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

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, 2016 1 / 15

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, 2016 2 / 15

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, 2016 3 / 15

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, 2016 4 / 15

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, 2016 4 / 15

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, 2016 4 / 15

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, 2016 5 / 15

Milliken space (FIN [ ],, r) FIN = {finite nonempty subsets of N}. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, 2016 6 / 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 N Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, 2016 6 / 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}. N Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, 2016 6 / 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. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, 2016 6 / 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. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, 2016 6 / 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, 2016 6 / 15

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, 2016 7 / 15

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

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, 2016 7 / 15

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, 2016 8 / 15

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, 2016 8 / 15

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, 2016 9 / 15

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, 2016 10 / 15

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, 2016 11 / 15

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, 2016 12 / 15

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, 2016 13 / 15

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, 2016 13 / 15

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, 2016 13 / 15

References I J. E. Baumgartner and R. Laver (1979) Iterated perfect-set forcing. Ann. Math. Logic, 17(3):271-288. A. Blass (1988) Selective ultrafilters and homogeneity. Ann. Pure Appl. Logic, 38(3):215-255. 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, 1659-1684. 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, 4627-4659. Yuan Yuan Zheng (University of Toronto) Selective ultrafilters October 9, 2016 14 / 15

References II R. Laver (1984) Products of infinitely many perfect trees. J. London Math. Soc. (2), 29(3):385-396. J. Mijares (2007) A notion of selective ultrafilter corresponding to topological Ramsey spaces. MLQ Math. Log. Q., 53(3):255-267. 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, 2016 15 / 15