Abstract. In [6] W. T. Gowers formulated and proved a Ramsey- type result which lies at the heart of his famous dichotomy for Banach spaces. He defines the ...
CODING INTO RAMSEY SETS ´ JORDI LOPEZ-ABAD Abstract. In [6] W. T. Gowers formulated and proved a Ramseytype result which lies at the heart of his famous dichotomy for Banach spaces. He defines the notion of weakly Ramsey set of block sequences of an infinite dimensional Banach space and shows that every analytic set of block sequences is weakly Ramsey. We show here that Gowers’ result follows quite directly from the fact that all Gδ sets are weakly Ramsey, if the Banach space does not contain c0 , and from the fact that all Fσδ sets are weakly Ramsey, in the case of an arbitrary Banach space. We also show that every result obtained by the application of Gowers’ theorem to an analytic set can also be obtained by applying the Theorem to a Fσδ set (or to a Gδ set if the space does not contain c0 ). This fact explains why the only known applications of this technique are based on very low-ranked Borel sets (open, closed, Fσ , or Gδ ).
Version: October 2003
1. Introduction Let E = (E, k · k) be an infinite dimensional Banach space with a fixed Schauder basis (en )n . We write S(E) to denote the unit sphere of E. Recall the following definitions and notations from [6]: The support of a vector x is the set supp x = {m : e∗m (x) 6= 0}. Let Σ(E) be the set of infinite sequences ((xn , λn ))n , where kxn k = 1, xn has finite support, max supp xn < min supp xn+1 , and λn ∈ [0, 1] for every n. A sequence ((xn , λn ))n is called a block sequence of E. Notice that Σ(E) ⊆ (S(E) × [0, 1])N is a closed subset and hence Σ(E) is a Polish space. For a block sequence X = ((xn , λn ))n , let [X] be the set of block subsequences of X, i.e., [X] = {((yn , µn ))n ∈ Σ(E) : ym ∈ hxn in for all m}, where hxn in denotes the linear span of {xn }n . The subset of [X] consisting of all sequences ((yn , µn ))n where µn = 1 for every n is denoted by [X]1 . Given a set of block sequences σ and X ∈ Σ(E), we say that σ is large for [X] iff σ ∩ [Y ] 6= ∅ for every Y ∈ [X]. For σ a set of block sequences, the game aσ [X] is defined as follows: There are two players, I and II. I 2000 Mathematics Subject Classification. Primary 03E15,46B25; Secondary 05D10. This Research was supported through a European Community Marie Curie Fellowship. 1
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starts playing Y0 ∈ [X], then II chooses x0 ∈ Y0 and some λ0 ∈ [0, 1]. Then player I plays Y1 ∈ [X], and II chooses y1 ∈ Y1 with y1 > y0 , and λ1 ∈ [0, 1], and so on. Notice that when we write y ∈ Y = ((yn , λn ))n we understand that y belongs to the unit sphere of the linear span of {yn }n . II wins the game iff ((yn , λn ))n ∈ σ. Otherwise I wins. A strategy for II (in X) is a function Φ : [X]f × [X] → X × [0, 1] satisfying that Φ(((y0 , λ0 ), . . . , (yn−1 , λn−1 )), Y ) ∈ Y × [0, 1], where [X]f denotes the set of finite block subsequences ((y0 , µ0 ), . . . , (yn , µn ))n of X, i.e., yi ∈ hXi, max supp yi < min supp yi+1 , and µi ∈ [0, 1] for every 0 ≤ i ≤ n. A strategy S for player II is a winning strategy if whenever II plays according to S, then he wins the game. The set σ is strategically large for [X] iff Player II has a winning strategy for aσ [X]. σ is called weakly Ramsey iff for every X ∈ Σ(E), if σ is large for [X] then for every ∆ > 0 there is some Y ∈ [X] such that σ∆ is strategically large for σ(Y ), where for a decreasing sequence of strictly positive reals ∆ = (δn )n , the ∆-expansion σ∆ of σ is the set of block sequences Y = ((yn , λn ))n such that there is some X = ((xn , µn ))n ∈ σ such that max{kxn − yn k, |λn − µn |} ≤ δn for every n, which is denoted in short by d(X, Y ) ≤ ∆. Recall the following result due to W. T. Gowers. Theorem 1.1. [6] Every analytic set of block sequences is weakly Ramsey. ¤ This theorem has several consequences regarding the geometry of Banach spaces. Maybe the best known is Gowers’ dichotomy: Every infinitedimensional Banach space has a subspace which either has an unconditional basis or is hereditarily indecomposable. However, all the known consequences of Theorem 1.1 do not use the full strength of the result, but only that the Gδ sets are weakly Ramsey. One of the aims of this paper is to determine if there are applications which need the full strength of the theorem. We show that every result that is obtained by applying Theorem 1.1 to an analytic set of block sequences can also be obtained from the fact that a very simple Borel set (Gδ sets if the space does not contain c0 and Fσδ for arbitrary spaces) is weakly Ramsey. Moreover, Gowers’ Theorem 1.1 itself follows quite directly from the fact that these Borel sets are weakly Ramsey, which also gives an alternative proof of the Theorem. Let us make some comments about why this phenomenon occurs. Let Σ1 (E) be the set of normalized block sequences ((xn , 1))n , that we denote by (xn )n . A set A ⊆ S(hen in ) is called asymptotic iff A ∩ S(hxn in ) 6= ∅ for every block sequence (xn )n in Σ(E). For example the sets Ai = {x ∈ S(hen in ) : (−1)i e∗min supp x (x) > 0}
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are clearly asymptotic for i = 0, 1. These sets are responsible for the fact that the stronger Ramsey-like property defined for σ ⊆ Σ1 (E) by: “σ is Ramsey iff for every normalized block sequence X, if σ is large for [X]1 , then there is some normalized block subsequence Y of X such that [Y ]1 ⊆ σ” fails to be true even for Fσ sets of block sequences, since for example the set σ0 = {(xn )n ∈ Σ1 (E) : x0 ∈ A0 } and its complement σ1 = {(xn )n ∈ Σ1 (E) : x0 ∈ A1 } are both Fσ and large for every [X]. Notice that (σ0 )∆ = (σ1 )∆ = Σ1 (E) for every ∆ > 0, which suggests that the weaker form of Ramsey property defined by: “σ is almost-Ramsey iff for every X, if σ is large for [X]1 then for every ∆ > 0 there is some normalized block subsequence Y of X such that [Y ]1 ⊆ σ∆ ” could be good enough. This turns out to be case for the space c0 since it was shown by Gowers [6] that every analytic set of normalized block sequences of c0 is almost-Ramsey. However, the almost-Ramsey property fails to be true for open sets of normalized block sequences for spaces E which are not c0 -saturated, namely there must be some normalized block sequence X with no normalized block subsequence Y equivalent to c0 , and hence two asymptotic sets A0 and A1 of Y which are separated, i.e., d(A0 , A1 ) = δ, some δ > 0 (see [9]). Then the set σ0 = {(xn )n ∈ Σ1 (E) : x0 ∈ (A0 )δ/2 } is open and large for [Y ] but (σ0 )(δ/2,δ/2,... ) does not contain [Z]1 for any Z ∈ [Y ]1 . But the existence of two disjoint asymptotic sets (separated or not) is not only an obstacle to have a stronger Ramsey property but it also allows to “code information” as follows. Recall that an analytic set σ of block sequences is the first projection of a closed set C of Σ(E) × N ↑ where N ↑ denotes the space of strictly increasing sequences of natural numbers. In other words, to know whether a block sequence (xn )n is in σ we need to know that (((xn , λn ))n , (kn )n ) ∈ C for some sequence (kn )n of integers. We are going to use the existence of two asymptotic sets to code in a single block sequence ((x0 , λ0 ), (y0 , 1), (x1 , λ1 ), (y1 , 1), . . . ) both ((xn , λn ))n and (kn )n in such a way that the corresponding set τ of block sequences coding pairs in C is of low Borel complexity, depending on whether the asymptotic sets are separated or not. If they are separated, then τ is a Gδ set. And, roughly speaking, all the consequences derived from the fact that σ is weakly Ramsey (or almost Ramsey in the case of c0 ) will follow from the fact that τ is weakly Ramsey (respectively almost Ramsey in the case of c0 ). A further application of our coding technique for decreasing the topological complexity of a set of block sequences, is to show that it is not possible to prove that the class of weakly Ramsey sets is closed under complements. More precisely, we show that there is a coanalytic set which fails to have the weakly Ramsey property. As it is shown in the last Section, in order to
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provide such a set it is necessary to use some extra set-theoretical assumptions. A similar result holds for the almost Ramsey property in the case of c0 . 2. Basic definitions and results Definition 1. Given a finite block sequence s = ((x0 , λ0 ), . . . , (xk , λk )) (which can be empty) and an infinite one A = ((an , µn ))n , let [s; A] be the set of block sequences s a (bn )n = ((x0 , λ0 ), . . . , (xk , λk ), (b0 , ρ0 ), (b1 , ρ1 ), . . . ) such that xk < b0 and bn is in the unit sphere S(A) of A for every n. Notice that [∅; A] = [A]. In other words, [s, A] is the set of all block sequences that result from the “concatenation” of s with a block sequence of elements of A. lh(s) denotes the length of s, i.e., lh(s) = k+1 if s = ((x0 , λ0 ), . . . , (xk , λk )). Given a finite or infinite block sequence α = ((xn , λn ))n≤N (N ≤ ∞) let min α = x0 , and A \ s = ((an , λn ))n≥n0 where n0 is the minimal n such that a n > xk . Let the D-topology on Σ(E) be that with basic open sets [((x0 , λ0 ), . . . , (xk , λk )); E] = {((x0 , λ0 ), . . . , (xk , λk )) a ((an , µn ))n : a0 > xk }. Given a finite block sequence s, let cs : Σ(E) → Σ(E) be defined by cs (A) = s a (A \ s). Notice that cs is continuous. For σ ⊆ Σ(E), let σ s = c−1 s σ. The N -topology on Σ(E) is the the topology inherited from (S(E) × [0, 1])N . Given a set of block sequences σ and s < A, we say that σ is (strategically) large for [s; A] iff c−1 s σ is (strategically) large for [A]. If Φ is an strategy for Player II and (Yn )n is an infinite run for player I we denote by Φ ∗ (Yn )n the block sequence (Φ(Y0 ), Φ((Φ(Y0 )), Y1 ), Φ((Φ(Y0 ), Φ((Φ(Y0 )), Y1 )), Y2 ), . . . ) Definition 2. Given a block sequence X we denote by OX (σ) the union of str the basic D-open sets [t; X] such that σ is large for [t; X], and by OX (σ) the union of the basic D-open sets [t; X] such that σ is strategically large for [t; X]. For n ∈ N, let Onstr (σ) be the union of [t; X] such that |t| > n and σ is strategically large for [t; X]. Notice all these sets introduced here are D-open subsets of [X]. Proposition 2.1. Let σ ⊆ Σ, and ∆ > 0. Then for every block sequence X, str str 1. OX (σ)∆ ⊆ OX (σ∆ ) and OX (σ)∆ ⊆ OX (σ∆ ). str 2. For every Y ∈ [X], if OX (σ) is strategically large for [Y ], then σ is also strategically large for [Y ]. Proof. 1. is trivial. We show 2.: For suppose that for some Y ∈ [X] OX (σ)∆ is strategically large for [Y ], i.e., there is a winning strategy Φ for player II in the game aOX (σ)∆ [Y ]. We sketch a winning strategy for player II in the game aσ∆ [Y ]. Player II follows Φ until he obtains some s ∈ [Y ]f
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such that σ is strategically large for [s; X], and then he chooses a winning strategy Φ0 in the game aσ [s; X], hence for the game aσ [s; Y ]. After this point he plays according to Φ0 in order to reach some block sequence Z such that t a Z ∈ σ. ¤ Recall the following result from [6]. Proposition 2.2. Every D-open set of block sequences is weakly Ramsey. ¤ From this, one easily shows the following. str Proposition 2.3. Suppose that OX (σ) is large for [X]. Then for every ∆ > 0 there is some Y ∈ [X] such that σ∆ is strategically large for [Y ]. str str Proof. If OX (σ) is large for [X], since OX (σ) is a D-open subset of str [X], then OX (σ)∆ is strategically large for [Y ], for some Y ∈ [X], an by Proposition 2.1 we are done. ¤
Proposition 2.4. Let σ ⊆ Σ(E). The following conditions are equivalent: 1. σ is weakly Ramsey. 2. σ ∩ [X] is weakly Ramsey for every X ∈ Σ(E). 3. For every block sequence X there is a block subsequence Y ∈ [X] such that σ ∩ [Y ] is weakly Ramsey. Proof. 1 ⇒ 2. Fix a block sequence Y such that σ ∩ [X] is large for [Y ], and fix ∆ > 0. Then we can choose Z ∈ [Y ] ∩ (σ ∩ [X]). Since σ is large for [Z], there is some W ∈ [Z] such that σ∆ is strategically large for [W ], or equivalently (σ ∩ [W ])∆ is strategically large for [W ]. But (σ ∩ [W ])∆ ⊆ (σ ∩ [X])∆ , and hence (σ ∩ [X])∆ is strategically large for [W ]. 2 ⇒ 3 is trivial. Let us show that 3 ⇒ 1: For suppose that σ is large for [X]. Fix ∆ > 0. By assumption, we can find Y ∈ [X] such that σ ∩ [Y ] is weakly Ramsey. Since σ ∩ [Y ] is large for [Y ], there is Z ∈ [Y ] such that (σ ∩ [Y ])∆ is strategically large for [W ], hence σ∆ is strategically large for [W ]. ¤ In the case of normalized block sequences of c0 we have a stronger Ramsey-like result. Theorem 2.1. [6] Every analytic set of normalized block sequences of c0 is almost-Ramsey. ¤ A Banach space E is called c0 -saturated if every infinite dimensional closed subspace of E contains a closed subspace isomorphic to c0 . Equivalently, every normalized block sequence X ∈ Σ1 (E) contains a normalized block subsequence (yn )n which is equivalent to the natural basis of c0 .
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Corollary 2.1. If E is a c0 -saturated space then every analytic set of normalized block sequences of E is almost-Ramsey. Proof. Suppose E is c0 -saturated and σ is an analytic subset of Σ1 (E). Suppose that σ is large for some [X]. To simplify the notation we identify a normalized block sequence ((xn , 1))n with (xn )n . Since E is c0 we can find some block subsequence Y = (yn )n of X such that (yn )n is C-equivalent to the natural basis (vn )n of c0 for some C > 0, i.e., the natural function h : hyn i → c0 extending h(yn ) = vn is an isomorphism such that khk, kh−1 k ≤ C. This equivalence defines naturally a homeomorphism H : [Y ]1 → Σ1 (c0 ) defined by µ ¶ h(zn ) H((zn )n ) = (1) kh(zn )k∞ n Fix ∆ = (δn )n > 0. Since σ 0 = H(σ) ⊆ Σ1 (c0 ) is analytic and large for 0 [(vn )n ] we can find W ∈ Σ1 (c0 ) such that [W 0 ]1 ⊆ σ(δ , and hence n /C)n [H −1 (W )]1 ⊆ σ∆ . ¤ 3. Coding with asymptotic sets Definition 3. An asymptotic pair of E is a pair A = {A0 , A1 } where A0 and A1 are disjoint asymptotic sets of E. We denote by ΣA (E) the set of sequences ((xn , λn ))n such that x2n+1 ∈ A0 ∪ A1 and λ2n+1 = 1 for A every n. Let ΣA 1 (E) be the set of block sequences ((xn , λn ))n of Σ (E) such that x2n+1 ∈ A1 for infinitely many n’s. By definition, a sequence ((xn , λn ))n ∈ ΣA 1 (E) naturally codes the pair (((x2n , λ2n ))n , (kn )n ), where {kn }n is the increasing enumeration of the set {k : x2k+1 ∈ A1 }. Con↑ defined by sider the corresponding mapping ΛA : ΣA 1 (E) → Σ(E) × N ΛA (((xn , λn ))n ) = (((x2n , λ2n ))n , (kn )n ) for every ((xn , λn ))n ∈ ΣA 1 (E), where N ↑ denotes the set of strictly increasing sequences of positive integers as a topological subspace of the Baire space N = NN . Fix an asymptotic pair A of E. It is well known that every analytic (and in particular every Borel) subset σ of Σ(E) is the first projection of a closed subset of Σ(E) × N ↑ . Given a subset C ⊆ Σ(E) × N ↑ we denote the first projection of C by σ(C). Let A τA (C) = Λ−1 A C = {X ∈ Σ1 (E) : ΛA (X) ∈ C}
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Given a family C of subsets of Σ(E) × N ↑ we denote by σ(C) = {σ(C) : C ∈ C} and τA (C) = {τA (C) : C ∈ C}. For example, if C is the family of closed sets of Σ(E) × N ↑ then σ(C) is the point-class Σ11 of analytic subsets ∼ of Σ.
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Proposition 3.1. For every block sequence X there is some block subsequence Y of X such that A [Y ] × N ↑ ⊆ ΛA ”([X] ∩ ΣA 1 ) = {ΛA (Z) : Z ∈ [X] ∩ Σ1 }
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i.e., every pair (Z, ~ε) ∈ [Y ] × N ↑ is coded by some block subsequence of X. Proof. Fix a block sequence X = ((xn , λn ))n . Let n0 be the first integer n n−1 such that S(hxi in−1 i=0 ) ∩ A0 and S(hxi ii=0 ) ∩ A1 are non empty, which is well defined since A0 and A1 are asymptotic sets and, by definition, every element of them has finite support. Once defined nk , let nk+1 be the first integer n−1 n > nk such that S(hxi ii=n ) ∩ A0 and S(hxi in−1 i=nk +1 ) ∩ A1 are non empty. k +1 Set Y = ((xnk , 1))k . We are going to show that [Y ]×N ↑ ⊆ ΛA ”[X]∩ΣA 1 (E). Fix a block sequence Z = ((zn , λn ))n ∈ [Y ] and ~ε = (εn )n ∈ N ↑ . Let kr be the minimal integer k such that zr ∈ S(hxn0 , . . . , xnk i) for every r. Choose ( nkr +1 −1 S(hxj ij=n ) ∩ A1 if r ∈ {εn }n k +1 wr ∈ (4) nkr−1r+1 S(hxj ij=nkr +1 ) ∩ A0 if r ∈ / {εn }n for every r. Then the block sequence W 0 = ((z0 , λ0 ), (w0 , 1), (z1 , λ1 ), 0 ε). ¤ (w1 , 1), . . . ) is in [X] ∩ ΣA 1 (E), and clearly ΛA (W ) = (Z, ~ Corollary 3.1. If σ(C) is large for [X] then τA (C) is also large for [X]. Proof. Suppose that σ(C) is large for [X] and fix some block subsequence X 0 of X. From Proposition 3.1 there is a block subsequence Y of X 0 such that [Y ] × N ↑ ⊆ ΛA ”(Σ(X 0 ) ∩ ΣA 1 (E)). Since σ(C) is large for [X] there is some Z ∈ [Y ] ∩ σ(C). Fix ~ε ∈ N ↑ such that (Z, ~ε) ∈ C and choose W ∈ ε). Clearly W ∈ τA (C) ∩ Σ(X 0 ). ¤ Σ(X 0 ) ∩ ΣA 1 (E) such that ΛA (W ) = (Z, ~ Proposition 3.2. If τA (C)∆ is strategically large for [X] then σ(C)∆ is also strategically large for [X]. Proof. Suppose that there is some X such that Player II has a winning strategy Φ for the game a(τA (C))∆ [X]. Let us describe a winning strategy Φ0 for Player II for the game aσ(C)∆ [X]: Start the game with Player I choosing X0 ∈ [X]. Then Player II splits X0 into two subsequences Y0 and Z0 and he picks (y0 , λ0 ) = Φ0 (X0 ) = Φ(Y0 ). Suppose that the next choice of Player I is X1 ∈ [X]. Then player II splits X1 into two subsequences Y1 and Z1 and he chooses (y1 , λ1 ) = Φ0 (X0 , X1 ) = Φ(Y0 , Z0 , Y1 ), and so on. At the end of the game the block sequence ((y0 , λ0 ), (y1 , λ1 ), . . . ) = Φ0 ∗ (Yn )n ∈ (σ(C))∆
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since ((y0 , λ0 ), (z0 , 1), (y1 , λ1 ), (z1 , 1), . . . ) = Φ ∗ (Y0 , Z0 , Y1 , Z1 , . . . ) ∈ (τA (C))∆
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Let G(E) be the family of weakly Ramsey subsets of Σ(E), and let Ga (E) be the family of almost Ramsey subsets of Σ1 (E). Corollary 3.2. Fix a family C of subsets of Σ(E)×N ↑ . Then τA (C) ⊆ G(E) implies that σ(C) ⊆ G(E), where G(E) denotes the family of weakly Ramsey subsets of Σ(E). ¤ Remark 3.1. In the case of the property of being almost Ramsey for a c0 saturated space E we have the analogous result showing that τA (C) ⊆ Ga (E) implies σ(C) ⊆ Ga (E). Then reason is that if τA (C)∆ is very large for [(xn )n ]1 , then σ(C)∆ is also very large for [(x2n )n ]1 . Next we compute the Borel complexity of the set ΣA 1 (E) and the mapping ΛA depending on the complexity of A. We will not discuss the situation for arbitrary asymptotic pairs but only for two particular cases. Definition 4. Let A = {A0 , A1 } be an asymptotic pair of E. Let δ(A) = d(A0 , A1 ) = inf{ka0 − a1 k : a0 ∈ A0 , a1 ∈ A1 }. A is called discrete if A0 and A1 are Fσ subsets of S(E), i.e. countable unions of closed sets of S(E), and A is called separated if (a) δ(A) > 0 and (b) A0 and A1 are closed subsets of S(hen in ), i.e., if both are the intersection of a closed set of E with S(hen in ). Remark 3.2. 1. As it was discussed in the Introduction, discrete asymptotic pairs always exist: Let Ai = {x ∈ S(hen in ) : (−1)i e∗min supp x (x) > 0} for i = 0, 1. A0 and A1 are Fσ sets since [ Ai = {x ∈ S(E) : for every l¡Le∗l (x) = 0 L∈N,N ∈N
1 } (7) N which is clearly a countable union of closed sets. Separated asymptotic pairs will exist if E is not c0 -saturated. 2. If the space E does not contain an isomorphic copy of c0 , then for every block sequence X ∈ Σ(E) there is a block subsequence Y of X with a separated asymptotic pair of Y (see Theorem 13.17 in [3]). and (−1)i e∗L (x) ≥
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Proposition 3.3. 1. Suppose that A is a discrete asymptotic pair. Then ΣA 1 (E) is a Fσδ -set and ΛA is a Baire class 1 function (i.e., the pre-image by ΛA of an open set is a countable union of closed sets). Hence τA (C) is a Fσδ set for every C ⊆ Σ(E) × N ↑ . 2. Suppose that A is a separated asymptotic pair. Then ΣA 1 (E) is a Gδ subset of Σ(E) and ΛA is continuous. Hence τA (C) is a Gδ set for every C ⊆ Σ(E) × N ↑ . Moreover τA (C) is the intersection of a ΣA 1 (E) and a N-closed subset of Σ(E). Proof. Notice that a block sequence ((xn , λn ))n ∈ ΣA 1 (E) iff x2n+1 ∈ A0 ∪ A1 , λ2n+1 = 1 for every n, and x2n+1 ∈ A1 for infinitely many n. Clearly this implies that ΣA 1 (E) is an Fσδ -set if A is a discrete asymptotic pair, and it is a Gδ set provided that A is a separated asymptotic pair. Fix a basic open set U = B(((z0 , λ0 ), . . . , (zn , λn )), ε) × hk0 , . . . , kn i of Σ(E) × N ↑ , where B(((z0 , λ0 ), . . . ,(zn , λn )), ε) = {((xi , µi ))i ∈ Σ(E) : max{kxi − zi k, |λi − µi |} < ε for every i = 0, . . . , n} and hk0 , . . . , kn i is the set of all sequences (αi )i ∈ N ↑ such that αi = ki , for every i = 0, . . . , n. Then A Λ−1 A U =Σ1 (E) ∩ {((xi , µi ))i ∈ Σ(E) : for every i ≤ n/2
max{kx2i − zi k, |µ2i − λi |} < εand x2i+1 ∈ A1 iff i ∈ {k0 , . . . , kn }} From this one can easily prove the desired conclusions.
¤
Recall that subsets of Polish spaces can be classified according to their topological complexity. This yields the so-called Projective (or Lusin) hierarchy of pointclasses (see for example [7]). We shall use the following standard notation: Σ11 is the class of analytic sets, i.e., the continuous im∼ ages of Borel sets. Π11 is the class of coanalytic sets, i.e., the complements 1∼ of analytic sets. Σn+1 is the class of the continuous images of Π1n sets, and ∼ ∼ Π1 is the class of complements of Σ1n+1 sets. ∼ ∼n+1 The following proposition provides direct proofs that certain projective subsets of Σ(E) are weakly Ramsey, assuming that so are much simpler sets. Proposition 3.4. 1. If all Fσδ sets are weakly Ramsey then so are all analytic subsets of Σ(E). Moreover, if E does not contain c0 , then if all Gδ subsets of Σ(E) are weakly Ramsey, so are all analytic subsets of Σ(E).
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2. If every coanalytic subset of Σ(E) is weakly Ramsey, then so is every Σ1 subset Σ(E). More generally, for every n ≥ 1, if every Π1n subset ∼2 ∼ of Σ(E) is weakly Ramsey, then so is every Σ1n+1 subset of Σ(E). ∼ 3. The corresponding results for the almost Ramsey property of c0 -saturated spaces are also true. Proof. 1. Recall that the class of all analytic sets of Σ(E) is exactly σ(C0 ) where C0 denotes the class of closed subsets of Σ(E) × N ↑ . Fix a discrete asymptotic pair A of E. Notice that by Proposition 3.3 every element of τA (C0 ) is a Fσδ set. Corollary 3.2 gives the desired result. Suppose now that E does not contain an isomorphic copy of c0 , and fix an analytic set σ = σ(C) ⊆ Σ(E) where C ⊆ Σ(E) × N ↑ is closed. By Proposition 2.4, σ(C) is weakly Ramsey iff every block sequence X as a block subsequence Y such that σ ∩ [Y ] is weakly Ramsey. Fix a block sequence X and let Y be a block subsequence of X with a separated asymptotic pair U = {A0 , A1 } of Y . Notice that σ∩[Y ] = σ(CY ), where CY = C ∩([Y ]×N ↑ ) is a closed set. Then τA (CY ) ⊆ [Y ] is a Gδ set of [Y ] and hence of Σ(E). By our assumption τA (C) is weakly Ramsey and by Corollary 3.2 σ ∩ [Y ] = σ(CY ) is also weakly Ramsey. For 2. we use the fact that the class Σ1n+1 is just the class σ(Π1n ), and ∼ ∼ ¤ that τA (Π1n ) ⊆ Π1n . ∼ ∼ Remark 3.3. Let us make a metamathematical comment at this point. Typically, Theorem 1.1 is used as follows. We consider two properties P and Q of subspaces of E, characterized by a set σ of block sequences, a family F of infinite sequences (Yn )n of block sequences and ∆ > 0 so that: 1. If X = ((xn , λn ))n is such that [X] ∩ σ = ∅, then (xn )n has the property P . 2. If X = ((xn , λn ))n is such that for every (Yn )n ∈ F with each Yn ∈ [X] there exists a block sequence ((yn , λn ))n with (a) ((yn , λn ))n ∈ σ∆ and (b) yn ∈ S(Yn ) for every n, then (xn )n has the property Q. It is clear that Gowers’ Theorem shows that if σ is analytic then every Banach space E has a subspace X with the property P or with the property Q. Let C be a closed subset of Σ(E) × N ↑ such that σ = σ(C), and let τA = τA (C). Then Gowers’ Theorem for τA also shows that there must be some X ∈ Σ(E) with the property P or with the property Q: Indeed, if τA ∩ [((xn , λn ))n ] = ∅ then by Corollary 3.1 σ ∩ [((xn , λn ))n ] = ∅ which implies, by 1., that (xn )n has the property P . Otherwise, by Gowers’ Theorem for τA , we can find some X = ((xn , λn ))n ∈ Σ(E) and some winning strategy Φ for Player II in the game a(τA )∆ [X]. Now given (Yn )n ∈ F with each
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Yn ∈ [X] we can split Y2n into two pieces Zn and Wn so that ((y0 , λ0 ), (z0 , µ0 ), (y1 , λ1 ), (z1 , µ1 ), . . . ) = Φ ∗ (Z0 , W0 , Z1 , W1 , . . . ) ∈ (τA )∆ hence ((yn , λn ))n ∈ σ∆ and yn ∈ S(Zn ) ⊆ S(Yn ) for every n. By 2. this implies that (xn )n has the property Q. Moreover, in general we may assume that τA (C) is a Fσδ set. Further, if E is not c0 -saturated then we may even assume that τA (C) is a Gδ -set: First we find X ∈ Σ(E) such that every subspace of X is not isomorphic to c0 , hence there is some block subsequence Y of X with a separated asymptotic pair A = {A0 , A1 } of Y . Next, we replace E by Y and we are done since τA (C ∩ [Y ] × N ↑ ) is a Gδ set. 4. Another proof that analytic sets are weakly Ramsey We present an alternative proof of the fact that all analytic sets are weakly Ramsey for a space E not containing c0 , which is based in the fact that every block sequence of E has a block subsequence with a separated asymptotic pair. More precisely, we will show rather directly that if A is a separated asymptotic pair and τ is an N-closed subset of Σ(E), then ↑ τ ∩ ΣA 1 (E) is weakly Ramsey. But if C ⊆ Σ(E) × N is closed, then τA (C) is one these intersections, hence it follows that σ(C) is weakly Ramsey. Let us point out that the way in which it is usually proved in infinite Ramsey theory that a Gδ set σ is Ramsey is to find a set A for which σ restricted to [A] is clopen. T approach does not work here, T However, this mainly because in general ( n σn )∆ and n (σn )∆ are different (see Remark 6.1). Definition 5. Fix an asymptotic pair A = {A0 , A1 } of some X ∈ Σ(E). Let ΣA 1,f (E) be the set of finite block sequences t = ((x0 , λ0 ), ..., (xn , λn )) such that (a) x2i+1 ∈ A0 ∪ A1 and λ2i+1 = 1 for every i ≤ n/2 and 0 we say that N ⊆ Σ1,f (E) is a (A, δ)admissible net iff N is countable and closed under concatenation, and for every ((x0 , λ0 ), ..., (xn , λn )) ∈ ΣA x0 , µ0 ), ..., (e xn , µn )) ∈ 1,f (E) there is some ((e N such that (a) max{kxi − x ei k, |λi − µi |} ≤ δ and supp xi = supp x ei for every i ≤ n, and (b) x2i+1 ∈ A1 iff x e2i+1 ∈ A1 for every i ≤ n/2. N is called a A-admissible net if N is a (A, δ)-admissible net for every δ > 0. For a given A-admissible net N of X, σ, and ∆ > 0, let [ N OX (σ, ∆) = {[t; X] : d(t, e t) ≤ ∆ for some e t∈N such that σ is large for [e t; X]}
(8)
12
´ JORDI LOPEZ-ABAD
Remark 4.1. 1. A-admissible nets exist for every A: Suppose that the Schauder basis (en )n has basic constant C. Fix n and a finite subset F ⊆ N, and let M be such that #F/M ≤ min{2−n , δ(A)}. Consider a covering {I0 , ..., IL } of [−2C, 2C] with each Ii an interval of diameter ≤ 1/M . Now for each f : F → {0, ..., L} such that there is some normalized vector x with supp x = F and such that e∗n (x) ∈ If (n) for every n ∈ F we pick a (n) normalized xf with this property. In addition, if there exists such an x (n)
as above in Aε (ε = 0, 1) then we choose xf ∈ Aε . Notice that it is not possible to have two x ∈ A0 , and x0 ∈ A1 as above (because #F/M ≤ S (n) min{2−n , δ(A)}). In this manner the set F ⊆N finite {xf : f ∈ F } is a −n countable 2 -cover of S(hen in ) with the extra properties that for every S (n) x ∈ S(hen in ) there is x0 ∈ F ⊆N finite {xf : f ∈ F } such that kx − x0 k ≤ −n 0 2 , supp x = supp x , and if in addition x ∈ Aε (ε = 0, 1), then x0 ∈ Aε . From here it is easy to build a A-admissible net. 2. If N is a A-admissible net, then for every δ and every ((x0 , λ0 ), . . . , (xn , λn )) there is a finite (A, δ)-admissible net N 0 ⊆ N of the set of A-admissible block subsequences of (x1 , . . . , xn ). This fact has a proof similar than 1. Lemma 4.1. Let A be an asymptotic pair of some X, and let N be a A-admissible net. Suppose that σ = τ ∩ ΣA 1 (E) is large for [s; X]. Then for every ∆ > 0 and every Y ∈ [X] there is some A-admissible block subsequence t of Y \ s, e t ∈ N , and some Z ∈ [Y ] such that (a) d(t, e t) ≤ ∆/2 and (b) A t; Z]. σ∆/2 ∩ Σ1 (E) is large for [s a e Proof. For suppose not. Let σ = τ ∩ ΣA 1 (E) be large for [s; X], fix ∆ > 0 and Y ∈ [X], and suppose that for every A-admissible block subsequence t of Y there is some Z such that for every e t ∈ N , if d(t, e t) ≤ ∆/2 then ae t ; W ] = ∅. This (E) ∩ [s there is some W ∈ [Z] such that σ∆/2 ∩ ΣA 1 condition allows to find recursively for every n a finite block sequence s < sn = ((y0 , 1), ..., (yn , 1)) and an infinite block sequence Yn such that for every n, (a) Y0 = Y , (b) yn = min Yn , Yn+1 ∈ [Yn \ yn ], and (c) for every e t ∈ Nn such that there is some A-admissible block subsequence t of ((y0 , λ0 ), . . . , (yn , λn )) with d(t, e t) ≤ ∆/2, we have σ∆/2 ∩ ΣA 1 (E) ∩ ae [s t; Yn+1 ] = ∅, where Nn ⊆ N is a finite (A, δn /2)- admissible net of the set of A-admissible block subsequences of ((y0 , λ0 ), . . . , (yn , λn )). Then, setting Y∞ = (min Yn )n ∈ [Y ], we obtain that σ ∩[s; Y∞ ] = ∅, since otherwise let s a ((wn , λn ))n ∈ σ∩[s; Y∞ ], and let k be the minimal integer m such that ((wi , λi ))m i=0 is A-admissible, and let n be the minimal integer such n e that ((wi , λi ))m i=0 is a block subsequence of ((yi , 1))i=0 . Let t ∈ Nn ⊆ N m a a f = s e be such that d(t, ((wi , λi ))i=0 ) ≤ ∆/2. Then W t ((wi , λi ))i>m ∈
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f ΣA 1 (E), ((wi , λi ))i>m ∈ [Yn+1 ], and clearly W ∈ σ∆/2 , a contradiction with the properties of ((yi , 1))i . ¤ Given two block sequences X and Y = ((yn , µn ))n , we write Y ∈∗ [X] iff ((yn , µn ))n>k ∈ [X] for some k. Lemma 4.2. 1. If σ is (strategically) large for [X] then σ is (strategically) large for [Y ], for every Y ∈∗ [X]. 2. For every family {σn }n of sets of block sequences and every block sequence X there is some X0 ∈ [X] such that for every n and every Y ∈ [X0 ], if σn is (strategically) large for [Y ], then σn is (strategically) large for [X0 ]. Proof. 1. is trivial. Let us show 2.: Fix a family {σn }n of sets of block sequences, and X. We can find inductively a sequence (Xn )n of block sequences such that (a) X0 = X, Xn+1 is a block subsequence of Xn \ min Xn and (b) if σn is (strategically) large for [Z] for some Z ∈ [Xn ], then σn is (strategically) large for [Xn+1 ]. Then the block sequence Y = (min Xn )n satisfies the desired result. We find this sequence (Xn )n inductively. Suppose defined Xn . If there is some block subsequence Z of Xn such that σn is (strategically) large for [Z], then let Xn+1 ∈ [Xn \ min Xn ] be one of these. If not, let Xn+1 = Xn \ min Xn . ¤ Lemma 4.3. Let τ be a N-closed subset of Σ(E), let A be a closed asymptotic pair of X and suppose that σ = τ ∩ ΣA 1 (E) is large for [X]. Then for every ∆ > 0 there exists Y ∈ [X] such that σ∆ is strategically large for [Y ]. Proof. Fix a A-admissible net N . We assume that ∆ = (δn )n < δ(A). Set ∆n = (2n − 1)/2n+1 ∆ for every n. Let X0 ∈ [X] be the result of the application of 2. of Lemma 4.2 to X and the family of block sequences {(τn )s : n ∈ N, s ∈ N }, and X1 ∈ [X0 ] the result of the same application to X0 and the family à ! µ ¶ δk N s {OX0 (τn ) , : n ∈ N, s ∈ N } 2 k>lh(s) A where τn = σ∆n ∩ ΣA 1 (E) for every n. Notice that (τn )(∆n+1 −∆n ) ∩ Σ1 (E) ⊆ τn+1 and τn ⊆ σ∆/2 for every n. We claim that σ∆ is strategically large for [X1 ]. We proceed to sketch a winning strategy for player II in the ∅ game for aσ∆ [X1 ]: By hypothesis, σ∆ is large for [X0 ]. By Lemma 4.1, for 0 every Y ∈ [X0 ] there is some block subsequence t of Y and some Z ∈ [X0 ] such that τ1 is large for [e t; Z] for some e t ∈ N with d(e t, t) ≤ ∆/4. By the properties of X0 , this is equivalent to saying that τ1 is large for [e t; X0 ]. So, N OX (τ , ∆/4) is large for [X ], and hence there is some Y ∈ [X ] 1 0 1 such that 0
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´ JORDI LOPEZ-ABAD
N N (OX (τ1 , ∆/4))∆/4 is strategically large for [Y ]. Since (OX (τ1 , ∆/4))∆/4 ⊆ 0 0 N (OX0 (τ1 , ∆/2)), by the properties of X1 we can fix a winning strategy Φ1 for N (τ ,∆/2) [X1 ]. Player II will follow Φ1 until he player II for the game aOX 1 0 reaches some t1 for which there is s1 ∈ N such that (a) d(t1 , s1 ) ≤ ∆/2 and (b) σ∆1 is strategically large for [s1 ; X0 ]. By Lemma 4.1, for every Y ∈ [X0 ] there is some block subsequence t > s1 of Y and some Z ∈ [X0 ] such that ae (τ1 )(∆2 −∆1 )\s1 ∩ ΣA t; Z] for some e t ∈ N satisfying 1 (E) is large for [s1 e t > s1 and d(e t, t) ≤ (δn /4)n>lh(s1 ) . Since (τ1 )(∆2 −∆1 )\s1 ∩ ΣA 1 (E) ⊆ τ2 and s1 N (τ , (δ /4) by the properties of X0 , we have that OX n n>lh(s1 ) ) is large for 2 0 [X0 ], and hence player II can follow a winning strategy Φ2 for the game aOX [X1 ] until he reaches t2 > t1 , s1 such that there exists N (τ s1 ,(δ /2) n n>lh(s1 ) ) 2 0 s2 ∈ N such that (a) s2 > s1 , d(t2 , s2 ) ≤ (δn /2)n>lh(s1 ) , and (b) τ2 is large for [s1 a s2 ; X0 ]. And so on. In this way, at the end of the game Player II has produced the block sequence Z = t1 a t2 a · · · a tn a · · · and the auxiliary block sequence W = s1 a s2 a · · · a sn a · · · such that (a) d(Z, W ) ≤ ∆/2, (b) τn is large for [s1 a · · · a sn ; X0 ]. Notice that (b) implies that σ∆/2 is large for [s1 a · · · a sn ; X0 ] for every n, which implies, using that τ ∩ ΣA (E) f ∈ τ ∩ ΣA (E) such that d(W, W f ) ≤ ∆/2. is N -closed, that there is some W f ∈ ΣA (E), hence W ∈ σ∆/2 , Since δn /2 ≤ δ(A) for every n, we have that W 1 and then Z ∈ σ∆ as desired. ¤
This gives another proof of Gowers’ Theorem 2.1 for spaces not containing c0 . Corollary 4.1. Suppose that E does not contain c0 . Then every analytic set is weakly Ramsey. Proof. Let σ be an analytic set of Σ(E), for E not containing c0 , and fix ∆ > 0. Suppose that σ is large for [X]. Choose Y ∈ [X] and a closed asymptotic pair A of Y . Let C ⊆ [Y ] × N ↑ be a closed set such that σ ∩ [Y ] = σ(C). We know that τA (C) = τ ∩ ΣA 1 (E) for some N-closed set τ of [Y ] and that τA (C) is large for [Y ]. By Lemma 4.3, there is some Z ∈ [Y ] such that τA (C)∆ is strategically large for [Z], and hence σ∆ is strategically large for [Z] as desired. ¤ 5. Normalized block sequences We will now study the relationship between the weakly Ramsey property for sets of arbitrary block sequences ((xn , λn ))n , as we have being considering up to this point, and the weakly Ramsey property for sets of normalized block sequences ((xn , 1))n , as it was used in [1]. For the latter, in the game aσ [X] player II only chooses normalized vectors, as opposed to pairs consisting of a normalized vector and a scalar. Somehow, the notion of weakly
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Ramsey for sets of arbitrary block sequences ((xn , λn ))n is the “parameterized” version of the normalized notion from [1]. In can be easily shown that for sets of normalized block sequences both notions are equivalent. But in principle, the notion of weakly Ramsey set from [1] is weaker. Nevertheless, we will show that for spaces not containing c0 they are equivalent. Proposition 5.1. Suppose that A is a separated asymptotic pair of E, and suppose that ∆ = (δn )n > 0 is such that δn < δ(A) for every n. Then for every subset σ ⊆ Σ(E) there is a subset τ ⊆ Σ1 (E) such that (a) if σ is large for [((xn , λn ))n ], then τ is large for [(xn )n ]1 and (b) if τ∆/2 is strategically large for [(xn )n ]1 , then σ∆ is strategically large for [(xn , 1)n ]. Moreover, if σ is analytic, then so is τ . Proof. Fix a separated asymptotic pair A = {A0 , A1 }, 0 < ∆ < δ(A), σ ⊆ Σ(E), and X = ((xn , λn ))n ∈ Σ(E). For each n ≥ 0 let Ln be the minimal integer l such that δn l ≥ 1 and let Mn be the minimal integer m (m) such that 2m ≥ Ln . Set M−1 = 0, and let h0 = M−1 +· · ·+Mm−1 +m+1 (m) (m) (m) and h1 = M−1 + · · · + Mm + Mm + m. Let Hm = [h0 , h1 ], and A let Σ∆ (E) be the set of normalized block sequences (xn )n such that for every m we have that xn ∈ A0 ∪ A1 for every n ∈ Hm . Notice that for every sequence (xn )n ∈ ΣA ∆ (E) and every m, the sequence (xi )i∈Hm codes a sequence in {0, 1}Mm , hence (xn )n naturally codes the block sequence ΛA,∆ ((xn )n ) = ((xh(m) −1 , λ((xi )i∈Hm )))m where 0 Ã ! X i−min Hm δm λ((xi )i∈Hm ) = εi 2 i∈Hm
and εi ∈ {0, 1} is such that xi ∈ Aεi . Let τ = Λ−1 A,∆/2 (σ∆/2 ). The proof of (a) for τ is quite similar to the proof of Corollary 3.1. Let us show (b): Suppose that Φ is a winning strategy for player II in the game aτ∆/2 [X] for some normalized block sequence X = (xn )n . We sketch a winning strategy Φ0 for player II in the game aσ∆ [((xn , 1))n ]. Suppose that the game starts (0) (0) with Player I picking X0 = ((xn , λn ))n . Let (0) (yn(0) )n = Φ ∗ ((x(0) n )n , (xn )n , . . . ) ∈ τ∆/2 (0)
(0)
(0)
and let (zn ) ∈ τ ⊆ ΣA ∆/2 (E) be such that d((yn , zn ))n ≤ ∆/2. Let (0)
Φ0 (X0 ) = (y0 , λ0 ) (0)
(9) (1)
where λ0 = λ((zi )i∈H0 ). Now suppose that Player I plays X1 = ((xn , (1) λn ))n , and let
´ JORDI LOPEZ-ABAD
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(M0 +1)
}| { z (0) (1) (1) (yn(1) )n = Φ ∗ ((x(0) n )n , . . . , (xn )n , (xn )n , . . . , (xn )n , ..) ∈ τ∆ and let
(1) (zn )
∈ τ be such that 0
Φ
(1) (1) d((yn , zn ))n
(0) ((y0 , λ0 ), X1 )
=
(10)
≤ ∆/2. Let
(1) (yM0 +1 , λ1 )
(11)
(1)
where λ1 = λ((zi )i∈H1 ), and so on. We show that Φ0 is winning, i.e. Φ0 ∗ (X0 , X1 , . . . ) = ((xn , λn ))n ∈ σ∆
(12)
Notice that (0)
(0)
(1)
(1)
W =(y0 , . . . , yM0 , yM0 +1 , . . . , yM0 +M1 +1 , . . . ) = (M0 +1)
(13)
(M1 +1)
}| { z }| { z (0) (1) (1) =Φ ∗ ((x(0) ) , . . . , (x ) , (x ) , . . . , (x ) n n n n n n n n , . . . ) ∈ τ∆
(14)
hence we can find Z = (zn )n such that d(Z, W ) ≤ ∆ and Z ∈ τ , i.e. ΛA,∆/2 (Z) ∈ σ∆/2 . The proof will be finished if we show that d(((xn , λn ))n , ΛA,∆/2 (Z)) ≤ ∆/2. Notice that for every m and every i ∈ Hm (m)
kzi − zi
(m)
k ≤ kzi − yi
(m)
k + kyi
(m)
− zi
k ≤ δi < δ(A)
(15)
Hence (m)
λ((zi )i∈Hm ) = λ((zi
)i∈Hm ) = λm
(16)
for every m. Notice also that for every m kzh(m) −1 − yh(m) −1 k ≤ δh(m) −1 /2 0
0
0
(17)
which implies that d(((xn , λn ))n , ΛA,∆/2 (Z)) ≤ ∆/2 since ∆ is decreasing. ¤ Corollary 5.1. Suppose that E does not contain c0 . Then for every subset σ ⊆ Σ(E) there is τ ⊆ Σ1 (E) such that σ is weakly Ramsey iff τ is weakly Ramsey. Moreover, if σ is analytic, then τ is analytic. Proof. This is a consequence of Proposition 2.4 and the localized version of Proposition 5.1. ¤ 6. Weakly Ramsey property and complements We give an example of a Σ12 set (i.e. a projection of a complement of an ∼ analytic set) which is not weakly Ramsey. Since, as we show in the next section, there are standard set theoretical models in which all the sets are weakly Ramsey, we are forced to use some extra assumptions to find such a set.
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Theorem 6.1. If there is a good Σ12 well ordering of the reals (see [7]), ∼ then there is a Σ12 non-weakly Ramsey set of block sequences in any Banach ∼ space. Proof. We reproduce the example of a non-weakly Ramsey set of normalized block sequences given in [1], and we sketch how, from the assumptions of the Theorem one can make it a Σ12 set. This is done in the context of nor∼ malized block sequences, but as we mentioned at the end of the introduction to section 5, for sets of normalized block sequences the notions of weakly Ramsey for sets of normalized block sequences and the general notion of weakly Ramsey coincide. If there is a Σ12 well ordering
