Selective ultrafilters in N [ ], FIN [ ], and R α

Size: px

Start display at page:

Download "Selective ultrafilters in N [ ], FIN [ ], and R α"

Edwin Williams
6 years ago
Views:

1 Selective ultrafilters in N [ ], FIN [ ], and R α Yuan Yuan Zheng University of Toronto yyz22@math.utoronto.ca October 9, 2016 Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

2 Overview 1 Selective Ultrafilters in N [ ] 2 Selective Ultrafilters in FIN [ ] 3 Selective Ultrafilters in R α Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

3 Overview 1 Selective Ultrafilters in N [ ] 2 Selective Ultrafilters in FIN [ ] 3 Selective Ultrafilters in R α Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

4 Ellentuck space (N [ ],, r) In [1] and [5], Baumgartner and Laver showed that selective ultrafilters on N are preserved under both side-by-side and iterated Sacks forcing. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

5 Ellentuck space (N [ ],, r) In [1] and [5], Baumgartner and Laver showed that selective ultrafilters on N are preserved under both side-by-side and iterated Sacks forcing. U selective on N: {A 0 A 1 } U X U such that n X X /n A n. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

6 Ellentuck space (N [ ],, r) In [1] and [5], Baumgartner and Laver showed that selective ultrafilters on N are preserved under both side-by-side and iterated Sacks forcing. U selective on N: {A 0 A 1 } U X U such that n X X /n A n. Sacks forcing (P, ): P = {perfect subsets of R = 2 N }. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

7 Ellentuck space (N [ ],, r) In [1] and [5], Baumgartner and Laver showed that selective ultrafilters on N are preserved under both side-by-side and iterated Sacks forcing. U selective on N: {A 0 A 1 } U X U such that n X X /n A n. Sacks forcing (P, ): Side-by-side Sacks forcing (P κ, ): P = {perfect subsets of R = 2 N }. P κ = { functions p : κ P such that dom(p) is countable}, p q if dom(p) dom(q) and p(α) q(α) for all α dom(q). Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

8 Overview 1 Selective Ultrafilters in N [ ] 2 Selective Ultrafilters in FIN [ ] 3 Selective Ultrafilters in R α Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

9 Milliken space (FIN [ ],, r) FIN = {finite nonempty subsets of N}. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

10 Milliken space (FIN [ ],, r) FIN = {finite nonempty subsets of N}. FIN [ ] = {infinite block sequences X = (x n ) n ω }. X : x 1 x 2 x 3 N Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

11 Milliken space (FIN [ ],, r) FIN = {finite nonempty subsets of N}. FIN [ ] = {infinite block sequences X = (x n ) n ω }. X : x 1 x 2 x 3 FIN [< ] = {finite block sequences}. N Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

12 Milliken space (FIN [ ],, r) FIN = {finite nonempty subsets of N}. FIN [ ] = {infinite block sequences X = (x n ) n ω }. X : x 1 x 2 x 3 FIN [< ] = {finite block sequences}. [X ] is the sublattice of FIN generated by X, [X ] = {x n0 x nk : k ω, n 0 < n k }. N Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

13 Milliken space (FIN [ ],, r) FIN = {finite nonempty subsets of N}. FIN [ ] = {infinite block sequences X = (x n ) n ω }. X : x 1 x 2 x 3 FIN [< ] = {finite block sequences}. [X ] is the sublattice of FIN generated by X, [X ] = {x n0 x nk : k ω, n 0 < n k }. N For Y = (y n ) FIN [ ], Y X if y n [X ] for all n. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

14 Milliken space (FIN [ ],, r) FIN = {finite nonempty subsets of N}. FIN [ ] = {infinite block sequences X = (x n ) n ω }. X : x 1 x 2 x 3 FIN [< ] = {finite block sequences}. [X ] is the sublattice of FIN generated by X, [X ] = {x n0 x nk : k ω, n 0 < n k }. N For Y = (y n ) FIN [ ], Y X if y n [X ] for all n. [, X ] is the set of all infinite Y X. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

15 Milliken space (FIN [ ],, r) FIN = {finite nonempty subsets of N}. FIN [ ] = {infinite block sequences X = (x n ) n ω }. X : x 1 x 2 x 3 FIN [< ] = {finite block sequences}. [X ] is the sublattice of FIN generated by X, [X ] = {x n0 x nk : k ω, n 0 < n k }. N For Y = (y n ) FIN [ ], Y X if y n [X ] for all n. [, X ] is the set of all infinite Y X. Finite approximation r is given by r n (X ) = {x 1,..., x n }. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

16 Selective ultrafilters in FIN [ ] Mijares [6] defined selective ultrafilters for topological Ramsey spaces. We have the following equivalent definition. Definition (selective in FIN [ ] ) An ultrafilter U on the base set FIN is selective if U is generated by elements of the form [B](B FIN [ ] ), and for every set {[A a ] : a FIN [< ] } of elements in U, there exists [X ] U such that for all a X X /a [A a ]. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

17 Selective ultrafilters in FIN [ ] Mijares [6] defined selective ultrafilters for topological Ramsey spaces. We have the following equivalent definition. Definition (selective in FIN [ ] ) An ultrafilter U on the base set FIN is selective if U is generated by elements of the form [B](B FIN [ ] ), and for every set {[A a ] : a FIN [< ] } of elements in U, there exists [X ] U such that for all a X X /a [A a ]. X : x 1 x 2 x 3 x 4 x 5 x 6 x 7 N

18 Selective ultrafilters in FIN [ ] Mijares [6] defined selective ultrafilters for topological Ramsey spaces. We have the following equivalent definition. Definition (selective in FIN [ ] ) An ultrafilter U on the base set FIN is selective if U is generated by elements of the form [B](B FIN [ ] ), and for every set {[A a ] : a FIN [< ] } of elements in U, there exists [X ] U such that for all a X X /a [A a ]. X : x 1 x 2 x 3 x 4 x 5 x 6 x 7 N a : N a 1 = x 2 a 2 = x 4 x 5

19 Selective ultrafilters in FIN [ ] Mijares [6] defined selective ultrafilters for topological Ramsey spaces. We have the following equivalent definition. Definition (selective in FIN [ ] ) An ultrafilter U on the base set FIN is selective if U is generated by elements of the form [B](B FIN [ ] ), and for every set {[A a ] : a FIN [< ] } of elements in U, there exists [X ] U such that for all a X X /a [A a ]. X : x 1 x 2 x 3 x 4 x 5 x 6 x 7 N a : N a 1 = x 2 a 2 = x 4 x 5 X /a : x 6 x 7 N Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

20 FIN [ ] : Parametrised Milliken Theorem Theorem (Parametrised Milliken Theorem, Todorcevic [7]) For every finite Souslin-measurable colouring of FIN [ ] R N there exist X FIN [ ] and a sequence (P i ) i<ω of nonempty perfect subsets of R such that [, X ] i<ω P i is monochromatic. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

21 FIN [ ] : Parametrised Milliken Theorem Theorem (Parametrised Milliken Theorem, Todorcevic [7]) For every finite Souslin-measurable colouring of FIN [ ] R N there exist X FIN [ ] and a sequence (P i ) i<ω of nonempty perfect subsets of R such that [, X ] i<ω P i is monochromatic. Corollary (YYZ) Let U be a selective ultrafilter on FIN. For every finite Souslin-measurable colouring of FIN [ ] R N, there exist [X ] U and a sequence (P i ) i<ω of nonempty perfect subsets of R such that [, X ] i<ω P i is monochromatic. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

22 FIN [ ] : Parametrised Milliken Theorem Theorem (Parametrised Milliken Theorem, Todorcevic [7]) For every finite Souslin-measurable colouring of FIN [ ] R N there exist X FIN [ ] and a sequence (P i ) i<ω of nonempty perfect subsets of R such that [, X ] i<ω P i is monochromatic. Corollary (YYZ) Let U be a selective ultrafilter on FIN. For every finite Souslin-measurable colouring of FIN [ ] R N, there exist [X ] U and a sequence (P i ) i<ω of nonempty perfect subsets of R such that [, X ] i<ω P i is monochromatic. The proof uses combinatorial forcing on U-trees, introduced by Blass [2]. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

23 FIN [ ] : Ultrafilters under Sacks forcing Corollary Let U be a selective ultrafilter on FIN in the ground model, and V a P κ -name for the upward closure {Y FIN : [X ] U [X ] Y } of U. Then Pκ V is a selective ultrafilter on FIN. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

24 FIN [ ] : Ultrafilters under Sacks forcing Corollary Let U be a selective ultrafilter on FIN in the ground model, and V a P κ -name for the upward closure {Y FIN : [X ] U [X ] Y } of U. Then Pκ V is a selective ultrafilter on FIN. The previous Corollary is used to prove V is ultra : Corollary (previous one) For every finite Souslin colouring of FIN [ ] R N, there exist [X ] U and a sequence (P i ) i<ω of nonempty perfect subsets of R such that [, X ] i<ω P i is monochromatic. Lemma ( ultra ) If p P κ and p τ FIN, then there exist [X ] U and q p such that q [X ] τ or q [X ] τ c. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

25 Overview 1 Selective Ultrafilters in N [ ] 2 Selective Ultrafilters in FIN [ ] 3 Selective Ultrafilters in R α Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

26 Spaces (R α,, r) In [3] and [4], Dobrinen and Todorcevic developed a new hierarchy of topological Ramsey spaces R α, α < ω 1, and corresponding ultrafilters U α. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

27 Spaces (R α,, r) Space R α Roughly speaking, members X R α are subtrees of T α with the same shape as T α. e.g. R 0 is the Ellentuck space (N [ ],, r). Ultrafilter U α U α is an ultrafilter on the base set [T α ], the set of maximal nodes in T α, which is Ramsey, hence selective. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

28 R α : Ultrafilters U α under Sacks forcing We follow the same recipe to show that the selectivity of U α is also preserved under the Sacks forcing. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

29 R α : Ultrafilters U α under Sacks forcing We follow the same recipe to show that the selectivity of U α is also preserved under the Sacks forcing. Theorem (Parametrised R α theorem, YYZ) For every finite Souslin-measurable colouring of R α R there exist X R α and p P such that [, X ] [p] is monochromatic. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

30 R α : Ultrafilters U α under Sacks forcing We follow the same recipe to show that the selectivity of U α is also preserved under the Sacks forcing. Theorem (Parametrised R α theorem, YYZ) For every finite Souslin-measurable colouring of R α R there exist X R α and p P such that [, X ] [p] is monochromatic. Corollary For every finite Souslin-measurable colouring of R α R there exists [X ] U α such that [, X ] is monochromatic. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

31 R α : Ultrafilters U α under Sacks forcing We follow the same recipe to show that the selectivity of U α is also preserved under the Sacks forcing. Theorem (Parametrised R α theorem, YYZ) For every finite Souslin-measurable colouring of R α R there exist X R α and p P such that [, X ] [p] is monochromatic. Corollary For every finite Souslin-measurable colouring of R α R there exists [X ] U α such that [, X ] is monochromatic. Corollary Let V be a P-name for the upward closure {Y [T α ] : [X ] U α [X ] Y } of U α. Then P V is a selective ultrafilter on [T α ]. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

32 References I J. E. Baumgartner and R. Laver (1979) Iterated perfect-set forcing. Ann. Math. Logic, 17(3): A. Blass (1988) Selective ultrafilters and homogeneity. Ann. Pure Appl. Logic, 38(3): N. Dobrinen and S. Todorcevic (2014) A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, Part 1. Trans. Amer. Math. Soc., 366 (2014), no. 3, N. Dobrinen and S. Todorcevic (2015) A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, Part 2. Trans. Amer. Math. Soc., 367 (2015), no. 7, Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

33 References II R. Laver (1984) Products of infinitely many perfect trees. J. London Math. Soc. (2), 29(3): J. Mijares (2007) A notion of selective ultrafilter corresponding to topological Ramsey spaces. MLQ Math. Log. Q., 53(3): S. Todorcevic. (2010) Introduction to Ramsey Spaces (AM-174). Annals of Mathematics Studies, Princeton University Press. Thank you. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, / 15

Selective Ultrafilters on FIN

Selective Ultrafilters on FIN Yuan Yuan Zheng Abstract We consider selective ultrafilters on the collection FIN of all finite nonempty subsets of N. If countablesupport side-by-side Sacks forcing is applied,