e-mail: atsarp@math.uoa.gr

Original Paper

# Some combinatorial principles for trees and applications to tree families in Banach spaces

Article first published online: 8 FEB 2014

DOI: 10.1002/malq.201300029

© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Additional Information

#### How to Cite

Poulios, C. and Tsarpalias, A. (2014), Some combinatorial principles for trees and applications to tree families in Banach spaces. Mathematical Logic Quarterly, 60: 70–83. doi: 10.1002/malq.201300029

#### Publication History

- Issue published online: 8 FEB 2014
- Article first published online: 8 FEB 2014
- Manuscript Accepted: 8 OCT 2013
- Manuscript Revised: 30 JUL 2013
- Manuscript Received: 18 MAY 2013

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### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Some combinatorial principles for the dyadic tree
- 3 Certain definable sets
- 4 Tree-families of continuous functions
- 5 The case of nearly unconditionality
- 6 The case of convex unconditionality
- 7 A dichotomy result for more general tree-families
- Acknowledgements
- References

Suppose that is a normalized family in a Banach space indexed by the dyadic tree *S*. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree-families. More precisely, assuming that for any infinite chain β of *S* the sequence is weakly null, we prove that there exists a subtree *T* of *S* such that for any infinite chain β of *T* the sequence is nearly (resp., convexly) unconditional. In the case where is a family of continuous functions, under some additional assumptions, we prove the existence of a subtree *T* of *S* such that for any infinite chain β of *T*, the sequence is unconditional. Finally, in the more general setting where for any chain β, is a Schauder basic sequence, we obtain a dichotomy result concerning the semi-boundedly completeness of the sequences .

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Some combinatorial principles for the dyadic tree
- 3 Certain definable sets
- 4 Tree-families of continuous functions
- 5 The case of nearly unconditionality
- 6 The case of convex unconditionality
- 7 A dichotomy result for more general tree-families
- Acknowledgements
- References

In a well-known example Maurey and Rosenthal [7] showed that if is a normalized weakly null sequence in a Banach space then we could not expect that admits an unconditional subsequence. Further, Gowers and Maurey [5] exhibited a Banach space not containing any unconditional basic sequence.

Despite the aforementioned constructions there are some positive results where either some special sequences are considered or weaker forms of unconditionality appear. More precisely, Rosenthal proved the following theorem.

Theorem 1.1. Let *K* be a compact Hausdorff space and let , , be a sequence of non-zero, continuous, characteristic functions. If converges pointwise to zero, then it contains an unconditional basic subsequence.

Although the initial proof uses transfinite induction, the nature of the previous result is purely combinatorial. Indeed, the proof of Theorem 1.1 (cf. [[1], [8]]) can be obtained from the next result which in turn depends on the infinite Ramsey theorem. In the following, if *M* is an infinite subset of , denotes the set of all infinite subsets of *M*.

Theorem 1.2. Let be a compact family of finite subsets of . Then for any , there exists such that the family is hereditary (that is, if and , then ).

As a matter of fact, it was infinite Ramsey theory which led to a series of positive results. One of them was obtained by Elton [4] (cf. also [8]).

Theorem 1.3. Every normalized weakly null sequence in a Banach space contains a nearly unconditional subsequence.

The notion of nearly unconditionality concerns the unconditional behavior of linear combinations with coefficients bounded away from zero. The precise definition is the following: A normalized sequence in a Banach space is called *nearly unconditional* if for every there exists such that for any , any scalars and any ,

Using Ramsey's theory in a very elegant way, Elton proved the following principle from which Theorem 1.3 is obtained.

Theorem 1.4. Let *F* be a weakly compact subset of the unit ball of *c*_{0} and let and be given. Then for every , there exists such that for every , and with there exists such that

- [where ]
- , where .

The subsequences of a weakly null sequence have also been investigated with respect to the property of convex unconditionality. A normalized sequence in a Banach space is called *convexly unconditional* if for every there exists such that for any absolutely convex combination with and any sequence of signs,

The next result, concerning the case of convex unconditionality, has been proved by Argyros, Mercourakis and Tsarpalias [2].

Theorem 1.5. Every normalized weakly null sequence in a Banach space contains a convexly unconditional subsequence.

As in the previous cases, the proof of this theorem is based on the next combinatorial principle.

Theorem 1.6. Let *F* be a weakly compact subset of *c*_{0} and let and be given. Then for every , there exists such that for every , and with there exists satisfying the conditions

- .

Finally, the combinatorial proof of Rosenthal's theorem has been expanded and some stronger results have been obtained. As pointed out in [3], Theorem 1.1 can not be extended in the case where the range of is a finite set of arbitrarily large cardinality. However, Arvanitakis [3] expanded this theorem in the case where the cardinality of the range of is finite and uniformly bounded by some positive integer.

Theorem 1.7. Let *K* be a Hausdorff compact space, *X* a Banach space and , , a normalized sequence of continuous functions. We assume that converges pointwise to zero and that the range of 's is of finite cardinality uniformly bounded by some positive integer *J*. Then contains an unconditional subsequence.

The following result, also proved in [3], concerns the case where the space *X* in the above theorem is finite dimensional.

Theorem 1.8. Let *K* be a Hausdorff compact space and , , a uniformly bounded sequence of continuous functions which converges pointwise to zero. We also assume that there are a null sequence of positive numbers and a positive real number μ such that for every and any either or . Then contains an unconditional subsequence.

The above theorems are derived by the following combinatorial principle (cf. also [3]) extending Theorem 1.2.

Theorem 1.9. Assume that *I* is a set, *n* a positive integer and for any , are finite subsets of such that setting , the closure of the family in the pointwise topology contains only finite sets. Then for any , there exists such that the following holds:

Given , , and then there exists such that and for any .

Throughout this paper *S* denotes the standard dyadic tree, that is the set of all finite sequences in , including the empty sequence denoted by . The elements are called *nodes*. If *s* is a node and , we say that *s* is on the *n**-th level of* *S*. We denote the level of a node *s* by lev(*s*). The initial segment partial ordering on *S* is denoted by ⩽ and we write if and . If , we say is a *follower* of *s* while if are nodes such that neither nor then *s* and are called incomparable. We also say that the nodes and are the *successors* of the node *s*.

A partially ordered set *T* is called a *dyadic tree* if it is order isomorphic to . A *subtree* of *S* is any subset of *S* which has a single minimal element and any element of has exactly two successors. In the sequel we mean by a tree always a dyadic tree. If *T* is a tree, a *chain* of *T* is an infinite linearly ordered subset of *T*. Throughout this paper, denotes the set of all chains of *T*. This notation should not be confused with C([0, 1]). The latter stands for the space of continuous functions on [0, 1] and it is used mainly in Section 'Certain definable sets'.

The set is endowed with the relative topology of the product topology of . A sub-basis of this topology consists of the sets and where *t* varies over the elements of *T*. Clearly the sets are open and closed. It is also known (cf. [10]) that is a subset of . Therefore (cf. [6]) the topology of is induced by a complete metric.

Considering on the topology described above, Stern [10] proved the following Ramsey-type theorem for the dyadic tree. Furthermore, Stern (cf. also [10]) applied his combinatorial result in the theory of Banach spaces and extended Rosenthal's ℓ_{1}-theorem [9] to the case of tree-families.

Theorem 1.10. Let *T* be a tree and let be an analytic set of chains. There exists a subtree of *T* such that either or .

Stern's initial proof uses forcing methods. However, Henson (cf. [8]) observed that the above mentioned theorem follows from some significant results of Ramsey theory.

In this paper, using Stern's theorem we prove in Section 'Some combinatorial principles for the dyadic tree' some combinatorial principles for the dyadic tree. These results combine combinatorial methods with concepts and techniques coming from analysis and extend Theorems 1.2, 1.4, 1.6 and 1.9 mentioned in this introduction.

In Section 'Certain definable sets', given a normalized family of elements of a Banach space indexed by the dyadic tree *S*, we verify that the set of all chains with a certain property is Borel or at the most analytic or co-analytic. These verifications allows us to apply Stern's theorem and to obtain the results of the next sections. Especially, in Sections 'Tree-families of continuous functions', 'The case of nearly unconditionality' and 'The case of convex unconditionality', we consider normalized tree-families such that for any chain β of *S* the sequence is weakly null. Our aim is to investigate the unconditional behaviour of the sequences for . More precisely, in Section 'Tree-families of continuous functions', we prove that in some special cases there exists a subtree *T* of *S* such that for any chain β of *T* the sequence is unconditional. These results extend Rosenthal's and Arvanitakis' theorems. In Section 'The case of nearly unconditionality', we show that there always exists a subtree *T* of *S* such that all the sequences , , are nearly unconditional. In Section 'The case of convex unconditionality', we also prove the existence of a subtree *T* of *S* such that all the sequences , , are convexly unconditional. These results extend Theorems 1.3 and 1.5 respectively.

Finally, in Section 'A dichotomy result for more general tree-families' we consider the more general case where is a normalized tree-family such that for any chain , is a Schauder basic sequence. In this framework we prove the following dichotomy result (cf. Theorem 7.1): there always exists a subtree *T* of *S* such that either (a) for any chain , is semi-boundedly complete, or (b) for no chain , is semi-boundedly complete. Recall that a normalized Schauder basic sequence in a Banach space is called semi-boundedly complete if for every sequence , the condition implies that . Furthermore, if we assume that is weakly null for any chain , then we can combine the above dichotomy with the results of Section 'The case of nearly unconditionality' and we obtain the next stronger result (cf. Theorem 7.2): there always exists a subtree *T* of *S* such that either (a) for any chain , is semi-boundedly complete, or (b) for any chain , is *C*-equivalent to the unit vector basis of *c*_{0}, where is a common constant. It is worth mentioning that the proof of Theorem 7.1 uses analytic sets. Actually, this is the only point where we appeal to the full strength of Stern's theorem. In all the other cases, the sets appearing in the proofs are Borel sets.

In order to prove the main results of Sections 'Tree-families of continuous functions', 'The case of nearly unconditionality' and 'The case of convex unconditionality', we can rely on the combinatorial principles of Section 'Some combinatorial principles for the dyadic tree' and transfer the arguments of [3], [4] and [2] respectively in the more complicated setting of tree-families. However the proofs obtained are quite long and technical and they will not be presented. Instead our approach uses the corresponding results for sequences and Stern's theorem and provides us with short proofs of the theorems contained in this paper. Although we do not use the principles of Section 'Some combinatorial principles for the dyadic tree', we think that they are of independent interest and they point out the underlying combinatorial nature of the main results of this work.

In conclusion, the purpose of the present work is to extend important results from sequences to tree-families. The essential attitude lying in the core of the paper is that in general this passage from sequences to tree-families often has a fundamental effect in analysis, set theory and logic. This passage is not trivial and usually requires new ideas and techniques. We further believe that the ideas contained in our proofs can be applied in more general concepts.

### 2 Some combinatorial principles for the dyadic tree

- Top of page
- Abstract
- 1 Introduction
- 2 Some combinatorial principles for the dyadic tree
- 3 Certain definable sets
- 4 Tree-families of continuous functions
- 5 The case of nearly unconditionality
- 6 The case of convex unconditionality
- 7 A dichotomy result for more general tree-families
- Acknowledgements
- References

In the following bcn(*S*) denotes the set of all functions such that *f* is bounded and for any chain β of *S* the sequence converges to zero. Clearly, bcn(*S*) is a linear subspace of the space of bounded real functions defined on *S*. Further, we consider on bcn(*S*) the topology of pontwise convergence, that is the relative topology of the product topology of . Since *S* is countable, bcn(*S*) is metrizable.

In this section, we first prove the following theorem which expands Elton's combinatorial principle.

Theorem 2.1. Suppose that *F* is a compact subset of bcn(*S*), with , where and let and be given. Then there exists a subtree *T* of *S* which satisfies the following property:

for any , any , any and any with , there exists such that

- [where ],
- , where or .

Roughly speaking, for any chain β of *T* and any we find a function which preserves the positive ℓ_{1} mass of *f* on the finite set *I* and is very close to zero on the other 's. If we could find a function such that for and for , , then it would follow (cf. [8]) that for any normalized family so that is weakly null for any there is a subtree *T* of *S* such that is unconditional for any , which of course is not true.

Proof of Theorem 2.1. Let *F*_{0} be a countable dense subset of *F*. Consider the set of all chains which satisfy the following property: for any , any and any with , , and , there is such that

- ,
- , where or .

Claim 2.2. The set is a Borel subset of .

Indeed, we have

where

Clearly, and are open subsets of ; therefore is a Borel set.

Now Stern's theorem implies that there is a subtree *T* of *S* such that either (a) or (b) . However, the case (b) can be excluded. Indeed, let us assume that and let β be any chain of *T*. Applying Theorem 1.4, we find a subchain of β such that for any , any and any with , there is such that and , where or . Since *F*_{0} is dense in *F*, it follows that we can find satisfying the above properties. Hence, and we have reached a contradiction. Therefore, .

Since *F*_{0} is dense in *F*, we can easily verify that the subtree *T* satisfies the conclusion of the theorem.

In a similar method we also prove the next combinatorial theorem for the dyadic tree.

Theorem 2.3. Suppose that *F* is a compact subset of bcn(*S*) which is bounded with respect to the supremum norm and let and be given. Then there exists a subtree *T* of *S* satisfying the following property:

for any chain of *T*, any , any and any with , there exists such that

- .

Proof. Let *F*_{0} be a countable dense subset of *F* and let be the set of all chains which satisfy the following: for any , any and any with there is such that and . Then,

where

It follows that is a Borel subset of . Stern's theorem implies that there is a subtree *T* of *S* such that either (a) or (b) . By Theorem 1.6, the case (b) is excluded. Therefore, and the result follows.

Finally, we expand Theorem 1.9 to obtain a combinatorial theorem for trees.

Theorem 2.4. Assume that *I* is a set, *n* a positive integer and for every , are finite subsets of *S*. For any we set and let . We also assume that for any *F* in the closure of and for any chain , the set is finite. Then there exists a subtree *T* of *S* satisfying the following property:

for any chain , any , any , any and any there exists such that

- for any .

Proof. The powerset endowed with the product topology is a compact metric space. Therefore, is separable. It follows that for any there is a countable subset such that is a dense subset of . We set . Clearly, *J* is countable and for every , is dense in .

We consider the set of all chains which satisfy the following property: for any , any , any and any there is such that and for any . Then we have

where

The sets and are open; therefore is a Borel subset of . By Stern's theorem, there is a subtree *T* of *S* such that either (a) or (b) . By Theorem 1.9 we must have (a). Since is dense in for any , the tree *T* satisfies the desired property.

### 3 Certain definable sets

- Top of page
- Abstract
- 1 Introduction
- 2 Some combinatorial principles for the dyadic tree
- 3 Certain definable sets
- 4 Tree-families of continuous functions
- 5 The case of nearly unconditionality
- 6 The case of convex unconditionality
- 7 A dichotomy result for more general tree-families
- Acknowledgements
- References

The results of this paper are based on applications of the Ramsey-type theorem of Stern for trees. However, to apply Stern's theorem (cf. Theorem 1.10) it is necessary first to verify that given a tree-family in a Banach space, the set of all chains that satisfy a certain property is a definable set, i.e., it is Borel or at the most analytic or co-analytic. We explain here the idea on which the verifications of that sort are based. Each of the properties of interest is described exclusively by conditions with finitely many parameters that are running in countable sets and the quantifiers “for all” and “exists”. If one considers the relevant concept in this way, it is easily understood that the sets which appear in the corresponding theorems are Borel and in only one case we take after all the quantifiers a continuous projection and finally the complement of its image, hence the resulting set is co-analytic. Nevertheless, for the proofs to be completely exact the technical details are rather complicated mainly because of the inevitable long symbolism. However, having in mind the idea we previously described, one can easily read the exact proofs.

Initially, we limit ourselves in subsets of the Polish space that are described as above, and then using a suitable mapping we transfer to the space of chains. Characteristically we give a detailed proof in a certain case that the corresponding set is Borel, while in the other cases we quote only the description of the corresponding sets. More precisely, we define the following subsets of :

Then, we have the next proposition.

Proposition 3.1. The sets , , , are Borel subsets of , while is a co-analytic subset of .

Proof. We prove in detail that is a Borel set. Firstly, the property “ is a normalized sequence” is obviously closed and we shall not refer to it in what follows. We observe now that for any constant , , and the set

is a closed subset of . We also have that:

Therefore

and obviously is a Borel set.

Similarly, we can see that

Hence, is a Borel set.

In order to prove that is Borel, we set

where , , , and . The sets of the form are closed, while the sets of the form are open. Moreover, we have

and this implies that is a Borel set.

Similarly, for the set , we consider the following

where , , , . For constant values of the parameters the sets and are closed subsets of . Hence, is a Borel set, as it is written as follows

Finally, as far as the set , we consider the following set

It is easily verified that is a Borel subset of the product . Hence, is a co-analytic set being the complement of the first projection of .

We assume now that is a normalized tree-family in a Banach space *X*. We consider the subspace *Y* generated from this family, i.e., . The space *Y* is a separable Banach space. It is well-known that any separable Banach space embeds isometrically in C([0, 1]). Therefore, *Y* can be viewed as a subspace of C([0, 1]). We now define the mapping which in every chain associates the sequence as an element of . Obviously, φ is an embedding. Further, we set

Clearly, , , , , are the inverse images via the map φ of the sets , , , , respectively. Therefore, by Proposition 3.1, we immediately have the following corollary.

Corollary 3.2. The sets , , , are Borel subsets of and is a co-analytic subset of .

### 4 Tree-families of continuous functions

- Top of page
- Abstract
- 1 Introduction
- 2 Some combinatorial principles for the dyadic tree
- 3 Certain definable sets
- 4 Tree-families of continuous functions
- 5 The case of nearly unconditionality
- 6 The case of convex unconditionality
- 7 A dichotomy result for more general tree-families
- Acknowledgements
- References

In this section we consider tree-families of continuous functions. Then, under some conditions, we show that there exists a subtree *T* of *S* such that for any chain , the sequence is unconditional. First, we prove the following general result.

Theorem 4.1. Let be a family in a Banach space *X* such that , . Suppose that for any chain β of *S* the sequence contains an unconditional subsequence. Then there exist a subtree *T* of *S* and a constant such that for any chain β of *T*, the sequence is *C*-unconditional.

Proof. We consider the set . By Corollary 3.2, we have that is a Borel subset of . Stern's theorem implies now that there exists a subtree of *S* such that either (a) or (b) . By our hypotheses, if β is any chain of , then there is a subchain such that is an unconditional sequence. Therefore, and the case (b) is impossible. Hence, we have that , that is for any chain β of the sequence is unconditional.

It remains to prove that we can find a subtree *T* of such that the sequences , , share the same unconditional constant *C*. To avoid introducing additional notation, we assume that for any chain β of the original tree *S*, is unconditional, that is . Then, we have:

where is the set of all chains β such that is *C*-unconditional. Following the ideas of Section 'Certain definable sets' it is easy to verify that is a closed set and, hence, is an set. Therefore, the Baire category theorem implies that there exists a constant *C* such that the set has non-empty interior. This means that there are finitely many nodes such that for any chain β beginning with , the sequence is *C*-unconditional. Let *T* be the subtree consisting of the node and all its followers. Clearly, .

Combining Theorem 4.1 with Theorems 1.7 and 1.8, we obtain the following results.

Theorem 4.2. Let *K* be a Hausdorff compact space, *X* a Banach space and , , a normalized family of continuous functions. We assume that for any maximal chain , the sequence converges pointwise to zero and that there exists a positive integer such that for any . Then there exist a subtree *T* of *S* and a constant such that for any chain β of *T*, the sequence is *C*-unconditional.

Theorem 4.3. Let *K* be a Hausdorff compact space and let , be a family of continuous functions. We assume that for any maximal chain β of *S*, the sequence is uniformly bounded and converges pointwise to zero. Furthermore, we assume that there are a null sequence of positive real numbers and a constant such that for any and any either or . Then there exist a subtree *T* of *S* and a constant such that for any chain β of *T*, is a *C*-unconditional sequence.

### 5 The case of nearly unconditionality

- Top of page
- Abstract
- 1 Introduction
- 2 Some combinatorial principles for the dyadic tree
- 3 Certain definable sets
- 4 Tree-families of continuous functions
- 5 The case of nearly unconditionality
- 6 The case of convex unconditionality
- 7 A dichotomy result for more general tree-families
- Acknowledgements
- References

In this section we prove the analogous to Elton's theorem for the case of tree-families. Further, as in Theorem 4.1, we obtain a uniformity of the constants on the chains. More precisely, we have the following.

Theorem 5.1. Let be a normalized family in a Banach space *X*. Assume that for every chain , the sequence is weakly null. Then there exists a subtree *T* of *S* with the following property: for every there exists such that for any chain of *T*, any , any and any ,

That is, for any chain , the sequence is nearly unconditional and the constant is independent of the chain β.

It is well-known that any normalized weakly null sequence in a Banach space contains a Schauder basic subsequence. The proof of this result can be easily transferred to tree-families. Thus we obtain the next lemma whose proof is omitted.

Lemma 5.2. Suppose that is a normalized family in a Banach space, such that for any chain β of *S* the sequence is weakly null. Then, for every there exists a subtree *T* of *S* such that for any chain β of *T*, is -basic.

Proof of Theorem 5.1. We may assume, by passing to a subtree if necessary, that for any chain β of *S*, is a basic sequence with basis constant , where *D* is an absolute constant. We next consider the set . By Corollary 3.2, we know that is a Borel subset of . Therefore, Stern's theorem implies that there is a subtree of *S* such that either (a) or (b) . By Theorem 1.3 every normalized, weakly null sequence contains a nearly unconditional subsequence; therefore the case (b) is impossible. Thus , that is for any chain β of the sequence is nearly unconditional.

Now assume that for the original tree *S* we have . It remains to show that there is a subtree *T* of *S* such that for every there is such that for any chain of *T*, any , any and any , . That is, the constant *C* is independent of β.

We start with the following observation. We fix some positive number δ. Since for any , is nearly unconditional, it follows that

where, for fixed and , denotes the set of all chains such that for any , any and any , . It is easy to verify that is a closed subset of . Therefore, as in the proof of Theorem 4.1, the Baire category theorem implies that there exist a positive constant and a subtree *T* of *S* such that . Therefore the subtree *T* satisfies the desired property, however for the specific number δ.

In order to obtain the general result for arbitrary , we consider a null sequence and we apply a diagonal-type argument for the dyadic tree. The desired subtree *T* is constructed inductively. We quote the first steps.

Let δ be equal to 1. By the previous observation there are a subtree of *S* and a positive constant *R*(1) such that for any chain , any , any and any , . Let be the minimum element of and the nodes placed on the first level of . Then is the minimum node of *T* and complete the first level of *T*.

Let δ be equal to 1/2 and let be the subtree of which contains the node *t*_{0} and all its followers in . By the previous observation, we find a subtree and a constant such that for any chain of *T*_{0}, any , any and any , . Let be the nodes placed on the first level of *T*_{0}. Then are the successors of *t*_{0} in *T*.

The subtree , the constant and the nodes are defined in a similar way. We also set and the second level of *T* has been completed.

We inductively construct a subtree *T* of *S* and positive constants , , satisfying the following property: for any chain of *T* with , any , any and any

Claim 5.3. The subtree *T* satisfies the conclusion of the theorem.

For any level , let be an enumeration of the maximal linearly ordered subsets of *T* which contain nodes of level less or equal to *r*. We set , where is the unconditional constant of the finite sequence . Therefore, for any maximal chain of *T*, any and any scalars we have

Suppose now that *k* is a positive integer. We show that there is a constant depending only on *k* such that for any chain , any , any and any , . It suffices to consider only the maximal chains of *T*. So, if is maximal then

The choice completes the proof.

### 6 The case of convex unconditionality

- Top of page
- Abstract
- 1 Introduction
- 2 Some combinatorial principles for the dyadic tree
- 3 Certain definable sets
- 4 Tree-families of continuous functions
- 5 The case of nearly unconditionality
- 6 The case of convex unconditionality
- 7 A dichotomy result for more general tree-families
- Acknowledgements
- References

We now use the techniques of the previous sections in the case of convex unconditionality. As a result, we prove the analogous to Theorem 1.5 for tree-families.

Theorem 6.1. Let be a normalized tree-family in a Banach space *X*. Assume that for each chain , the sequence is weakly null. Then, there exists a subtree *T* of *S* with the following property: for every there exists a constant such that for any chain of *T*, any absolutely convex combination with and any sequence of signs,

That is, for any chain , is a convexly unconditional sequence and the constant depends only on δ.

Proof. The proof follows the lines of the proof of Theorem 5.1. We assume that for any chain β of *S* the sequence is *D*-basic. We consider the set . By Corollary 3.2, is a Borel subset of . Stern's theorem implies that there is a subtree of *S* such that either (a) or (b) . By Theorem 1.5 we must have (a), that is for any chain β of the sequence is convexly unconditional.

We next assume that for the original tree *S* we have and we show that there is a subtree *T* of *S* such that for any there exists depending only on δ such that for any chain of *T*, any absolutely convex combination with and any sequence of signs, .

For a fixed , as in the proof of Theorem 4.1, we find a subtree which satisfies the desired property for the specific δ. We next consider the sequence and using repeatedly the previous observation we inductively construct a subtree *T* of *S* and positive constants , , such that for any , any chain of *T* with , any absolutely convex combination with and any , .

Claim 6.2. The subtree *T* satisfies the conclusion of the theorem.

Let . We show that there is a constant depending on *k* such that for any maximal chain , any absolutely convex combination with and any sequence of signs, . We distinguish the following two cases.

*Case 1*. Suppose that . Then we have

where the constant has been defined in the proof of Theorem 5.1.

*Case 2*. Suppose that . We set and . Then and *x* is an absolutely convex combination of such that . By the construction of *T*, it follows that . Therefore

The choice completes the proof.

### 7 A dichotomy result for more general tree-families

- Top of page
- Abstract
- 1 Introduction
- 2 Some combinatorial principles for the dyadic tree
- 3 Certain definable sets
- 4 Tree-families of continuous functions
- 5 The case of nearly unconditionality
- 6 The case of convex unconditionality
- 7 A dichotomy result for more general tree-families
- Acknowledgements
- References

In this section, we consider the more general setting, where is a normalized tree-family such that for any chain β of *S*, is a Schauder basic sequence. For such families we prove the following dichotomy theorem.

Theorem 7.1. Let be a normalized tree-family in a Banach space *X* such that for any chain , the sequence is Schauder basic. Then there exists a subtree *T* of *S* such that either

- for any chain , the sequence is semi-boundedly complete; or
- for no chain , the sequence is semi-boundedly complete.

Proof. We consider the set is a semi-boundedly complete basic sequence }. By Corollary 3.2 we know that is a co-analytic subset of and, hence, we can apply Stern's theorem. It follows that there exists a subtree *T* of *S* such that either or , that is either

- for any chain , the sequence is semi-boundedly complete; or
- for no chain , the sequence is semi-boundedly complete.

Suppose now that is a normalized tree-family such that for any chain , is weakly null. In this case, using Theorem 5.1, we can improve the result of Theorem 7.1 and we obtain the following dichotomy.

Theorem 7.2. Let be a normalized tree-family in a Banach space *X*. We assume that for any chain , the sequence is weakly null. Then there exists a subtree *T* of *S* such that either

- for any chain , the sequence is semi-boundedly complete; or
- for any chain , the sequence is
*C*-equivalent to the unit vector basis of*c*_{0}, where is a common constant.

Proof. We may assume that for any chain , the sequence is *D*-basic. In view of Theorem 7.1, it suffices to consider only the case where no sequence , , is semi-boundedly complete and then we have to prove that there exists a subtree *T* of *S* such that for any chain , is equivalent to the unit vector basis of *c*_{0}. Furthermore, by Theorem 5.1 we may assume that for any chain , the sequence is nearly unconditional.

Our first step is to show that any chain β of *S* contains a subchain , such that is equivalent to the unit vector basis of *c*_{0}. The proof of this fact is essentially contained in [8]. However we shall give a brief description.

Since is not semi-boundedly complete, it follows that there exists a bounded sequence not converging to 0 so that . Clearly, we may assume that for every . Let and be such that for all . Then we set and we claim that is equivalent to the unit vector basis of *c*_{0}.

Firstly, by Theorem 5.1, we have

and further, for any signs ,

(where the constant is given by Theorem 5.1). Therefore, for any , any and any , , we obtain

It follows that

So far we have shown that any chain contains a subchain such that is equivalent to the unit vector basis of *c*_{0}. Now, we proceed as follows. We consider the set is equivalent to the unit vector basis of . By Corollary 3.2 the set is a Borel subset of . Therefore, by Theorem 1.10 there exists a subtree of *S* such that either (a) or (b) . However, the case (b) must be excluded, since for any chain the sequence contains a subsequence equivalent to the basis of *c*_{0}. Finally, as in the proof of Theorem 4.1, an application of the Baire category theorem shows that we can find a further subtree *T* of and a constant such that for any chain , is *C*-equivalent to the unit vector basis of *c*_{0}.

### Acknowledgements

- Top of page
- Abstract
- 1 Introduction
- 2 Some combinatorial principles for the dyadic tree
- 3 Certain definable sets
- 4 Tree-families of continuous functions
- 5 The case of nearly unconditionality
- 6 The case of convex unconditionality
- 7 A dichotomy result for more general tree-families
- Acknowledgements
- References

The authors would like to thank the referee for their valuable suggestions which improve the presentation of the results.

### References

- Top of page
- Abstract
- 1 Introduction
- 2 Some combinatorial principles for the dyadic tree
- 3 Certain definable sets
- 4 Tree-families of continuous functions
- 5 The case of nearly unconditionality
- 6 The case of convex unconditionality
- 7 A dichotomy result for more general tree-families
- Acknowledgements
- References

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- 3Weakly null sequences with an unconditional subsequence, Proc. Amer. Math. Soc. 134, 67–74 (2005).,
- 4Weakly null normalized sequences in Banach space, Ph. D. thesis (Yale University, 1978).,
- 5 and ,
- 6Classical Descriptive Set Theory, Graduate Texts in Mathematics Vol. 156 (Springer-Verlag, New York, 1995).,
- 7Normalized weakly null sequence with no unconditional subsequence, Studia Math. 61, 77–98 (1977).and ,
- 8Applications of Ramsey theorems to Banach space theory, in: Notes in Banach spaces, edited by H. E. Lacey (University of Texas Press, 1980), pp. 379–404.,
- 9A characterization of Banach spaces containing ℓ,
^{1}, Proc. Nat. Acad. Sci. USA 71, 2411–2413 (1974). - 10A Ramsey theorem for trees with an application to Banach spaces, Isr. J. Math. 29, 179–188 (1978).,