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Selective but not Ramsey
Topology and its Applications 202 (2016) 61–69
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Topology and its Applications www.elsevier.com/locate/topol
Selective but not Ramsey Timothy Trujillo Colorado School of Mines, Department of Applied Mathematics and Statistics, Golden, CO 80401-1887, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 19 September 2013 Received in revised form 25 October 2015 Accepted 31 December 2015 Available online 12 January 2016
We give a partial answer to the following question of Dobrinen: For a given topological Ramsey space R, are the notions of selective for R and Ramsey for R equivalent? Every topological Ramsey space R has an associated notion of Ramsey ultrafilter for R and selective ultrafilter for R (see [1]). If R is taken to be the Ellentuck space then the two concepts reduce to the familiar notions of Ramsey and selective ultrafilters on ω; so by a well-known result of Kunen the two are equivalent. We give the first example of an ultrafilter on a topological Ramsey space that is selective but not Ramsey for the space. We show that for the topological Ramsey space R1 from [2], the notions of selective for R1 and Ramsey for R1 are not equivalent. In particular, we prove that forcing with a closely related topological Ramsey space using almost-reduction, adjoins an ultrafilter that is selective but not Ramsey for R1 . © 2016 Elsevier B.V. All rights reserved.
MSC: 05D10 03E02 54D80 03E05 Keywords: Ramsey space Selective ultrafilter
1. Introduction This paper is concerned with giving an example of a topological Ramsey space R and an ultrafilter that is selective for R but not Ramsey for R. The first result of topological Ramsey theory was the infinite dimensional extension of the Ramsey theorem known as the Ellentuck theorem (see [3]). Ellentuck proved this theorem in order to give a proof of Silver’s theorem stating that analytic sets have the Ramsey property. In order to state the Ellentuck theorem it is necessary to introduce the Ellentuck space. We denote the infinite subsets of ω by [ω]ω and the finite subsets of ω by [ω]<ω . If B ∈ [ω]ω and {b0 , b1 , b2 , . . . } is its increasing enumeration, then for each i < ω, we let ri (B) denote the set {b0 , b1 , b2 , . . . , bi−1 } and call it the ith approximation of B. The Ellentuck space is the set [ω]ω of all infinite subsets of ω with the topology generated by the basic open sets, [a, B] = {A ∈ [ω]ω : A ⊆ B & (∃i)ri (A) = a} where a ∈ [ω]<ω and B ∈ [ω]ω . E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2015.12.088 0166-8641/© 2016 Elsevier B.V. All rights reserved.
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Recall that a subset of a topological space is nowhere dense if its closure has empty interior and meager if it is the countable union of nowhere dense sets. A subset X of a topological space has the Baire property if and only if X = OΔM for some open set O and some meager set M. A subset X of the Ellentuck space is Ramsey if for every ∅ = [a, A], there is a B ∈ [a, A] such that [a, B] ⊆ X or [a, B] ∩ X = ∅. The next theorem is the infinite-dimensional version of the Ramsey theorem. Theorem 1 (Ellentuck theorem). ([3]) Every subset of the Ellentuck space with the Baire property is Ramsey. Topological Ramsey spaces are spaces that have enough structure in common with the Ellentuck space that an abstract version of the Ellentuck theorem can be stated and proved. The Ellentuck space leads naturally to the notion of a selective ultrafilter on ω. Definition 2. Let U be a nonprincipal ultrafilter on ω. If i < ω and A is an infinite subset of ω, i.e. in the Ellentuck space, then we let A/i = A \ ri (A).
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U is selective, if for each decreasing sequence A0 ⊇ A1 ⊇ . . . of members of U there exists X = {x0 , x1 , . . . } ∈ U enumerated in increasing order such that for all i < ω, A/i ⊆ Ai .
(3)
The next theorem, due to Kunen, characterizes selective ultrafilters as those which are minimal with respect to the Rudin–Keisler ordering. Theorem 3. ([4]) Let U be an ultrafilter on ω. The following conditions are equivalent: 1. U is selective. 2. For each partition of the two-element subsets of ω into two parts, there is a set X ∈ U all of whose two-element subsets lie in one part of the partition. 3. Every function on ω is constant or one-to-one on some set in U. An ultrafilter that satisfies the second item is called a Ramsey ultrafilter on ω. Generalizations of the previous theorem have been studied in many contexts. For example, the notions of selective coideal (see [5]) and semiselective coideals (see [6]) have been shown to also satisfy similar Ramsey properties. In [1], Mijares generalizes the notion of selective ultrafilter on ω to a notion of selective ultrafilter on an arbitrary topological Ramsey space R. Mijares also generalizes the notion of Ramsey ultrafilter on ω to a notion of Ramsey ultrafilter for R and shows that if an ultrafilter is Ramsey for R then it is also selective for R. If one takes R to be the Ellentuck space then the two generalizations reduce to the concepts of selective and Ramsey ultrafilter. The theorem of Kunen above shows that the notions of selective for the Ellentuck space and Ramsey for the Ellentuck space are equivalent. This leads to the following question asked by Dobrinen about the generalizations from selective and Ramsey to arbitrary topological Ramsey spaces. Question 4. For a given topological Ramsey space R, are the notions of selective for R and Ramsey for R equivalent? Ramsey for R ultrafilters have also been studied by Dobrinen and Todorcevic in [2] and [7]. Motivated by Tukey classification problems, the authors develop a hierarchy of topological Ramsey spaces Rα , α ≤ ω1 . Associated to each space Rα is an ultrafilter Uα , which is Ramsey for Rα . The space R0 is taken to be the
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Ellentuck space; therefore, Ramsey for R0 is equivalent to selective for R0 . We show that there is a triple (R1 , ≤, r) such that forcing with the space using almost-reduction, adjoins an ultrafilter that is selective for R1 but not Ramsey for R1 . In this article we use the methods of forcing but all of our constructions can be carried out using CH or MA. We work with σ-closed partial orders and all of the constructions only require 2ℵ0 conditions to be met. For example, assuming CH we can guarantee the conditions hold at successor stages and use σ-closure at limit stages. The author would like to thank Natasha Dobrinen and the two referees that carefully read the article for their valuable comments and suggestions that helped make this article and its proofs more readable. 2. Partition relation notation for trees on ω For each set X, we let X <ω denote the collection of all finite sequences of elements of X. For each finite sequence s, we let |s| denote the length of s. For each i ≤ |s|, πi (s) denotes the sequence of the first i + 1 elements of s and si denotes the (i + 1)th element of the sequence. For each pair of sequences s and t, we say that s is an initial segment of t and write s t if there exists i ≤ |t| such that s = πi (t). The closure of T ⊆ X <ω (denoted by cl(T )) is the set of all initial segments of elements of T . A subset T of X <ω is a tree on X, if cl(T ) = T . A maximal node of T , is a sequence s in T such that for each t ∈ T , s t ⇒ s = t. We let [T ] denote the set of all maximal nodes of T . The height of T is the smallest ordinal greater than or equal to the length of each element of T . The lexicographical order of ω <ω is defined as follows: s is lexicographically less than t if and only if s t or |s| = |t| and the least i on which s and t disagree, si ≤ ti . If S and T are trees on ω, then S is isomorphic to T , if there exists a bijection h : S → T which preserves the lexicographic ordering. A subtree of T is a tree S such that S ⊆ T . If S is a subtree of T then we will write S ≤ T . Given two trees S and T on ω, we let TS denote the set of all subtrees of T that are isomorphic to S. Suppose S, T and U are trees on ω, the partition relation T → (S)U means that for each partition of the partition.
T U
into two parts there exists V ∈
(4) T S
such that
V U
lies in one part of
3. Selective but not Ramsey for R1 We begin this section with the definition of the triple (R1 , ≤, r). This space was first defined by Dobrinen and Todorcevic in [2]. The construction of R1 in [2] was inspired and motivated by the work of Laflamme in [8] which uses forcing to adjoin a weakly-Ramsey ultrafilter satisfying complete combinatorics over HOD(R). The purpose of this section is to introduce a closely related triple (R1 , ≤, r) and show that forcing with R1 using almost-reduction, adjoins an ultrafilter that is selective but not Ramsey for R1 . Definition 5 ((R1 , ≤, r)). ([2]) For each i < ω, let
T1 (i) = Let T1 =
i<ω
(i + 1)(i + 2) i(i + 1) ≤j<
, i , i, j : 2 2
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T1 (i) and (R1 , ≤, r) denote the triple (R(T1 ), ≤, r). Fig. 1 includes a graph of the tree T1 .
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Fig. 1. Graph of T1 and T1 .
Let R1 denote the set of all subtrees of T1 isomorphic to T1 , i.e. TT11 . For each S ∈ R1 we let { kiS : i < ω} denote the lexicographically increasing enumeration of π0 [S]. For each i < ω, let S(i) = cl({s ∈ [S] : π0 (s) = kiS }), ri (S) = S(j).
(6) (7)
j
Let AR1 = i<ω {ri (S) : S ∈ R1 } and define r : ω × R1 → AR1 by letting r(i, S) = ri (S). Here we let ≤ denote the subtree relation. For S, S ∈ R1 almost-reduction is defined as follows: S ≤∗ S if and only if there exists i < ω such that S \ ri (S) ⊆ S . The Ellentuck topology on R1 is the topology generated by the sets, [a, B] = {A ∈ R1 : A ⊆ B & (∃i)ri (A) = a}
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where a ∈ AR1 and B ∈ R1 . A subset X of R1 is Ramsey if for every ∅ = [a, A], there is a B ∈ [a, A] such that [a, B] ⊆ X or [a, B] ∩ X = ∅. The next result is Theorem 3.9 of Dobrinen and Todorcevic in [2]. Theorem 6. ([2]) (R1 , ≤, r) forms a topological Ramsey space. That is, every subset of R1 with the Baire property with respect to the Ellentuck topology on R1 is Ramsey. Next following Dobrinen and Todorcevic in [2] we introduce a generalization of the notion of Ramsey and selective for R1 . Definition 7. Let U be an ultrafilter on [T1 ]. 1. We say that U is generated by G ⊆ R1 , if {[S] : S ∈ G} is cofinal in (U, ⊇). 2. An ultrafilter U generated by G ⊆ R1 is selective for R1 if and only if for each decreasing sequence S0 ≥ S1 ≥ S2 ≥ . . . of elements of G, there exists another S ∈ G such that for all i < ω, S \ ri (S) ⊆ Si .
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3. An ultrafilter U generated by G ⊆ R1 is Ramsey for R1 if and only if for each i < ω and each partition S of riT(T11 ) into two parts there exists S ∈ G such that ri (T lies in one part of the partition. 1) The next result addresses the existence of Ramsey ultrafilters for R1 . We omit its proof as it follows by applying Lemma 3.3 of Mijares in [1] to topological Ramsey spaces of the form R(T ). Theorem 8. ([1]) Forcing with (R1 , ≤∗ ) adjoins no new elements of (AR1 )ω , and if G is a (R1 , ≤∗ )-generic filter over some ground model V , then G generates a Ramsey for R1 ultrafilter in V [G]. The next result is a consequence of Theorem 3.5 of Mijares in [9] applied to the topological Ramsey space R1 . Lemma 9. ([9]) For each pair of positive integers k and n with k ≤ n there exists m < ω such that rm (T1 ) → (rn (T1 ))rk (T1 ) .
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Next we define R1 . The space is constructed in a similar way to R1 using a modified version of the tree T1 . The modified tree T1 has height 3 and for each i < ω, the maximal nodes of T1 (i) are in one-to-one correspondence with the maximal nodes of T1 (i). These two properties of T1 are used below to show that forcing with R1 using almost-reduction adjoins an ultrafilter that is selective but not Ramsey for R1 . The one-to-one correspondence is used to show that the adjoined ultrafilter is selective for R1 and the extra level of T1 is used to show that the adjoined ultrafilter fails to be Ramsey for R1 . Definition 10 ((R1 , ≤, r)). For each i < ω, let T1 (i) = cl({ i, j, k : k ≤ i & j, k ∈ T1 }).
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Let T1 = i<ω T1 (i) and (R1 , ≤, r) denote the triple (R(T1 ), ≤, r). Fig. 1 includes a graph of the tree T1 . Let R1 denote the set of all subtrees of T1 isomorphic to T1 , i.e. TT1 . For each S ∈ R1 we let { kiS : 1 i < ω} denote the lexicographically increasing enumeration of π0 [S]. For each i < ω, let S(i) = cl({s ∈ [S] : π0 (s) = kiS }), S(j). ri (S) =
(11) (12)
j
Let AR1 = i<ω {ri (S) : S ∈ R1 } and define r : ω × R1 → AR1 by letting r(i, S) = ri (S). Here we let ≤ denote the subtree relation. For S, S ∈ R1 almost-reduction is defined as follows: S ≤∗ S if and only if there exists i < ω such that S \ ri (S) ⊆ S . The next two partition properties are needed to show that forcing with (R1 , ≤∗ ) adjoins an ultrafilter on [T1 ]. In fact, all that is needed is the case for k = 0. However, we prove the more general lemma as it is the crux of the proof that R1 has enough in common with the Ellentuck space that it forms a topological Ramsey space. Lemma 11. For each pair of positive integers k and n with k ≤ n, there exists m < ω such that
T1 (m) → (T1 (n))T1 (k) .
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Proof. For each i < ω, let i be the smallest natural number such that for all j ≥ i , T1 (i) is isomorphic to a subtree of cl({ j s : s ∈ ri (T1 )}). By Lemma 9 there exists m < ω such that rm (T1 ) → (rn (T1 ))T1 (k ) . Note that it must be the case that m is greater than both k and n . 1) . For each j < 2, let Πj = {S ∈ rrm (T : cl({ m s : s ∈ Suppose that {Π0 , Π1 } is a partition of TT1(m) k (T1 ) 1 (k) ˆ ∈ Π } where [S] ˆ consists of the lexicographically first k elements of [S]. {Π , Π } forms a partition of [S]}) 0 1 rm (T1 ) j rm (T1 ) S . Hence, there exists j < 2 and S ∈ such that ⊆ Π . If we let S := cl({ m s : j rk (T1 ) rk (T1 ) rn (T1 ) ˆ ˆ consists of the lexicographically first n elements of [S], then S ∈ T1(m) , and ∈ Πj where [S] s ∈ [S]}) T1 (n) S T (k) ⊆ Πj . 2 1
Lemma 12. For each positive integer k,
T1 → (T1 )T1 (k) .
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Proof. Lemma 11 shows that there exists a strictly increasing sequence (mi )i<ω such that for each i < ω, 1 and (S0 , S1 , . . . ) be a sequence of trees such T1 (mi ) → (T1 (i))T1 (k) . Let {Π0 , Π1 } be a partition of T T (k) 1 T Si that for each i < ω, Si ∈ T 1(i) and T (k) is contained in one piece of the partition {Π0 , Π1 }. By the 1 1 pigeonhole principle there exists j < 2 and a strictly increasing sequence (i0 , i1 , . . . ) such that for all l < ω, T (mi ) Si l Sil ∈ 1T (i)l and T (k) ⊆ Πj . Let S = l<ω Sil . If S is any element of TS then T S(k) ⊆ Πj . 2 1
1
1
1
Remark 13. The Ellentuck topology on R1 is the topology generated by the sets, [a, B] = {A ∈ R1 : A ⊆ B & (∃i)ri (A) = a}
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where a ∈ AR1 and B ∈ R1 . A subset X of R1 is Ramsey if for every ∅ = [a, A], there is a B ∈ [a, A] such that [a, B] ⊆ X or [a, B] ∩ X = ∅. It can be shown that (R1 , ≤, r) forms a topological Ramsey space. That is, every subset of R1 with the Baire property with respect to the Ellentuck topology is Ramsey. The proof uses the Abstract Ellentuck Theorem from [10]. It is enough to show that (R1 , ≤, r) satisfies A.1–A.4 from the statement of the Abstract Ellentuck Theorem in [10] and forms a closed subspace of (AR1 )ω . The proof that R1 is a closed subspace of (AR1 )ω and satisfies axioms A.1–A.3 follows by trivial modifications to the proofs of the same facts for the space R1 in [2]. By definition of R1 , A.4 from [10] is equivalent to Lemma 12. Hence A.4 holds for R1 . By the Abstract Ellentuck Theorem, R1 forms a topological Ramsey space. Lemma 14. For each sequence S0 ∗ ≥ S1 ∗ ≥ S2 . . . of elements of R1 there exists S ∈ R1 such that for all i < ω, S \ ri (S) ⊆ Si . Proof. Let S0 ∗ ≥ S1 ∗ ≥ S2 . . . be a decreasing sequence in R1 . Hence, there is a strictly increasing sequence (ki )i<ω of natural numbers such that for all i < ω and all j < i, Si+1 \ rki (Si+1 ) ⊆ Sj . For each i < ω, let i ,Si+1 ) S(i) be an element of Si+1 \r(k . Let S = i<ω S(i). Then S ∈ R1 and for all i < ω, S \ri (S) ⊆ Si . 2 T (i) 1
Lemma 15. For all S, T ∈ R1 there exists S ≤ S such that either S ≤ T or S ∩ T = { }. Proof. Define the partition {Π0 , Π1 } of T T 1(0) by letting Π0 = {U ∈ T T 1(0) : [U ] ∈ [T ]} and Π1 = 1 1 T 1 \ Π0 . By Lemma 12, there exists S ∈ TS and j < 2 such that T S(0) ⊆ Πj . If j = 0 then T1 (0) 1 1 [S ] = U ∈ S [U ] ⊆ [T ]. So, S ≤ T . On the other hand, if j = 1 then [T ] ∩ [S ] = [T ] ∩ U ∈ S [U ] = ∅. T1 (0)
So, S ∩ T = { }. 2
T1 (0)
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Fig. 2. Graph of S ∈ R 1 and Γ(S) ∈ R1 .
Lemma 16. Let (S0 , S1 , S2 , . . . ) be a sequence of elements of R1 . For all T ∈ R1 there exists S ≤ T such that one of the following holds: ∃i < ω, Si ∩ S = { },
(16)
∀i < ω, S \ ri (S) ⊆ Si .
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Proof. Without loss of generality we can assume that for all S ≤ T and for all i < ω, Si ∩ S = { }. By Lemma 15 there exists S0 ≤ T such that S0 ≤ S0 . By Lemma 15 applied again, there exists S1 ≤ S0 ≤ T such that S1 ≤ S1 . Repeatedly applying Lemma 15, there exists a sequence T ≥ S0 ≥ S1 ≥ S2 ≥ . . . such that for all i < ω, Si ≤ Si . By Lemma 14, there exists S such that for all i < ω, S \ ri (S) ⊆ Si ⊆ Si . Since r0 (S) = ∅, S = S \ r0 (S) ⊆ S0 ⊆ T . 2 Next we define maps γ and Γ that will be used to transfer an ultrafilter on [T1 ] generated by a subset of to an ultrafilter on [T1 ] generated by a subset of R1 .
R1
Definition 17. Let {t0 , t1 , t2 , . . . } and {s0 , s1 , s2 , . . . } be the lexicographically increasing enumeration of [T1 ] and [T1 ], respectively. Let γ : [T1 ] → [T1 ] such that for all i < ω, γ(si ) = ti .
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Γ(S) = cl(γ [S]).
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Let Γ : R1 → R1 be the map given by
For a graphical interpretation of γ and Γ see Fig. 2.
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Remark 18. γ is bijective and Γ is injective but not surjective. Notice also that [T1 ] is in bijective corre spondence with T T 1(0) via the map 1
s ∈ [T1 ] → cl(γ
−1
T1 . T1 (0)
(s)) ∈
Theorem 19. (R1 , ≤∗ ) is σ-closed, and if G is a generic filter for (R1 , ≤∗ ) over some ground model V , then Γ G generates an ultrafilter on [T1 ] that is selective for R1 but not Ramsey for R1 in V [G]. Proof. By Lemma 14, (R1 , ≤∗ ) is a σ-closed partial order. Suppose that G is a generic filter for (R1 , ≤∗ ) and X ⊆ [T1 ]. By the previous remarks, X is in the ground model V . Since [T1 ] is in bijective correspondence with T T 1(0) , Lemma 12 shows that for each T ∈ R1 there exists S ∈ TT such that either γ [S] ⊆ X or 1 1 γ [S] ∩ X = ∅. Hence, ΔX = {S ∈ R1 : γ [S] ⊆ X or γ [S] ∩ X = ∅}
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is dense in (R1 , ≤∗ ). Since G is generic, for each X ⊆ [T1 ], G ∩ ΔX = ∅. In particular, for each X ⊆ [T1 ] there exists S ∈ G such that [γ(S)] ⊆ X or [γ(S)] ∩ X = ∅. Therefore Γ G generates an ultrafilter on [T1 ]. Let V1 denote the ultrafilter on [T1 ] generated by Γ G. Let (S0 , S1 , . . . ) be a sequence of elements of R1 and Δ1 =
{S ∈ R1 : S ∩ Si = { }},
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{S ∈ R1 : S \ ri (S) ⊆ Si }.
(22)
i<ω
Δ2 =
i<ω
Let Δ(S0 ,S1 ,... ) = Δ1 ∪ Δ2 . Since (R1 , ≤∗ ) is σ-closed, Δ(S0 ,S1 ,... ) is in the ground model V . By Lemma 16, Δ(S0 ,S1 ,... ) is dense in (R1 , ≤∗ ). So, G ∩ Δ(S0 ,S1 ,... ) = ∅. Claim 20. V1 is selective for R1 . Proof. Suppose that the sequence Γ(S0 ) ≥ Γ(S1 ) ≥ Γ(S2 ) . . . consists of members Γ G. In this case, G ∩ Δ1 = ∅. Thus, there exists S ∈ G ∩ Δ2 . That is, for all i < ω, Γ(S) \ ri (Γ(S)) ⊆ Γ(Si ). 2 Next we construct a partition of r2T(T1 1 ) and show that the partition witnesses that V1 is not Ramsey T1 for R1 . For each s ∈ r2 (T1 ) , we let s and s denote the two lexicographically smallest elements of [s \r1 (s)]. Notice that for each s ∈ r2T(T1 1 ) the length of the longest common initial segment of γ −1 (s ) and γ −1 (s ) is either 1 or 2. For each j < 2, let Πj denote the set of all s ∈ r2T(T1 1 ) such that length of the longest common S is neither a subset of Π0 nor Π1 . initial segment of γ −1 (s ) and γ −1 (s ) is 1 + j. For each S ∈ Γ G, r2 (T 1) Therefore V1 is not a Ramsey for R1 ultrafilter on [T1 ]. 2 4. Conclusion We have shown that there is a topological Ramsey space R where the notions of selective for R and Ramsey for R as introduced by Mijares in [1] are not equivalent. The next natural question is the following: Is this phenomena pervasive among topological Ramsey spaces or is R1 just a pathological example? It is a straight forward exercise to extend the methods used in this paper to the topological Ramsey spaces Rn where n < ω introduced by Dobrinen and Todorcevic in [7]. For each space Rn it is possible to construct a
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second topological Ramsey space Rn in much the same way that R1 is constructed from R1 . Forcing with these spaces gives ultrafilters that are selective but not Ramsey for these spaces. Dobrinen and Todorcevic in [7] have also introduced the spaces Rα where ω ≤ α < ω1 . These spaces are constructed from trees of infinite height. In these cases it is possible to construct a modified version of Rα for ω ≤ α < ω1 . However, the methods used in this article fail for these spaces. In particular the next question remains open. Question 21. For α between ω and ω1 , are the notions of selective for Rα and Ramsey for Rα equivalent? For each positive integer n, let Hn denote the space introduced by Dobrinen, Mijares and Trujillo in [11]. The methods used in this paper can be used to construct topological Ramsey spaces (H )n that give rise to ultrafilters that are Ramsey but not selective for these spaces. However, the methods fail for the space Hω from [11]. So our next question also remains open. Question 22. Are the notions of selective for Hω and Ramsey for Hω equivalent? All the spaces studied by the author except the Ellentuck space either support ultrafilters which are selective but not Ramsey for the space or it is unknown if the notions of selective and Ramsey for the space are equivalent. In fact the following question is still open. Question 23. Is the Ellentuck space the only topological Ramsey space for which the notions of selective and Ramsey for the space are equivalent? A different approach to answering this type of question is proposed by Trujillo in [12]. It involves characterizing the Dedekind cuts in the nonstandard models of arithmetic that arise from ultrafilter mappings from selective and Ramsey ultrafilters for a given topological Ramsey space. If it can be shown that the two cuts arising from the two types of ultrafilters are different then it follows that the two notions of selective and Ramsey for the space are not equivalent. Interestingly, even though selective and Ramsey for R1 are not equivalent, it is shown in [12] that the Dedekind cuts that arise from selective and Ramsey for R1 ultrafilters are identical. In another direction, Di Prisco, Mijares and Nieto in [13] develop abstract local Ramsey theory and obtain many interesting results by considering a notion of selective ultrafilter that differs from the definitions found in this article and in the articles [1,2] and [7]. An added benefit of the approach is a natural generalization of the notion of a semiselective coideal to the setting of topological Ramsey spaces. References [1] J.G. Mijares, A notion of selective ultrafilter corresponding to topological Ramsey spaces, Math. Log. Q. 53 (2007) 255–267. [2] N. Dobrinen, S. Todorcevic, A new class of Ramsey-classification theorems and their application in the Tukey theory of ultrafilters, part 1, Trans. Am. Math. Soc. 366 (2014) 1659–1684. [3] E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symb. Log. (1974) 163–165. [4] D. Booth, Ultrafilters on a countable set, Ann. Math. Log. 2 (1970) 1–24. [5] A.R. Mathias, Happy families, Ann. Math. Log. 12 (1977) 59–111. [6] I. Farah, Semiselective coideals, Mathematika 45 (1998) 79–103. [7] N. Dobrinen, S. Todorcevic, A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, part 2, Trans. Am. Math. Soc. 367 (2015) 4627–4659. [8] C. Laflamme, Forcing with filters and complete combinatorics, Ann. Pure Appl. Log. 42 (1989) 125–163. [9] J.G. Mijares, On Galvins lemma and Ramsey spaces, Ann. Comb. 16 (2012) 319–330. [10] S. Todorcevic, Introduction to Ramsey Spaces (AM-174), Princeton University Press, 2010. [11] N. Dobrinen, J.G. Mijares, T. Trujillo, Topological Ramsey spaces from Fraisse classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points, arXiv preprint, arXiv:1401.8105, 2014. [12] T. Trujillo, Ramsey for R1 ultrafilter mappings and their Dedekind cuts, Math. Log. Q. 61 (2015) 263–273. [13] C. Di Prisco, J.G. Mijares, J. Nieto, Local Ramsey theory, an abstract approach, 2015.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Selective but not Ramsey Timothy Trujillo Colorado School of Mines, Department of Applied Mathematics and Statistics, Golden, CO 80401-1887, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 19 September 2013 Received in revised form 25 October 2015 Accepted 31 December 2015 Available online 12 January 2016
We give a partial answer to the following question of Dobrinen: For a given topological Ramsey space R, are the notions of selective for R and Ramsey for R equivalent? Every topological Ramsey space R has an associated notion of Ramsey ultrafilter for R and selective ultrafilter for R (see [1]). If R is taken to be the Ellentuck space then the two concepts reduce to the familiar notions of Ramsey and selective ultrafilters on ω; so by a well-known result of Kunen the two are equivalent. We give the first example of an ultrafilter on a topological Ramsey space that is selective but not Ramsey for the space. We show that for the topological Ramsey space R1 from [2], the notions of selective for R1 and Ramsey for R1 are not equivalent. In particular, we prove that forcing with a closely related topological Ramsey space using almost-reduction, adjoins an ultrafilter that is selective but not Ramsey for R1 . © 2016 Elsevier B.V. All rights reserved.
MSC: 05D10 03E02 54D80 03E05 Keywords: Ramsey space Selective ultrafilter
1. Introduction This paper is concerned with giving an example of a topological Ramsey space R and an ultrafilter that is selective for R but not Ramsey for R. The first result of topological Ramsey theory was the infinite dimensional extension of the Ramsey theorem known as the Ellentuck theorem (see [3]). Ellentuck proved this theorem in order to give a proof of Silver’s theorem stating that analytic sets have the Ramsey property. In order to state the Ellentuck theorem it is necessary to introduce the Ellentuck space. We denote the infinite subsets of ω by [ω]ω and the finite subsets of ω by [ω]<ω . If B ∈ [ω]ω and {b0 , b1 , b2 , . . . } is its increasing enumeration, then for each i < ω, we let ri (B) denote the set {b0 , b1 , b2 , . . . , bi−1 } and call it the ith approximation of B. The Ellentuck space is the set [ω]ω of all infinite subsets of ω with the topology generated by the basic open sets, [a, B] = {A ∈ [ω]ω : A ⊆ B & (∃i)ri (A) = a} where a ∈ [ω]<ω and B ∈ [ω]ω . E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2015.12.088 0166-8641/© 2016 Elsevier B.V. All rights reserved.
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Recall that a subset of a topological space is nowhere dense if its closure has empty interior and meager if it is the countable union of nowhere dense sets. A subset X of a topological space has the Baire property if and only if X = OΔM for some open set O and some meager set M. A subset X of the Ellentuck space is Ramsey if for every ∅ = [a, A], there is a B ∈ [a, A] such that [a, B] ⊆ X or [a, B] ∩ X = ∅. The next theorem is the infinite-dimensional version of the Ramsey theorem. Theorem 1 (Ellentuck theorem). ([3]) Every subset of the Ellentuck space with the Baire property is Ramsey. Topological Ramsey spaces are spaces that have enough structure in common with the Ellentuck space that an abstract version of the Ellentuck theorem can be stated and proved. The Ellentuck space leads naturally to the notion of a selective ultrafilter on ω. Definition 2. Let U be a nonprincipal ultrafilter on ω. If i < ω and A is an infinite subset of ω, i.e. in the Ellentuck space, then we let A/i = A \ ri (A).
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U is selective, if for each decreasing sequence A0 ⊇ A1 ⊇ . . . of members of U there exists X = {x0 , x1 , . . . } ∈ U enumerated in increasing order such that for all i < ω, A/i ⊆ Ai .
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The next theorem, due to Kunen, characterizes selective ultrafilters as those which are minimal with respect to the Rudin–Keisler ordering. Theorem 3. ([4]) Let U be an ultrafilter on ω. The following conditions are equivalent: 1. U is selective. 2. For each partition of the two-element subsets of ω into two parts, there is a set X ∈ U all of whose two-element subsets lie in one part of the partition. 3. Every function on ω is constant or one-to-one on some set in U. An ultrafilter that satisfies the second item is called a Ramsey ultrafilter on ω. Generalizations of the previous theorem have been studied in many contexts. For example, the notions of selective coideal (see [5]) and semiselective coideals (see [6]) have been shown to also satisfy similar Ramsey properties. In [1], Mijares generalizes the notion of selective ultrafilter on ω to a notion of selective ultrafilter on an arbitrary topological Ramsey space R. Mijares also generalizes the notion of Ramsey ultrafilter on ω to a notion of Ramsey ultrafilter for R and shows that if an ultrafilter is Ramsey for R then it is also selective for R. If one takes R to be the Ellentuck space then the two generalizations reduce to the concepts of selective and Ramsey ultrafilter. The theorem of Kunen above shows that the notions of selective for the Ellentuck space and Ramsey for the Ellentuck space are equivalent. This leads to the following question asked by Dobrinen about the generalizations from selective and Ramsey to arbitrary topological Ramsey spaces. Question 4. For a given topological Ramsey space R, are the notions of selective for R and Ramsey for R equivalent? Ramsey for R ultrafilters have also been studied by Dobrinen and Todorcevic in [2] and [7]. Motivated by Tukey classification problems, the authors develop a hierarchy of topological Ramsey spaces Rα , α ≤ ω1 . Associated to each space Rα is an ultrafilter Uα , which is Ramsey for Rα . The space R0 is taken to be the
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Ellentuck space; therefore, Ramsey for R0 is equivalent to selective for R0 . We show that there is a triple (R1 , ≤, r) such that forcing with the space using almost-reduction, adjoins an ultrafilter that is selective for R1 but not Ramsey for R1 . In this article we use the methods of forcing but all of our constructions can be carried out using CH or MA. We work with σ-closed partial orders and all of the constructions only require 2ℵ0 conditions to be met. For example, assuming CH we can guarantee the conditions hold at successor stages and use σ-closure at limit stages. The author would like to thank Natasha Dobrinen and the two referees that carefully read the article for their valuable comments and suggestions that helped make this article and its proofs more readable. 2. Partition relation notation for trees on ω For each set X, we let X <ω denote the collection of all finite sequences of elements of X. For each finite sequence s, we let |s| denote the length of s. For each i ≤ |s|, πi (s) denotes the sequence of the first i + 1 elements of s and si denotes the (i + 1)th element of the sequence. For each pair of sequences s and t, we say that s is an initial segment of t and write s t if there exists i ≤ |t| such that s = πi (t). The closure of T ⊆ X <ω (denoted by cl(T )) is the set of all initial segments of elements of T . A subset T of X <ω is a tree on X, if cl(T ) = T . A maximal node of T , is a sequence s in T such that for each t ∈ T , s t ⇒ s = t. We let [T ] denote the set of all maximal nodes of T . The height of T is the smallest ordinal greater than or equal to the length of each element of T . The lexicographical order of ω <ω is defined as follows: s is lexicographically less than t if and only if s t or |s| = |t| and the least i on which s and t disagree, si ≤ ti . If S and T are trees on ω, then S is isomorphic to T , if there exists a bijection h : S → T which preserves the lexicographic ordering. A subtree of T is a tree S such that S ⊆ T . If S is a subtree of T then we will write S ≤ T . Given two trees S and T on ω, we let TS denote the set of all subtrees of T that are isomorphic to S. Suppose S, T and U are trees on ω, the partition relation T → (S)U means that for each partition of the partition.
T U
into two parts there exists V ∈
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such that
V U
lies in one part of
3. Selective but not Ramsey for R1 We begin this section with the definition of the triple (R1 , ≤, r). This space was first defined by Dobrinen and Todorcevic in [2]. The construction of R1 in [2] was inspired and motivated by the work of Laflamme in [8] which uses forcing to adjoin a weakly-Ramsey ultrafilter satisfying complete combinatorics over HOD(R). The purpose of this section is to introduce a closely related triple (R1 , ≤, r) and show that forcing with R1 using almost-reduction, adjoins an ultrafilter that is selective but not Ramsey for R1 . Definition 5 ((R1 , ≤, r)). ([2]) For each i < ω, let
T1 (i) = Let T1 =
i<ω
(i + 1)(i + 2) i(i + 1) ≤j<
, i , i, j : 2 2
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T1 (i) and (R1 , ≤, r) denote the triple (R(T1 ), ≤, r). Fig. 1 includes a graph of the tree T1 .
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Fig. 1. Graph of T1 and T1 .
Let R1 denote the set of all subtrees of T1 isomorphic to T1 , i.e. TT11 . For each S ∈ R1 we let { kiS : i < ω} denote the lexicographically increasing enumeration of π0 [S]. For each i < ω, let S(i) = cl({s ∈ [S] : π0 (s) = kiS }), ri (S) = S(j).
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j
Let AR1 = i<ω {ri (S) : S ∈ R1 } and define r : ω × R1 → AR1 by letting r(i, S) = ri (S). Here we let ≤ denote the subtree relation. For S, S ∈ R1 almost-reduction is defined as follows: S ≤∗ S if and only if there exists i < ω such that S \ ri (S) ⊆ S . The Ellentuck topology on R1 is the topology generated by the sets, [a, B] = {A ∈ R1 : A ⊆ B & (∃i)ri (A) = a}
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where a ∈ AR1 and B ∈ R1 . A subset X of R1 is Ramsey if for every ∅ = [a, A], there is a B ∈ [a, A] such that [a, B] ⊆ X or [a, B] ∩ X = ∅. The next result is Theorem 3.9 of Dobrinen and Todorcevic in [2]. Theorem 6. ([2]) (R1 , ≤, r) forms a topological Ramsey space. That is, every subset of R1 with the Baire property with respect to the Ellentuck topology on R1 is Ramsey. Next following Dobrinen and Todorcevic in [2] we introduce a generalization of the notion of Ramsey and selective for R1 . Definition 7. Let U be an ultrafilter on [T1 ]. 1. We say that U is generated by G ⊆ R1 , if {[S] : S ∈ G} is cofinal in (U, ⊇). 2. An ultrafilter U generated by G ⊆ R1 is selective for R1 if and only if for each decreasing sequence S0 ≥ S1 ≥ S2 ≥ . . . of elements of G, there exists another S ∈ G such that for all i < ω, S \ ri (S) ⊆ Si .
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3. An ultrafilter U generated by G ⊆ R1 is Ramsey for R1 if and only if for each i < ω and each partition S of riT(T11 ) into two parts there exists S ∈ G such that ri (T lies in one part of the partition. 1) The next result addresses the existence of Ramsey ultrafilters for R1 . We omit its proof as it follows by applying Lemma 3.3 of Mijares in [1] to topological Ramsey spaces of the form R(T ). Theorem 8. ([1]) Forcing with (R1 , ≤∗ ) adjoins no new elements of (AR1 )ω , and if G is a (R1 , ≤∗ )-generic filter over some ground model V , then G generates a Ramsey for R1 ultrafilter in V [G]. The next result is a consequence of Theorem 3.5 of Mijares in [9] applied to the topological Ramsey space R1 . Lemma 9. ([9]) For each pair of positive integers k and n with k ≤ n there exists m < ω such that rm (T1 ) → (rn (T1 ))rk (T1 ) .
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Next we define R1 . The space is constructed in a similar way to R1 using a modified version of the tree T1 . The modified tree T1 has height 3 and for each i < ω, the maximal nodes of T1 (i) are in one-to-one correspondence with the maximal nodes of T1 (i). These two properties of T1 are used below to show that forcing with R1 using almost-reduction adjoins an ultrafilter that is selective but not Ramsey for R1 . The one-to-one correspondence is used to show that the adjoined ultrafilter is selective for R1 and the extra level of T1 is used to show that the adjoined ultrafilter fails to be Ramsey for R1 . Definition 10 ((R1 , ≤, r)). For each i < ω, let T1 (i) = cl({ i, j, k : k ≤ i & j, k ∈ T1 }).
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Let T1 = i<ω T1 (i) and (R1 , ≤, r) denote the triple (R(T1 ), ≤, r). Fig. 1 includes a graph of the tree T1 . Let R1 denote the set of all subtrees of T1 isomorphic to T1 , i.e. TT1 . For each S ∈ R1 we let { kiS : 1 i < ω} denote the lexicographically increasing enumeration of π0 [S]. For each i < ω, let S(i) = cl({s ∈ [S] : π0 (s) = kiS }), S(j). ri (S) =
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j
Let AR1 = i<ω {ri (S) : S ∈ R1 } and define r : ω × R1 → AR1 by letting r(i, S) = ri (S). Here we let ≤ denote the subtree relation. For S, S ∈ R1 almost-reduction is defined as follows: S ≤∗ S if and only if there exists i < ω such that S \ ri (S) ⊆ S . The next two partition properties are needed to show that forcing with (R1 , ≤∗ ) adjoins an ultrafilter on [T1 ]. In fact, all that is needed is the case for k = 0. However, we prove the more general lemma as it is the crux of the proof that R1 has enough in common with the Ellentuck space that it forms a topological Ramsey space. Lemma 11. For each pair of positive integers k and n with k ≤ n, there exists m < ω such that
T1 (m) → (T1 (n))T1 (k) .
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Proof. For each i < ω, let i be the smallest natural number such that for all j ≥ i , T1 (i) is isomorphic to a subtree of cl({ j s : s ∈ ri (T1 )}). By Lemma 9 there exists m < ω such that rm (T1 ) → (rn (T1 ))T1 (k ) . Note that it must be the case that m is greater than both k and n . 1) . For each j < 2, let Πj = {S ∈ rrm (T : cl({ m s : s ∈ Suppose that {Π0 , Π1 } is a partition of TT1(m) k (T1 ) 1 (k) ˆ ∈ Π } where [S] ˆ consists of the lexicographically first k elements of [S]. {Π , Π } forms a partition of [S]}) 0 1 rm (T1 ) j rm (T1 ) S . Hence, there exists j < 2 and S ∈ such that ⊆ Π . If we let S := cl({ m s : j rk (T1 ) rk (T1 ) rn (T1 ) ˆ ˆ consists of the lexicographically first n elements of [S], then S ∈ T1(m) , and ∈ Πj where [S] s ∈ [S]}) T1 (n) S T (k) ⊆ Πj . 2 1
Lemma 12. For each positive integer k,
T1 → (T1 )T1 (k) .
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Proof. Lemma 11 shows that there exists a strictly increasing sequence (mi )i<ω such that for each i < ω, 1 and (S0 , S1 , . . . ) be a sequence of trees such T1 (mi ) → (T1 (i))T1 (k) . Let {Π0 , Π1 } be a partition of T T (k) 1 T Si that for each i < ω, Si ∈ T 1(i) and T (k) is contained in one piece of the partition {Π0 , Π1 }. By the 1 1 pigeonhole principle there exists j < 2 and a strictly increasing sequence (i0 , i1 , . . . ) such that for all l < ω, T (mi ) Si l Sil ∈ 1T (i)l and T (k) ⊆ Πj . Let S = l<ω Sil . If S is any element of TS then T S(k) ⊆ Πj . 2 1
1
1
1
Remark 13. The Ellentuck topology on R1 is the topology generated by the sets, [a, B] = {A ∈ R1 : A ⊆ B & (∃i)ri (A) = a}
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where a ∈ AR1 and B ∈ R1 . A subset X of R1 is Ramsey if for every ∅ = [a, A], there is a B ∈ [a, A] such that [a, B] ⊆ X or [a, B] ∩ X = ∅. It can be shown that (R1 , ≤, r) forms a topological Ramsey space. That is, every subset of R1 with the Baire property with respect to the Ellentuck topology is Ramsey. The proof uses the Abstract Ellentuck Theorem from [10]. It is enough to show that (R1 , ≤, r) satisfies A.1–A.4 from the statement of the Abstract Ellentuck Theorem in [10] and forms a closed subspace of (AR1 )ω . The proof that R1 is a closed subspace of (AR1 )ω and satisfies axioms A.1–A.3 follows by trivial modifications to the proofs of the same facts for the space R1 in [2]. By definition of R1 , A.4 from [10] is equivalent to Lemma 12. Hence A.4 holds for R1 . By the Abstract Ellentuck Theorem, R1 forms a topological Ramsey space. Lemma 14. For each sequence S0 ∗ ≥ S1 ∗ ≥ S2 . . . of elements of R1 there exists S ∈ R1 such that for all i < ω, S \ ri (S) ⊆ Si . Proof. Let S0 ∗ ≥ S1 ∗ ≥ S2 . . . be a decreasing sequence in R1 . Hence, there is a strictly increasing sequence (ki )i<ω of natural numbers such that for all i < ω and all j < i, Si+1 \ rki (Si+1 ) ⊆ Sj . For each i < ω, let i ,Si+1 ) S(i) be an element of Si+1 \r(k . Let S = i<ω S(i). Then S ∈ R1 and for all i < ω, S \ri (S) ⊆ Si . 2 T (i) 1
Lemma 15. For all S, T ∈ R1 there exists S ≤ S such that either S ≤ T or S ∩ T = { }. Proof. Define the partition {Π0 , Π1 } of T T 1(0) by letting Π0 = {U ∈ T T 1(0) : [U ] ∈ [T ]} and Π1 = 1 1 T 1 \ Π0 . By Lemma 12, there exists S ∈ TS and j < 2 such that T S(0) ⊆ Πj . If j = 0 then T1 (0) 1 1 [S ] = U ∈ S [U ] ⊆ [T ]. So, S ≤ T . On the other hand, if j = 1 then [T ] ∩ [S ] = [T ] ∩ U ∈ S [U ] = ∅. T1 (0)
So, S ∩ T = { }. 2
T1 (0)
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Fig. 2. Graph of S ∈ R 1 and Γ(S) ∈ R1 .
Lemma 16. Let (S0 , S1 , S2 , . . . ) be a sequence of elements of R1 . For all T ∈ R1 there exists S ≤ T such that one of the following holds: ∃i < ω, Si ∩ S = { },
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∀i < ω, S \ ri (S) ⊆ Si .
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Proof. Without loss of generality we can assume that for all S ≤ T and for all i < ω, Si ∩ S = { }. By Lemma 15 there exists S0 ≤ T such that S0 ≤ S0 . By Lemma 15 applied again, there exists S1 ≤ S0 ≤ T such that S1 ≤ S1 . Repeatedly applying Lemma 15, there exists a sequence T ≥ S0 ≥ S1 ≥ S2 ≥ . . . such that for all i < ω, Si ≤ Si . By Lemma 14, there exists S such that for all i < ω, S \ ri (S) ⊆ Si ⊆ Si . Since r0 (S) = ∅, S = S \ r0 (S) ⊆ S0 ⊆ T . 2 Next we define maps γ and Γ that will be used to transfer an ultrafilter on [T1 ] generated by a subset of to an ultrafilter on [T1 ] generated by a subset of R1 .
R1
Definition 17. Let {t0 , t1 , t2 , . . . } and {s0 , s1 , s2 , . . . } be the lexicographically increasing enumeration of [T1 ] and [T1 ], respectively. Let γ : [T1 ] → [T1 ] such that for all i < ω, γ(si ) = ti .
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Γ(S) = cl(γ [S]).
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Let Γ : R1 → R1 be the map given by
For a graphical interpretation of γ and Γ see Fig. 2.
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Remark 18. γ is bijective and Γ is injective but not surjective. Notice also that [T1 ] is in bijective corre spondence with T T 1(0) via the map 1
s ∈ [T1 ] → cl(γ
−1
T1 . T1 (0)
(s)) ∈
Theorem 19. (R1 , ≤∗ ) is σ-closed, and if G is a generic filter for (R1 , ≤∗ ) over some ground model V , then Γ G generates an ultrafilter on [T1 ] that is selective for R1 but not Ramsey for R1 in V [G]. Proof. By Lemma 14, (R1 , ≤∗ ) is a σ-closed partial order. Suppose that G is a generic filter for (R1 , ≤∗ ) and X ⊆ [T1 ]. By the previous remarks, X is in the ground model V . Since [T1 ] is in bijective correspondence with T T 1(0) , Lemma 12 shows that for each T ∈ R1 there exists S ∈ TT such that either γ [S] ⊆ X or 1 1 γ [S] ∩ X = ∅. Hence, ΔX = {S ∈ R1 : γ [S] ⊆ X or γ [S] ∩ X = ∅}
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is dense in (R1 , ≤∗ ). Since G is generic, for each X ⊆ [T1 ], G ∩ ΔX = ∅. In particular, for each X ⊆ [T1 ] there exists S ∈ G such that [γ(S)] ⊆ X or [γ(S)] ∩ X = ∅. Therefore Γ G generates an ultrafilter on [T1 ]. Let V1 denote the ultrafilter on [T1 ] generated by Γ G. Let (S0 , S1 , . . . ) be a sequence of elements of R1 and Δ1 =
{S ∈ R1 : S ∩ Si = { }},
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{S ∈ R1 : S \ ri (S) ⊆ Si }.
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i<ω
Δ2 =
i<ω
Let Δ(S0 ,S1 ,... ) = Δ1 ∪ Δ2 . Since (R1 , ≤∗ ) is σ-closed, Δ(S0 ,S1 ,... ) is in the ground model V . By Lemma 16, Δ(S0 ,S1 ,... ) is dense in (R1 , ≤∗ ). So, G ∩ Δ(S0 ,S1 ,... ) = ∅. Claim 20. V1 is selective for R1 . Proof. Suppose that the sequence Γ(S0 ) ≥ Γ(S1 ) ≥ Γ(S2 ) . . . consists of members Γ G. In this case, G ∩ Δ1 = ∅. Thus, there exists S ∈ G ∩ Δ2 . That is, for all i < ω, Γ(S) \ ri (Γ(S)) ⊆ Γ(Si ). 2 Next we construct a partition of r2T(T1 1 ) and show that the partition witnesses that V1 is not Ramsey T1 for R1 . For each s ∈ r2 (T1 ) , we let s and s denote the two lexicographically smallest elements of [s \r1 (s)]. Notice that for each s ∈ r2T(T1 1 ) the length of the longest common initial segment of γ −1 (s ) and γ −1 (s ) is either 1 or 2. For each j < 2, let Πj denote the set of all s ∈ r2T(T1 1 ) such that length of the longest common S is neither a subset of Π0 nor Π1 . initial segment of γ −1 (s ) and γ −1 (s ) is 1 + j. For each S ∈ Γ G, r2 (T 1) Therefore V1 is not a Ramsey for R1 ultrafilter on [T1 ]. 2 4. Conclusion We have shown that there is a topological Ramsey space R where the notions of selective for R and Ramsey for R as introduced by Mijares in [1] are not equivalent. The next natural question is the following: Is this phenomena pervasive among topological Ramsey spaces or is R1 just a pathological example? It is a straight forward exercise to extend the methods used in this paper to the topological Ramsey spaces Rn where n < ω introduced by Dobrinen and Todorcevic in [7]. For each space Rn it is possible to construct a
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second topological Ramsey space Rn in much the same way that R1 is constructed from R1 . Forcing with these spaces gives ultrafilters that are selective but not Ramsey for these spaces. Dobrinen and Todorcevic in [7] have also introduced the spaces Rα where ω ≤ α < ω1 . These spaces are constructed from trees of infinite height. In these cases it is possible to construct a modified version of Rα for ω ≤ α < ω1 . However, the methods used in this article fail for these spaces. In particular the next question remains open. Question 21. For α between ω and ω1 , are the notions of selective for Rα and Ramsey for Rα equivalent? For each positive integer n, let Hn denote the space introduced by Dobrinen, Mijares and Trujillo in [11]. The methods used in this paper can be used to construct topological Ramsey spaces (H )n that give rise to ultrafilters that are Ramsey but not selective for these spaces. However, the methods fail for the space Hω from [11]. So our next question also remains open. Question 22. Are the notions of selective for Hω and Ramsey for Hω equivalent? All the spaces studied by the author except the Ellentuck space either support ultrafilters which are selective but not Ramsey for the space or it is unknown if the notions of selective and Ramsey for the space are equivalent. In fact the following question is still open. Question 23. Is the Ellentuck space the only topological Ramsey space for which the notions of selective and Ramsey for the space are equivalent? A different approach to answering this type of question is proposed by Trujillo in [12]. It involves characterizing the Dedekind cuts in the nonstandard models of arithmetic that arise from ultrafilter mappings from selective and Ramsey ultrafilters for a given topological Ramsey space. If it can be shown that the two cuts arising from the two types of ultrafilters are different then it follows that the two notions of selective and Ramsey for the space are not equivalent. Interestingly, even though selective and Ramsey for R1 are not equivalent, it is shown in [12] that the Dedekind cuts that arise from selective and Ramsey for R1 ultrafilters are identical. In another direction, Di Prisco, Mijares and Nieto in [13] develop abstract local Ramsey theory and obtain many interesting results by considering a notion of selective ultrafilter that differs from the definitions found in this article and in the articles [1,2] and [7]. An added benefit of the approach is a natural generalization of the notion of a semiselective coideal to the setting of topological Ramsey spaces. References [1] J.G. Mijares, A notion of selective ultrafilter corresponding to topological Ramsey spaces, Math. Log. Q. 53 (2007) 255–267. [2] N. Dobrinen, S. Todorcevic, A new class of Ramsey-classification theorems and their application in the Tukey theory of ultrafilters, part 1, Trans. Am. Math. Soc. 366 (2014) 1659–1684. [3] E. Ellentuck, A new proof that analytic sets are Ramsey, J. Symb. Log. (1974) 163–165. [4] D. Booth, Ultrafilters on a countable set, Ann. Math. Log. 2 (1970) 1–24. [5] A.R. Mathias, Happy families, Ann. Math. Log. 12 (1977) 59–111. [6] I. Farah, Semiselective coideals, Mathematika 45 (1998) 79–103. [7] N. Dobrinen, S. Todorcevic, A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters, part 2, Trans. Am. Math. Soc. 367 (2015) 4627–4659. [8] C. Laflamme, Forcing with filters and complete combinatorics, Ann. Pure Appl. Log. 42 (1989) 125–163. [9] J.G. Mijares, On Galvins lemma and Ramsey spaces, Ann. Comb. 16 (2012) 319–330. [10] S. Todorcevic, Introduction to Ramsey Spaces (AM-174), Princeton University Press, 2010. [11] N. Dobrinen, J.G. Mijares, T. Trujillo, Topological Ramsey spaces from Fraisse classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points, arXiv preprint, arXiv:1401.8105, 2014. [12] T. Trujillo, Ramsey for R1 ultrafilter mappings and their Dedekind cuts, Math. Log. Q. 61 (2015) 263–273. [13] C. Di Prisco, J.G. Mijares, J. Nieto, Local Ramsey theory, an abstract approach, 2015.