nd coun table homogeneous structures, including undirected and directed graphs, and p osets. W e sho w that this n um b er is in most cases tw o or infinite, and b esides a few exceptions conjecture that this is so fo r all primitiv e homogeneous coun table structures. http://arxi v.org/abs/0804.4019 C. L aflamme, L. Nguyen V an Th´ e, N. W. Sauer 2.7. Indestructible colourings and rain b o w Ramsey t heorems. W e give a neg- ativ e answ er to a question of Erdos and Ha jnal: it is consisten t that GCH holds and there is a colouring c : [ ω 2 ] 2 → 2 establishing ω 2 6→ [( ω 1 ; ω )] 2 2 suc h that some colouring g : [ ω 1 ] 2 → 2 can not b e em b edded into c . It is also consisten t that 2 ω 1 is arbitrarily large, and a function g establishes 2 ω 1 6→ [( ω 1 , ω 2 )] 2 ω 1 suc h that t here is no uncoun t- able g -rain b o w subset o f 2 ω 1 . W e also sho w that for each k ∈ ω it is consisten t with Martin’s Axiom that the negativ e partition relation ω 1 6→ ∗ [( ω 1 ; ω 1 )] k − bdd holds. http://arxi v.org/abs/0804.4548 L ajos Soukup 4 S P M BULLETIN 25 (AUGUST 2008) 2.8. Pro ducts of Bo r el subgroups. W e in v estigate the Borelness o f the pro duct of t w o Borel subgroups in P olish groups. While the in tersection of t hese tw o subgroups is Polish able, the Borelness of their pro duct is confirmed. On t he other hand, we construct tw o ∆ 0 3 subgroups whose pro duct is not Borel in ev ery uncountable ab elian P olish gro up. www.ams.org /proc/0000-000-00/S0002-9939-08-09334-9 L ongyun Ding and Bingqing Li 2.9. Selection theorems and t reeabilit y . W e sho w that do ma ins of non-t r ivial Σ 1 1 trees hav e ∆ 1 1 mem b ers. Using this, we show that smo oth treeable equiv alence relations hav e Borel transv ersals, and essen tially coun table treeable equiv alence rela- tions hav e Borel complete coun table sections. W e show also that treeable equiv alence relations whic h are ccc idealistic, measured, or generated by a Borel a ction of a P olish group ha v e Borel complete coun table sections. http://www. ams.org/journal-getitem?pii=S0002-9939-08-09548-8 Gr e g Hjorth 2.10. Combina torial and mo del-theoretical principles related to r egularity of ultrafilters and compactness of top ological spaces, IV . W e extend to sin- gular cardinals the mo del-theoretical relation λ κ ⇒ µ introduced in P . Lipparini, The compactness sp ectrum of a bstract lo g ics, large cardinals and combinatorial princi- ples, Bo ll. Unione Matematica Italiana ser. VI I, 4-B 875–9 03 (1990). W e extend some results o btained in P art I I, finding equiv alen t conditions inv olving uniformity of ultrafilters and the existence of certain infinite matrices. Our pr esen t definition suggests a new compactness prop erty for abstract logics. http://arxi v.org/abs/0805.1548 Paolo Lip p arini 2.11. A prop erty of C p [0 , 1] . W e prov e that for eve ry finite dimensional compact metric space X there is an op en con tin uous linear surjection from C p [0 , 1] o nto C p ( X ). The pro of mak es use o f embeddings in tro duced by Kolmog o ro v a nd Sternfeld in con- nection with Hilb ert’s 13th problem. http://arxi v.org/abs/0806.2719 Michael L evin 2.12. A Dedekind Finite Borel Set. In this pa per we pro ve three theorems ab out the theory of Borel sets in mo dels of Z F without any form of the axiom of c hoice. W e pro v e that if B is a G δσ set, then either B is coun table or B con tains a p erfect subset. Second, w e prov e that if the real line is the countable union of coun table sets, t hen there exists an F σδ set whic h is uncoun table but contains no p erfect subset. Fina lly , w e construct a mo del of ZF in whic h w e hav e an infinite Dedekind finite set of reals whic h is F σδ . http://www. math.wisc.edu/~miller/res/ded.pdf S P M BULLETIN 25 (August 2008) 5 A rnold W. Mil ler 2.13. A ronsza jn Compacta. W e consider a class of compacta X suc h that the maps fro m X on to metric compacta define an Aronsza jn tree of closed subsets of X . http://arxi v.org/abs/0806.4499 Jo an E. Hart and Kenneth Kunen 2.14. A strong an tidiamond principle compatible with CH. A strong antidi- amond principle ( ∗ c ) is show n to b e consisten t with CH. This principle can b e stated as a “ P -ideal dic hotom y”: ev ery P -ideal o n omeg a − 1 (i.e. an ideal tha t is σ -directed under inclusion mo dulo finite) either has a closed un b ounded subset of ω 1 lo cally inside of it, or else has a stationary subset of ω 1 orthogonal to it. W e r ely on Shelah’s theory of parameterized prop erness for NNR iteratio ns, and make a con tribution to the theory with a metho d of constructing t he prop erness parameter sim ultaneously with the iteration. Our handling of the a pplication of the NNR itera t ion theory in- v olv es definability of forcing notions in third order ar it hmetic, analogous to Souslin forcing in second order arithmetic. http://arxi v.org/abs/0806.4220 James Hirsc h o rn 2.15. On the strength of Hausdorff ’s gap condition. Hausdorff ’s ga p condi- tion w as satisfied by his orig ina l 1936 construction of an ( ω 1 , ω 1 ) gap in P ( N ) /F in . W e solv e an op en problem in determining whether Hausdorff ’s condition is actually stronger tha n the more mo dern indestructibility condition, b y constructing an in- destructible ( ω 1 , ω 1 ) gap not equiv alent to any ga p satisfying Hausdorff ’s condition, from uncoun tably man y random reals. http://arxi v.org/abs/0806.4732 James Hirsc h o rn 2.16. N onhomogen eous analytic families of trees. W e consider a dic hotom y for analytic families of trees stating that either there is a colouring of the no des fo r whic h all but finitely man y lev els of ev ery tree are nonhomogeneous, or else the family con tains an uncoun table an tichain. This dic hotomy implies that ev ery non trivial Souslin p oset satisfying the coun table c hain condition adds a splitting real. W e then reduce the dichotom y to a conjecture o f Sperner Theory . This conjecture is concerning the a symptotic b ehav iour of the pro duct of the sizes of the m-shades o f pairs of cross- t-in tersecting fa milies. http://arxi v.org/abs/0807.0147 James Hirsc h o rn 2.17. R easona ble non-Radon-Nik o dym ideals. F o r an y abelian P olish σ - compact group H there exist a σ -ideal Z ov er N and a Borel Z -approx imate homomorphism f : H → H N whic h is not Z - a ppro ximable by a contin uous tr ue homomorphism g : H → H N . 6 S P M BULLETIN 25 (AUGUST 2008) http://arxi v.org/abs/0806.4760 Vladimir K a novei and V assily Lyub e tsky 2.18. σ -con tin uit y and related forcings. The Steprans fo rcing no tion arises as a quotien t of Borel sets mo dulo the ideal of σ - con tin uit y of a certain Borel not σ - con tin uous function. W e giv e a c haracterization of this forcing in the language of t rees and using t his characterization we establish suc h prop erties of the forcing as fusion and contin uous reading of names. Although the latter pro perty is usually implied by the fact that the asso ciated ideal is g enerated b y closed sets, w e sho w it is not the case with Steprans forcing. W e also establish a connection b et w een Steprans forcing and Miller forcing th us giving a new description of the latter. Ev en tually , we exhibit a v ariet y of forcing notions whic h do not ha ve con tinuous reading of names in an y presen tation. http://arxi v.org/abs/0807.1254 Mar cin Sab ok 2.19. A n exact Ramsey principle for blo c k sequences. W e pro ve an exact, i.e., form ulated without ∆-expansions, Ramsey principle for infinite blo c k sequences in v ector spaces o v er coun table fields, where the tw o sides of the dic hotomic principle are represen ted b y resp ectiv ely winning strategies in Gow ers’ blo c k sequence game and winning strategies in the infinite asymptotic ga me. This allows us to recov er Go we rs’ dic hotomy theorem for blo c k sequenc es in normed v ector spaces b y a simple application of the basic determinacy theorem for infinite asymptotic games. http://arxi v.org/abs/0807.2205 Christian R o sendal 2.20. B aire reflection. W e study reflection principles in volvin g no nmeager sets a nd the Baire Prop erty whic h are consequences of the generic sup ercompactness of ω 2 , suc h as t he principle a sserting that an y p oin t coun table Baire space has a stationary set of closed subspaces of w eigh t ω 1 whic h are also Baire spaces. These principles en tail the analo gous principles of stationa r y reflection but are incompatible with fo rcing axioms. Assuming MM, t here is a Baire metric space in whic h a club of closed subspaces of weigh t ω 1 are meager in themselv es. Unlik e stronger forms o f Game Reflection, these reflection principles do not decide CH, though t hey do giv e ω 2 as an upp er b ound for the size of the con tin uum. http://www. ams.org/tran/0000-000-00/S0002-9947-08-04503-0/home.html Stevo T o dor c evic and S tuart Zoble 2.21. T uk ey classes of ultrafilters on ω . Motiv ated by a question o f Isb ell, w e sho w tha t Jensen’s Diamond Principle implies there is a no n- P-p oin t ultrafilter U on ω suc h that U, whether ordered b y rev erse inclusion or rev erse inclusion mo d finite, is not T uk ey equiv alen t to the finite sets of reals ordered b y inclusion. W e also sho w that, for ev ery regular infinite k a ppa not greater than 2 ℵ 0 , if M A ( σ − center ed ) S P M BULLETIN 25 (August 2008) 7 holds, then some ultr a filter U on ω , ordered b y rev erse inclusion mo d finite, is T uk ey equiv alent to the sets of reals o f size less than k appa, ordered b y inclusion. W e also pro v e t w o negat iv e ZFC results ab out the p ossible T uke y classes of ultrafilters on ω . http://arxi v.org/abs/0807.3978 David Milovich 2.22. Countably determined compact ab elian groups. F o r an ab elian to po log- ical group G let b G b e the dual group of all con tinuous c haracters endo w ed with the compact op en top ology . G iv en a closed subset X of an infinite compact ab elian group G suc h that w ( X ) < w ( G ) and an op en neigh b ourho o d U of 0 in T , w e sho w that |{ π ∈ b G : π ( X ) ⊆ U }| = | b G | . (Here w ( G ) denotes t he w eigh t o f G .) A subgroup D of G determines G if the restriction ho mo mor phism b G → b D of the dual g r o ups is a top ological isomorphism. W e prov e that w ( G ) = min {| D | : D is a subgroup of G that determines G } for ev ery compact a b elian group G . In particular, an infinite com- pact ab elian gr oup determined b y its countable subgroup mus t b e metrizable. This giv es a negat ive answ er to questions of Comfor t , Hern´ andez, Macario, Raczk o wski and T rigos-Arrieta. As an application, w e furnish a short elemen tary pro of of the result that compact determined ab elian groups are metrizable. http://arxi v.org/abs/0807.3846 Dikr an Dikr anjan, D mitri Sh akhmatov 2.23. A t opological reflection principle equiv alen t to Shelah’s Strong Hy- p othesis. W e notice that Shelah’s Str ong Hyp othesis is equiv alen t to the following reflection principle: Supp ose h X , τ i is a first-coun table space whose density is a regu- lar cardinal, κ . If eve ry separable subspace of X is of cardina lity at most κ , then the cardinalit y o f X is κ . dx.doi.org/ 10.1090/S0002-9939-08-09411-2 Assaf Rinot 2.24. Sup erfilters, Ramsey theory , and v an der W aerden’s Theorem. Sup er- filters ar e generalized ultrafilters, whic h capture the underlying concept in Ramsey theoretic theorems suc h as v an der W aerden’s Theorem. W e establish sev eral prop- erties of sup erfilters, which generalize b oth Ra msey’s Theorem and its v arian t for ultrafilters o n the natural num b ers. W e use them to confirm a conjecture of Koˇ cinac and Di Maio, which is a generalization of a Ramsey theoretic result o f Schee p ers, concerning selections from op en cov ers. F ollow ing Bergelson and Hindman’s 19 8 9 Theorem, w e presen t a new sim ultaneous g eneralization of the theorems of Ramsey , v a n der W aerden, Sc h ur, F olkman-Ra do -Sanders, Ra do , and others, where the colored sets can b e muc h smaller than the full set o f natura l num bers. http://arxi v.org/abs/0808.1654 Nadav Samet and Bo az Tsab an 8 S P M BULLETIN 25 (AUGUST 2008) 3. Unsol ved problems from earlier issues Issue 1. I s  Ω Γ  =  Ω T  ? Issue 2. I s U fin ( O , Ω) = S fin (Γ , Ω) ? And if not, do es U fin ( O , Γ) imply S fin (Γ , Ω) ? Issue 4. D o es S 1 (Ω , T) imply U fin (Γ , Γ) ? Issue 5. I s p = p ∗ ? (Se e the defin ition of p ∗ in that issue.) Issue 6. D o es ther e exist (in ZFC) an unc ountable s et satisfying S fin ( B , B ) ? Issue 8. D o es X 6∈ NON ( M ) and Y 6∈ D imply that X ∪ Y 6∈ COF ( M ) ? Issue 9 ( CH) . Is Split (Λ , Λ ) pr eserve d under finite unions? Issue 10. I s cov ( M ) = od ? (S e e the definition of od in that issue.) Issue 11. D o es S 1 (Γ , Γ) always c ontain an element of c ar dinality b ? Issue 12 . Could ther e b e a Bair e me tric sp ac e M of weigh t ℵ 1 and a p artition U of M into ℵ 1 me ager sets wher e fo r e ach U ′ ⊂ U , S U ′ has the B air e pr op erty in M ? Issue 14. Do es ther e exist (in ZFC) a set of r e als X of c ar d i nality d such that al l finite p owers of X have Menger’s pr op erty S fin ( O , O ) ? Issue 15. C a n a Bor el no n- σ -c omp a ct g r oup b e gener ate d by a Hur ewicz s ubs p ac e? Issue 16 ( MA) . Is ther e an unc ountable X ⊆ R satisfying S 1 ( B Ω , B Γ ) ? Issue 17 (CH) . Is ther e a total ly imp erfe ct X satisfying U fin ( O , Γ) that c an b e mapp e d c ontinuously onto { 0 , 1 } N ? Issue 18 ( CH) . Is ther e a Hur ewic z X such that X 2 is Menger but not Hur ewicz? Issue 19. Do es the Pytke ev pr op erty of C p ( X ) imply that X has Menger’s pr op erty? Issue 20. D o es every her e ditarily Hur ewicz sp ac e satisfy S 1 ( B Γ , B Γ ) ? Issue 21 (CH) . Is ther e a R othb er ger-b ounde d G ≤ Z N such that G 2 is not Menger- b ounde d? Issue 22. L et W b e the va n d e r Waer den ide al. Ar e W -ultr a filters close d under pr o ducts? Issue 23. I s the δ -pr op erty e quivalent to the γ -pr op erty  Ω Γ  ? Previous issues. The previous issu es of this b ulletin are av ailable online at http://f ront.math.u cdavis.edu/search?&t=%22SPM+Bulletin%22 Con tributions. Announcements, discussions , and op en problems should b e emailed to tsaban@m ath.biu.ac. il Subscription. T o receiv e this bulletin (free) to y our e-mailb o x, e-mail u s.