2 Preliminaries

In this section, we recall some necessary definitions, namely, the Schreier families $(\mathcal {S}_n)_n$ [Reference Alspach and Argyros2] and the corresponding repeated averages $\{a(n,L):n\in \mathbb {N},\; L\in [\mathbb {N}]^{\infty }\}$ [Reference Argyros, Mercourakis and Tsarpalias11] which we call n-averages, as well as the notion of special convex combinations. For a more thorough discussion of the above, we refer the reader to [Reference Argyros and Tolias13]. We begin with some useful notation.

Finally, throughout the paper, we follow [Reference Lindenstrauss and Tzafriri23] (see also [Reference Albiac and Kalton1]) for standard notation and terminology concerning Banach space theory. For $x\in c_{00}(\mathbb {N})$ , we denote $\text {supp}(x)=\{n\in \mathbb {N}:x(n)\neq 0\}$ , and by $\operatorname {\mathrm {range}}(x)$ , the minimum interval of $\mathbb {N}$ containing $\text {supp}(x)$ . Moreover, for $x,y\in c_{00}(\mathbb {N})$ , we write $x<y$ to denote that $\operatorname {\mathrm {maxsupp}}(x)<\operatorname {\mathrm {minsupp}}(y)$ .

2.1 Schreier families

For a family $\mathcal {M}$ and a sequence $(E_i)_{i=1}^k$ of finite subsets of $\mathbb {N}$ , we say that $(E_i)_{i=1}^k$ is $\mathcal {M}$ -admissible if there is $\{m_1,\ldots ,m_k\}\in \mathcal {M}$ , such that $m_1\le E_1<m_2\le E_2<\cdots <m_k\le E_k$ . Moreover, a sequence $(x_i)_{i=1}^k$ in $c_{00}(\mathbb {N})$ is called $\mathcal {M}$ -admissible if $(\text {supp} (x_i))_{i=1}^k$ is $\mathcal {M}$ -admissible. In the case where $\mathcal {M}$ is a spreading family (i.e. whenever $E=\{m_1,\ldots ,m_k\}\in \mathcal {M}$ and $F=\{n_1<\ldots <n_k\}$ satisfy $m_i\le n_i$ , $i=1,\ldots ,k$ , then $F\in \mathcal {M}$ ), a sequence $(E_i)_{i=1}^k$ is $\mathcal {M}$ -admissible if $\{\min E_i:i=1,\ldots ,k\}\in \mathcal {M}$ , and thus a sequence of vectors $(x_i)_{i=1}^k$ in $c_{00}(\mathbb {N})$ is $\mathcal {M}$ -admissible if $\{\min \text {supp} (x_i):i=1,\ldots ,k\}\in \mathcal {M}$ .

For $\mathcal {M}$ , $\mathcal {N}$ families of finite subsets of $\mathbb {N}$ , we define the convolution of $\mathcal {M}$ and $\mathcal {N}$ as follows:

$$ \begin{align*} \mathcal{M}*\mathcal{N}=\Big\{ & E\subset\mathbb{N}:\text{ there exists an }\mathcal{M}\text{-admissible finite sequence }\\&(E_i)_{i=1}^k\text{ in }\mathcal{N},\text{ such that }E=\cup_{i=1}^kE_i\Big\}\cup\big\{\emptyset\big\}. \end{align*} $$

The Schreier families $(\mathcal {S}_n)_{n\in \mathbb {N}}$ are defined inductively as follows:

$$\begin{align*}\mathcal{S}_0=\big\{ \{k\}:k\in\mathbb{N} \big\}\cup\big\{\emptyset\big\}\quad\text{and}\quad \mathcal{S}_1=\big\{E\subset\mathbb{N}:\# E\le \min E\big\}\cup\{\emptyset\} \end{align*}$$

and if $\mathcal {S}_n$ , for some $n\in \mathbb {N}$ , has been defined, then

$$ \begin{align*} \mathcal{S}_{n+1}=\mathcal{S}_1*\mathcal{S}_n=\Big\{ &E\subset\mathbb{N} : E=\cup_{i=1}^kE_i\text{ where } E_1<\ldots<E_k\in\mathcal{S}_n\ \text{and }k\le\min E_1\Big\}\cup\{\emptyset\}. \end{align*} $$

It follows easily by induction that for every $n,m\in \mathbb {N}$ ,

$$\begin{align*}\mathcal{S}_n*\mathcal{S}_{m}=\mathcal{S}_{n+m}. \end{align*}$$

Furthermore, for each $n\in \mathbb {N}$ , the family $\mathcal {S}_n$ is regular. This means that it includes the singletons, it is hereditary, that is, if $E\in \mathcal {S}_n$ and $F\subset E$ , then $F\in \mathcal {S}_n$ , it is spreading and finally it is compact, identified as a subset of $\{0,1\}^{\mathbb {N}}$ .

For each $n\in \mathbb {N}$ , we also define the regular family

$$\begin{align*}\mathcal{A}_n= \big\{ E\subset\mathbb{N} : \#E\le n \big\}. \end{align*}$$

Then, for $n,m\in \mathbb {N}$ , we are interested in the family $\mathcal {S}_n*\mathcal {A}_m$ , that is, the family of all subsets of $\mathbb {N}$ of the form $E=\cup ^k_{i=1}E_i$ , where $E_1<\ldots <E_k$ , $\#E_i\le m $ for $i=1,\ldots ,k$ and $\{\min E_i:1\le i \le k\}\in \mathcal {S}_n$ . In fact, any such E is the union of at most m sets in $\mathcal {S}_n$ , and moreover, if $m\le E$ , then $E\in \mathcal {S}_{n+1}$ , as we show next.

Lemma 2.1. For every $n,m\in \mathbb {N}$ ,

1. (i) $\mathcal {S}_n*\mathcal {A}_m\subset \mathcal {A}_m*\mathcal {S}_n$ and

2. (ii) if $E\in \mathcal {S}_n*\mathcal {A}_m$ with $m\le E$ , then $E\in \mathcal {S}_{n+1}$ .

Remark 2.2. Let $k,m\in \mathbb {N}$ and F be a subset of $\mathbb {N}$ with $\#F\le km$ and $k\le F$ . Set $d=\max \{1,\lfloor \#F/m \rfloor \}$ , and define $F_j=\{F(n):n=(j-1)d+1,\ldots ,jd\}$ for each $j=1,\ldots ,m-1$ and $F_m=F\setminus \cup _{j=1}^{m-1}F_j$ . Then, it is immediate to check that $F_j\in \mathcal {S}_1$ for every $i=1,\ldots ,m$ .

Proof of Lemma 2.1.

Fix $n,m\in \mathbb {N}$ . We prove (i) by induction on $n\in \mathbb {N}$ . For $n=1$ , let $E\in \mathcal {S}_1*\mathcal {A}_m$ , that is, $E=\cup _{i=1}^kE_i$ with $k\le E_1<\ldots <E_k$ and $\#E_i\le m$ for every $i=1,\ldots ,k$ . Since $\#E\le km$ , Remark 2.2 yields a partition $E=\cup _{j=1}^m F_j$ with $F_j\in \mathcal {S}_1$ for every $j=1,\ldots ,m$ , and, hence, $E\in \mathcal {A}_m*\mathcal {S}_1$ .

Suppose that (i) holds for some $n\in \mathbb {N}$ and let $E\in \mathcal {S}_{n+1}*\mathcal {A}_m$ . Then $E=\cup _{i=1}^kE_i$ for an $\mathcal {S}_{n+1}$ -admissible sequence $(E_i)_{i=1}^k$ with $\#E_i\le m$ for every $i=1,\ldots ,m$ . Hence, $\{\min E_i:i=1,\ldots ,k\}=\cup _{j=1}^lF_j$ , where $F_j\in \mathcal {S}_n$ for every $j=1,\ldots ,l$ and $l\le F_1<\cdots <F_l$ . Define, for each $j=1,\ldots ,l$ ,

$$\begin{align*}G_j=\cup\{E_i:i=1,\ldots,k\text{ and } \min E_i\in F_j\},\end{align*}$$

and note that $G_j\in \mathcal {S}_{n}*\mathcal {A}_m$ since $F_j\in \mathcal {S}_n$ . Hence, for every $j=1,\ldots ,l$ , the inductive hypothesis implies that $G_j\in \mathcal {A}_m*\mathcal {S}_n$ , that is, $G_j=\cup _{i=1}^{m_j}G^j_i$ with $m_j\le m$ and $G^j_i\in \mathcal {S}_n$ for all $i=1,\ldots ,m_j$ . Define

$$\begin{align*}H=\{\min G^j_i:j=1,\ldots,l,\text{ and }i=1,\ldots,m_j\}.\end{align*}$$

Observe that $H\in \mathcal {S}_1*\mathcal {A}_m$ and apply Remark 2.2 to obtain a partition $H=\cup _{q=1}^mH_q$ , where $H_q\in \mathcal {S}_1$ for every $q=1,\ldots ,m$ . Finally, define

$$\begin{align*}\Delta_q=\cup\{G^j_i:j=1,\ldots,l,\;i=1,\ldots,m_j\text{ and } \min G^j_i\in H_q\},\end{align*}$$

for each $q=1,\ldots ,m$ , and observe that $E=\cup _{q=1}^m\Delta _q$ and that $\Delta _q\in \mathcal {S}_1*S_{n}=S_{n+1}$ since $H_q\in \mathcal {S}_1$ and $G^j_i\in \mathcal {S}_{n}$ . Thus, we conclude that $E\in \mathcal {A}_m*\mathcal {S}_{n+1}$ .

Finally, note that (ii) is an immediate consequence of (i).

2.2 Repeated averages

The notion of repeated averages was first defined in [Reference Argyros, Mercourakis and Tsarpalias11]. The notation we use below, however, is somewhat different, and we instead follow the one found in [Reference Argyros and Tolias13], namely, $\{a(n,L):n\in \mathbb {N},\; L\in [\mathbb {N}]^{\infty }\}$ . The $n-$ averages $a(n,L)$ are defined as elements of $c_{00}(\mathbb {N})$ in the following manner.

Let $(e_j)_j$ denote the unit vector basis of $c_{00}(\mathbb {N})$ and $L\in [\mathbb {N}]^{\infty }$ . For $n=0$ , we define $a(0,L)=e_{l_1}$ , where $l_1=\min L$ . Suppose that $a(n,M)$ has been defined for some $n\in \mathbb {N}$ and every $M\in [\mathbb {N}]^{\infty }$ . We define $a(n+1,L)$ in the following way: We set $L_1=L$ and $L_k=L_{k-1}\setminus \text {supp}( a(n,L_{k-1}))$ for $k=2,\ldots ,l_1$ and finally define

$$\begin{align*}a(n+1,L)=\frac{1}{l_1}\big( a(n,L_1) + \cdots + a(n,L_{l_1}) \big). \end{align*}$$

Remark 2.3. Let $n\in \mathbb {N}$ and $L\in [\mathbb {N}]^{\infty }$ . The following properties are easily established by induction.

1. (i) $a(n,L)$ is a convex combination of the unit vector basis of $c_{00}(\mathbb {N})$ .

2. (ii) $\|a(n,L)\|_{\ell _1}=1$ and $a(n,L)(k)\ge 0$ for all $k\in \mathbb {N}$ .

3. (iii) $\text {supp} (a(n,L))$ is the maximal initial segment of L contained in $\mathcal {S}_n$ .

4. (iv) $\|a(n,L)\|_{\infty }= l_1^{-n}$ , where $l_1=\min L$ .

5. (v) If $\text {supp}(a(n,L))=\{i_1<\ldots <i_d\}$ and $a(n,L)=\sum _{k=1}^da_{i_k}e_{i_k}$ , then we have that $a_{i_1}\ge \ldots \ge a_{i_d}$ .

A proof of the following proposition can be found in [Reference Argyros and Tolias13].

Proposition 2.4. Let $n\in \mathbb {N}$ and $L\in [\mathbb {N}]^{\infty }$ . For every $F\in \mathcal {S}_{n-1}$ , we have that

$$\begin{align*}\sum_{k\in F}a(n,L)(k)<\frac{3}{\min L}. \end{align*}$$

3 Asymptotic structures

Let us recall the definitions of the asymptotic notions that appear in the results of this paper and were mentioned in the Introduction. Namely, asymptotic models, joint spreading models and the notions of Asymptotic $\ell _p$ and Asymptotic $c_0$ spaces. For a more thorough discussion, including several open problems and known results, we refer the reader to [Reference Argyros and Motakis9, Section 3].

Definition 3.1 [Reference Halbeisen and Odell20].

An infinite array of sequences $(x^{i}_j)_j$ , $i\in \mathbb {N}$ , in a Banach space X, is said to generate a sequence $(e_i)_i$ , in a seminormed space E, as an asymptotic model if for every $\varepsilon>0$ and $n\in \mathbb {N}$ , there is a $k_0\in \mathbb {N}$ , such that for any natural numbers $k_0\leq k_1<\cdots <k_n$ and any scalars $a_1,\ldots ,a_n$ in $[-1,1]$ , we have

$$\begin{align*}\Big|\big\|\sum_{i=1}^na_ix_{k_i}^{i}\big\| - \big\|\sum_{i=1}^na_ie_{i}\big\|\Big| < \varepsilon.\end{align*}$$

A Banach space X is said to admit a unique asymptotic model with respect to a family $\mathscr {F}$ of normalised sequences in X if whenever two infinite arrays, consisting of sequences from $\mathscr {F}$ , generate asymptotic models, then those must be equivalent. Typical families under consideration are those of normalised weakly null sequences, denoted $\mathscr {F}_0(X)$ , normalised Schauder basis sequences, denoted $\mathscr {F}(X)$ , or the family of all normalised block sequences of a fixed basis of X, if it has one, denoted $\mathscr {F}_b(X)$ .

Definition 3.3 [Reference Argyros, Georgiou, Lagos and Motakis6].

A finite array of sequences $(x^{i}_j)_j$ , $1\leq i\leq l$ , in a Banach space X, is said to generate another array of sequences $(e_j^{i})_j$ , $1\leq i\leq l$ , in a seminormed space E, as a joint spreading model if for every $\varepsilon>0$ and $n\in \mathbb {N}$ , there is a $k_0\in \mathbb {N}$ , such that for any $(s_i)_{i=1}^l\in S$ - $Plm_{l}([\mathbb {N}]^n)$ with $k_0\le s_1(1)$ and for any $l\times n$ matrix $A=(a_{ij})$ with entries in $[-1,1]$ , we have that

$$\begin{align*}\Big|\big\|\sum_{i=1}^l\sum_{j=1}^na_{ij}x_{s_i(j)}^{i}\big\| - \big\|\sum_{i=1}^l\sum_{j=1}^na_{ij}e_j^{i}\big\|\Big|<\varepsilon.\end{align*}$$

A Banach space X is said to admit a uniformly unique joint spreading model with respect to a family of normalised sequences $\mathscr {F}$ in X if there exists a constant C, such that whenever two arrays $(x_j^{i})_j$ and $(y_j^{i})_j$ , $1\leq i\leq l$ , of sequences from $\mathscr {F}$ generate joint spreading models, then those must be C-equivalent. Moreover, a Banach space admits a uniformly unique joint spreading model with respect to a family $\mathscr {F}$ if and only if it admits a unique asymptotic model with respect to $\mathscr {F}$ (see, e.g. [Reference Argyros, Georgiou, Lagos and Motakis6, Remark 4.21] or [Reference Argyros and Motakis9, Proposition 3.12]). In particular, if a space admits a uniformly unique joint spreading model with respect to some family $\mathscr {F}$ satisfying certain conditions described in [Reference Argyros, Georgiou, Lagos and Motakis6, Proposition 4.9], then this is equivalent to some $\ell _p$ . In order to show that a space admits some $\ell _p$ as a uniformly unique joint spreading model, it may be more convenient to prove (ii) of the following lemma, thereby avoiding the use of plegma families.

Proof. Note that (i) implies that there are constants $c,C>0$ , such that for every array $(x^i_j)_j$ , $1\le i\le l$ , of sequences from $\mathscr {F}$ , there is $N=\{n_1<n_2<\ldots \}$ , an infinite subset of the naturals, such that for any k, any strict plegma family $(s_i)_{i=1}^l\in S\text {-}Plm_l([N]^k)$ with $n_k\le s_1(1)$ and any $l\times k$ matrix $A=(a_{ij})$ of scalars, we have that

$$\begin{align*}c \|(a_{ij})_{i=1,j=1}^{l,k}\|_p \le \big\| \sum_{i=1}^l\sum_{j=1}^k a_{ij} x^{i}_{s_i(j)} \big\| \le C \|(a_{ij})_{i=1,j=1}^{l,k}\|_p. \end{align*}$$

Let $N'=\{ n_{2kl}:k\in \mathbb {N} \}$ and observe that for $k_1,\ldots ,k_d\in \mathbb {N}$ , there is a strict plegma family $(s_i)_{i=1}^l\in S\text {-}Plm_l([N]^d )$ , such that $n_{2k_jl}\in \{s_i(j):i=1,\ldots ,l\}$ for all $j=1,\ldots , d$ . Hence, we may find $M\subset N'$ satisfying (ii) with constants $c,C$ . Finally, by repeating the sequences in the array, it follows easily that (ii) yields (i).

We recall the main result from [Reference Argyros, Georgiou, Lagos and Motakis6], stating that whenever a Banach space admits a uniformly unique joint spreading model with respect to some family satisfying certain stability conditions, then it satisfies a property concerning its bounded linear operators called the Uniform Approximation on Large Subspaces property (see [Reference Argyros, Georgiou, Lagos and Motakis6, Theorem 5.17] and [Reference Argyros, Georgiou, Lagos and Motakis6, Theorem 5.23]).

Although this is not the initial formulation, it is equivalent and follows from [Reference Maurey, Milman and Tomczak-Jaegermann24, Subsection 1.5]. The typical example of a nonclassical Asymptotic $\ell _p$ space is the Tsirelson space from [Reference Figiel and Johnson17]. This is a reflexive Asymptotic $\ell _1$ space, and it is the dual of Tsirelson’s original space from [Reference Tsirelson27] which is Asymptotic $c_0$ . Finally, whenever a Banach space is Asymptotic $\ell _p$ or Asymptotic $c_0$ , it admits a uniformly unique joint spreading model with respect to $\mathscr {F}_0(X)$ (see, e.g. [Reference Argyros, Georgiou, Lagos and Motakis6, Corollary 4.12]).

The above definition is the coordinate-free version of the notion of an asymptotic $\ell _p$ Banach space with a basis introduced by Milman and Tomczak-Jaegermann in [Reference Milman and Tomczak-Jaegermann25].

Definition 3.6 [Reference Milman and Tomczak-Jaegermann25].

Let X be a Banach space with a Schauder basis $(e_i)_i$ and $1\le p< \infty $ . We say that the Schauder basis $(e_i)_i$ of X is asymptotic $\ell _p$ if there exist positive constants $D_1$ and $D_2$ , such that for all $n\in \mathbb {N}$ , there exists $N(n)\in \mathbb {N}$ with the property that whenever $N(n)\le x_1< \cdots < x_n$ are vectors in X, then

$$\begin{align*}\frac{1}{D_1}(\sum_{i=1}^n\|x_i\|^p)^{\frac{1}{p}}\le \|\sum_{i=1}^nx_i\|\le D_2 (\sum_{i=1}^n\|x_i\|^p)^{\frac{1}{p}}. \end{align*}$$

Specifically, we say that $(e_i)_i$ is D-asymptotic $\ell _p$ for $D=D_1D_2$ . The definition of an asymptotic $c_0$ space is given similarly.

It is easy to show that if X has a Schauder basis that is asymptotic $\ell _p$ , then X is Asymptotic $\ell _p$ . Moreover, if X is Asymptotic $\ell _p$ , then it contains an asymptotic $\ell _p$ sequence. In particular, note that if X has a Schauder basis and Y is an Asymptotic $\ell _p$ subspace of X, then Y contains a further subspace that is isomorphic to an asymptotic $\ell _p$ block subspace.

A noteworthy remark is that sequential asymptotic properties, array asymptotic properties and global asymptotic properties of a Banach space X can alternatively be interpreted as properties of special weakly null trees. A collection $\{x_A:A\in [\mathbb {N}]^{\leq n}\}$ in X is said to be a normalised weakly null tree of height n, if for every $A\in [\mathbb {N}]^{\leq n-1}$ , $(x_{A\cup \{j\}})_{j>\max (A)}$ is a normalised weakly null sequence. Such a tree is said to originate from a sequence $(x_j)_j$ if for all $A = \{a_1,\ldots ,a_i\}$ , we have $x_A = x_{a_i}$ . Similarly, a tree $\{x_A:A\in [\mathbb {N}]^{\leq n}\}$ is said to originate from an array of sequences $(x^{(i)}_j)_j$ , $1\leq i\leq n$ if for all $A = \{a_1,\ldots ,a_i\}$ , we have $x_A = x^{(i)}_{a_i}$ . Then, X has a uniformly unique $\ell _p$ spreading model if and only if there exists $C>0$ , so that every tree $\{x_A:A\in [\mathbb {N}]^{\leq n}\}$ originating from a normalised weakly null sequence $(x_j)_j$ in X has a maximal branch that is C-equivalent to the unit vector basis of $\ell _p^n$ . Similarly, X has a unique $\ell _p$ asymptotic model if the same can be said about all trees originating from normalised weakly null arrays in X. Finally, a Banach space X is an Asymptotic $\ell _p$ space (or an Asymptotic $c_0$ space if $p=\infty $ ) if there exists $C>0$ , so that every normalised weakly null tree of height n has a maximal branch $x_{\{a_1\}}, x_{\{a_1,a_2\}},\ldots ,x_{\{a_1,a_2,\ldots ,a_n\}}$ that is C-equivalent to the unit vector basis of $\ell _p^n$ . For more details, see [Reference Baudier, Lancien, Motakis and Schlumprecht14, Remark 3.11].

4 Measures on countably branching well-founded trees

In this section, we recall certain results from [Reference Argyros, Georgiou and Motakis8] concerning measures on countably branching well-founded trees. These will be used to prove that for all $1\le p<\infty $ , the space $\mathfrak {X}^{(p)}_{\text {awi}}$ admits $\ell _p$ as a unique asymptotic model. In particular, Proposition 4.1 and Lemma 4.6 will be used to prove Lemma 7.2, which is one of the key ingredients in the proof of the main result, Theorem 1.4.

Let $\mathcal {T}=(A,<_{\mathcal {T}})$ , where A is a countably infinite set equipped with a partial order $<_{\mathcal {T}}$ . In the sequel, we use $t\in \mathcal {T}$ instead of $t\in A$ . We assume that $<_{\mathcal {T}}$ is such that there is a unique minimal element in $\mathcal {T}$ , and for each $t\in \mathcal {T}$ , the set $S_t=\{s\in \mathcal {T}: s\le _{\mathcal {T}} t \}$ is finite and totally ordered, that is, $\mathcal {T}$ is a rooted tree. We also assume that $\mathcal {T}$ is well founded, that is, it contains no infinite totally ordered sets, and countably branching, that is, every nonmaximal node has countably infinite immediate successors.

Observe that ${\widetilde {\mathcal {T}}}=(\{S_t :{t\in \mathcal {T}}\},<_{\widetilde {\mathcal {T}}})$ , where $<_{\widetilde {\mathcal {T}}}$ denotes inclusion, is also a tree, and that $\mathcal {T}$ is in fact isomorphic to ${\widetilde {\mathcal {T}}}$ via the mapping $t\mapsto S_t$ . Given $t\in \mathcal {T}$ , we will denote $S_t$ by ${\tilde {t}}$ , identifying it as an element of ${\widetilde {\mathcal {T}}}$ . For each ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , we denote by $S({\tilde {t}})$ the set of immediate successors of ${\tilde {t}}$ in ${\widetilde {\mathcal {T}}}$ . In particular, if ${\tilde {t}}$ is maximal, then $S({\tilde {t}})$ is empty. Moreover, for ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , we denote $V_{\tilde {t}} = \{\tilde {s}\in {\widetilde {\mathcal {T}}}:{\tilde {t}}\leq _{\widetilde {\mathcal {T}}} \tilde {s}\}$ and view ${\widetilde {\mathcal {T}}}$ as a topological space with the topology generated by the sets $V_{\tilde {t}}$ and ${\widetilde {\mathcal {T}}}\setminus V_{\tilde {t}}$ , ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , that is, the pointwise convergence topology. This is a compact metric topology, such that for each ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , the sets of the form $V_{\tilde {t}}\setminus (\cup _{\tilde {s}\in F}V_{\tilde {s}})$ , $F\subset S({\tilde {t}})$ finite, form a neighbourhood base of clopen sets for $\tilde t$ . We denote by $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ the cone of all bounded positive measures $\mu :\mathcal {P}({\widetilde {\mathcal {T}}})\to [0,+\infty )$ . For $\mu \in \mathcal {M}_+({\widetilde {\mathcal {T}}})$ , we define the support of $\mu $ to be the set $\mathrm {supp}(\mu ) = \{{\tilde {t}}\in {\widetilde {\mathcal {T}}}:\mu (\{{\tilde {t}}\})>0\}$ . Finally, we say that a subset $\mathcal {A}$ of $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ is bounded if $\sup _{\mu \in \mathcal {A}}\mu ({\widetilde {\mathcal {T}}})<\infty $ .

For the proof, we refer the reader to [Reference Argyros, Georgiou and Motakis8, Proposition 3.1].

Recall that a bounded sequence $(\mu _i)_i$ in $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ converges in the $w^*$ -topology to a $\mu \in \mathcal {M}_+({\widetilde {\mathcal {T}}})$ if and only if for all clopen sets $V\subset {\widetilde {\mathcal {T}}}$ , we have $\lim _i\mu _i(V) = \mu (V)$ if and only if for all ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , we have $\lim _i\mu _i(V_{\tilde {t}}) = \mu (V_{\tilde {t}})$ . In this case, we write $\mu = w^*\text {-}\lim _i\mu _i$ .

Although these limits are not necessarily the same, there is an explicit formula relating $\mathrm {succ}\text {-}\!\lim _i\mu _i$ to $w^*\text {-}\lim _i\mu _i$ .

Lemma 4.5. Let $(\mu _i)_i$ be a bounded and disjointly supported sequence in $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ , such that $w^*\text {-}\lim _i\mu _i = \mu $ exists, and for all ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , the limit $\nu (\{{\tilde {t}}\}) = \lim _i\mu _i(S({\tilde {t}})) $ exists as well. Then for every ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ and enumeration $({\tilde {t}}_j)_j$ of $S({\tilde {t}})$ , we have

(4.1) $$ \begin{align} \mu(\{{\tilde{t}}\}) = \nu(\{{\tilde{t}}\}) + \lim_j\lim_i\mu_i\Big(\cup_{k\geq j}(V_{{\tilde{t}}_k}\setminus\{{\tilde{t}}_{k}\})\Big). \end{align} $$

In particular, $\mu (\{{\tilde {t}}\}) = \nu (\{{\tilde {t}}\})$ if and only if the double limit in (4.1) is zero.

For the proofs, we refer the reader to [Reference Argyros, Georgiou and Motakis8, Lemma 4.10] and [Reference Argyros, Georgiou and Motakis8, Lemma 4.12].

5 Definition of the space $\mathfrak {X}_{\text {awi}}^{(1)}$

The method of saturation with asymptotically weakly incomparable constraints, that is used in the construction of both spaces presented in this paper, was introduced in [Reference Argyros, Georgiou and Motakis8], where it was shown that (b) $_1\not \Rightarrow $ (a) $_1$ . There, it was also used to prove an even stronger result, namely, the existence of a Banach space with a basis admitting $\ell _1$ as a unique asymptotic model, and in which any infinite subsequence of the basis generates a non-Asymptotic $\ell _1$ subspace. This method requires the existence of a well-founded tree defined either on the basis of the space or on a family of functionals of its norming set. In this section, we define the space $\mathfrak {X}_{\text {awi}}^{(1)}$ by introducing its norm via a norming set, which is a subset of the norming set of a Mixed Tsirelson space $\mathcal {T}[(m_j,\mathcal {S}_{n_j})_j]$ for an appropriate choice of $(m_j)_j$ and $(n_j)_j$ described below. The key ingredient in the definition of this norming set is the notion of asymptotically weakly incomparable sequences of functionals, which is also introduced in this section. This notion will allow the space $\mathfrak {X}_{\text {awi}}^{(1)}$ to admit $\ell _1$ as a unique asymptotic model, while at the same time, it will force the norm to be small on the branches of a tree, in every subspace of $\mathfrak {X}_{\text {awi}}^{(1)}$ , showing that the space does not contain Asymptotic $\ell _1$ subspaces.

5.1 Definition of the space $\mathfrak {X}_{\text {awi}}^{(1)}$

Define a pair of strictly increasing sequences of natural numbers $(m_j)_j$ , $(n_j)_j$ as follows:

$$ \begin{align*} m_1 &= 2 & n_1 &= 1 \\ m_{j+1}&=m_j^{m_j} & n_{j+1}&=2^{2m_{j+1}}n_j .\end{align*} $$

Definition 5.1. Let $V_{(1)}$ denote the minimal subset of $c_{00}(\mathbb {N})$ that

1. (i) contains 0 and all $\pm e_j^*$ , $j\in \mathbb {N}$ and

2. (ii) is closed under the operations $(m_j,\mathcal {S}_{n_j})_j$ , that is, if $j\in \mathbb {N}$ and $f_1<\ldots <f_n$ is an $\mathcal {S}_{n_j}$ -admissible sequence (see Section 2.1) in $V_{(1)}\setminus \{0\}$ , then $m_j^{-1}\sum _{i=1}^nf_i $ is also in $V_{(1)}$ .

Remark 5.2.

1. (i) If $f\in V_{(1)}\setminus \{0\}$ , then either $f\in \{\pm e^*_j:j\in \mathbb {N}\}$ , or it is of the form $f=m_j^{-1}\sum _{i=1}^nf_i$ with $f_1<\ldots <f_n$ an $\mathcal {S}_{n_j}$ -admissible sequence in $V_{(1)}$ for some $j\in \mathbb {N}$ .

2. (ii) As usual, we view the elements of $V_{(1)}$ as functionals acting on $c_{00}(\mathbb {N})$ , inducing a norm $\|\cdot \|_{V_{(1)}}$ . The completion of $(c_{00}(\mathbb {N}),\|\cdot \|_{V_{(1)}})$ is the Mixed Tsirelson space $\mathcal {T}[(m_j,\mathcal {S}_{n_j})_j]$ introduced for the first time in [Reference Argyros and Deliyanni5]. The first space with a saturated norm defined by a countable family of operations is the Schlumprecht space [Reference Schlumprecht26], which is a fundamental discovery and was used by Gowers and Maurey [Reference Gowers and Maurey19] to define the first hereditarily indecomposable (HI) space.

We now recall the notion of tree analysis which appeared for the first time in [Reference Argyros and Deliyanni4]. This has become a standard tool in proving upper bounds for the estimations of functionals on certain vectors in Mixed Tsirelson spaces. However, it is the first time where the tree analysis has a significant role in the definition of the norming set $W_{(1)}$ . Additionally, it is also a key ingredient in the proof that $\mathfrak {X}^{(1)}_{\text {awi}}$ contains no Asymptotic $\ell _1$ subspaces.

Let $\mathcal {A}$ be a rooted tree. For a node $\alpha \in \mathcal {A}$ , we denote by $S(\alpha )$ the set of all immediate successors of $\alpha $ , by $|\alpha |$ the height of $\alpha $ , that is, $|\alpha |=\#\{\beta \in \mathcal {A}:\beta <_{\mathcal {A}}\alpha \}$ , and finally, we denote by $h(\mathcal {A})$ the height of $\mathcal {A}$ , that is, the maximum height over its nodes.

Definition 5.3. Let $f\in V_{(1)}\setminus \{0\}$ . For a finite tree $\mathcal {A}$ , a family $(f_{\alpha })_{\alpha \in \mathcal {A}}$ is called a tree analysis of f if the following are satisfied.

1. (i) $\mathcal {A}$ has a unique root denoted by $0$ and $f_0=f$ .

2. (ii) Each $f_{\alpha }$ is in $V_{(1)}$ , and if $\beta <\alpha $ in $\mathcal {A}$ , then $\text {range}(f_{\alpha })\subset \text {range}(f_{\beta })$ .

3. (iii) For every maximal node $\alpha \in \mathcal {A}$ , we have that $|\alpha |=h(\mathcal {A})$ .

4. (iv) For every nonmaximal node $\alpha \in \mathcal {A}$ , either $f_{\alpha }$ is the result of some $(m_j,\mathcal {S}_{n_j})$ operation of $(f_{\beta })_{\beta \in S(\alpha )}$ , i.e., $f_{\alpha }=m_j^{-1}\sum _{\beta \in S(\alpha )} f_{\beta }$ , or $f_{\alpha }\in \{\pm e_j^*:j\in \mathbb {N}\}$ and $S(\alpha )=\{\beta \}$ with $f_{\beta }=f_{\alpha }$ .

5. (v) For every maximal node $\alpha \in \mathcal {A}$ , $f_{\alpha }\in \{\pm e^*_j:j\in \mathbb {N}\}.$

Remark 5.4.

1. (i) It follows by minimality that every f in $V_{(1)}\setminus \{0\}$ admits a tree analysis, but it may not be unique. For example, $f=(m_{1}m_2)^{-1}e^*_1$ admits two distinct tree analyses.

2. (ii) The standard definition of a tree analysis does not include 5.3 (iii). This property is included for technical reasons and is used below in the equality of Remark 5.8 (i).

Definition 5.5. Let $f\in V_{(1)}$ .

1. (i) If $f=0$ or $f\in \{\pm e^*_j:j\in \mathbb {N}\}$ , then we define the weight $w(f)$ of f as $w(f)=0$ and $w(f)=1$ , respectively.

2. (ii) If f is the result of an $(m_j,S_{n_j})$ -operation for some $j\in \mathbb {N}$ , then $w(f)=m_j$ .

Remark 5.6. It is not difficult to see that $w(f)$ , for $f\in V_{(1)}$ , is not uniquely determined, that is, f could be the result of more than one distinct $(m_j,\mathcal {S}_{n_j})$ -operation. However, if we fix a tree analysis $(f_{\alpha })_{\alpha \in \mathcal {A}}$ of f, then for $\alpha \in \mathcal {A}$ with $f_{\alpha }=(m_{j_{\alpha }})^{-1}\sum _{\beta \in S(\alpha )}f_{\beta }$ , the tree analysis determines the weight $w(f_{\alpha })$ , being equal to $m_{j_{\alpha }}$ . Thus, for $f\in V_{(1)}$ and a fixed tree analysis $(f_{\alpha })_{\alpha \in \mathcal {A}}$ of f, with $w(f_{\alpha })$ , we will denote the weight $m_{j_{\alpha }}$ determined by $(f_{\alpha })_{\alpha \in \mathcal {A}}$ , for every $\alpha \in \mathcal {A}$ . In addition, we will denote by $\bar {f}_{\alpha }$ the pair $(f_{\alpha },m_{j_{\alpha }})$ .

Definition 5.7. Let $f\in V_{(1)}$ and $(f_{\alpha })_{\alpha \in \mathcal {A}}$ be a tree analysis of f. Then for $\alpha \in \mathcal {A}$ , we define the relative weight $w_f(f_{\alpha })$ of $f_{\alpha }$ as

$$\begin{align*}w_f(f_{\alpha}) = \begin{cases} \prod_{\beta<\alpha}w(f_{\beta}) & \text{if }\alpha\neq0\\ 1 &\text{otherwise.} \end{cases} \end{align*}$$

Remark 5.8. Let $f\in V_{(1)}$ and $(f_{\alpha })_{\alpha \in \mathcal {A}}$ be a tree analysis of f.

1. (i) For every $k=1,\ldots ,h(\mathcal {A})$

   $$\begin{align*}f=\sum_{|a|=k}w_f(f_{\alpha})^{-1}f_{\alpha}. \end{align*}$$
   This can be proved by induction and essentially relies on the fact that $(f_{\alpha })_{\alpha \in \mathcal {A}}$ satisfies 5.3 (iii).

2. (ii) If $\mathcal {B}$ is a maximal pairwise incomparable subset of $\mathcal {A}$ , then

   $$\begin{align*}f=\sum_{\beta\in\mathcal{B}}w_f(f_{\beta})^{-1}f_{\beta}. \end{align*}$$

3. (iii) For every $\alpha \in \mathcal {A}$ , whose immediate predecessor $\beta $ in $\mathcal {A}$ (if one exists) satisfies $f_{\beta }\notin \{\pm e_j^*:j\in \mathbb {N}\}$ , we have $w_f(f_{\alpha })\ge 2^{|\alpha |}.$

Fix an injection $\sigma $ that maps any pair $(f,w(f))$ , for $f\in V_{(1)}$ and $w(f)$ a weight of f, to some $m_j$ with $m_j>\max \text {supp} (f)\: w(f)$ whenever $f\neq 0$ .

Definition 5.9. Define a partial order $<_{\mathcal {T}}$ on the set of all pairs $(f,w(f))$ for $f\in V_{(1)}$ and $w(f)$ a weight of f, as follows: $(f,w(f))<_{\mathcal {T}} (g ,w(g))$ either if $f=0$ or if there exist $f_1<\ldots <f_n \in V_{(1)}$ and weights $w(f_1),\ldots ,w(f_n)$ , such that

1. (i) $(f_i)_{i=1}^n$ is $\mathcal {S}_1$ -admissible,

2. (ii) $w(f_1)=\sigma (0,0)$ and $w(f_i)=\sigma (f_{i-1},w(f_{i-1}))$ for every $i=2,\ldots ,n$ ,

3. (iii) there are $1\le i_1<i_2\le n$ , such that $f=f_{i_1}$ and $g=f_{i_2}$ .

It is easy to see that $<_{\mathcal {T}}$ induces a tree structure rooted at $\bar {0}=(0,0)$ . Let us denote this tree by $\mathcal {T}$ , and observe that this is a countably branching well-founded tree, due to 5.9(i). For $t=(f,w(f))\in \mathcal {T}$ , we set $f_t=f$ and $w(t)=w(f)$ .

It is clear that unlike the case where the tree is defined on the basis of the space, here, incomparable segments need not necessarily have disjoint supports. This forces us to introduce the notion of essentially incomparable nodes, which was first defined in [Reference Argyros, Georgiou and Motakis8]. To this end, we first need to define an additional tree structure that is readily implied by $\mathcal {T}$ via the projection $(f,w(f))\mapsto w(f)$ .

Definition 5.10. Define a partial order $<_{\mathcal {W}}$ on $\{ m_j:j\in \mathbb {N} \}$ as follows: $m_i<_{\mathcal {W}} m_j$ if there exist $t_{1},t_{2}\in \mathcal {T}$ , such that $t_1<_{\mathcal {T}} t_2$ , $w(t_1)=m_i$ and $w(t_2)=m_j$ .

As an immediate consequence of the fact that $\mathcal {T}$ is a countably branching well-founded tree, we have that $<_{\mathcal {W}}$ also defines a tree structure. Let us denote this tree by $\mathcal {W}$ and note that it is also countably branching and well founded.

Remark 5.11. The above definition implies that if $t_1,t_2\in \mathcal {T}$ are such that $w(t_1)<_{\mathcal {W}} w(t_2)$ , then there exist $t_3,t_4\in \mathcal {T}$ , such that $t_3<_{\mathcal {T}} t_4$ , $w(t_3)=w(t_1)$ and $w(t_4)=w(t_2)$ . The tree structure of $\mathcal {T}$ implies that $t_3$ is uniquely defined, and we will say that $t_3$ generates $w(t_2)$ . This is not the case, however, for $t_4$ , and, moreover, it is not necessary that $t_3<_{\mathcal {T}} t_2$ .

Definition 5.12.

1. (i) A subset A of $\mathcal {T}\setminus \{\bar {0}\}$ is called essentially incomparable if whenever $t_1,t_2\in A$ are such that $w(t_1)<_{\mathcal {W}} w(t_2)$ , then for the unique $t_3\in \mathcal {T}$ with $w(t_3)=w(t_1)$ that generates $w(t_2)$ , we have that $f_{t_3}<f_{t_1}$ .

2. (ii) A subset A of $\mathcal {T}$ is called weight incomparable if for any $t_1\neq t_2$ in A, $w(t_1)\neq w(t_2)$ and the weights $w(t_1)$ and $w(t_2)$ are incomparable in $\mathcal {W}$ .

3. (iii) A sequence $(A_j)_j$ of subsets of $\mathcal {T}$ is called pairwise weight incomparable if for every $j_1\neq j_2$ in $\mathbb {N}$ , $t_1\in A_{j_1}$ and $t_2\in A_{j_2}$ , $w(t_1)\neq w(t_2)$ and the weights $w(t_1)$ and $w(t_2)$ are incomparable in $\mathcal {W}$ .

Remark 5.13.

1. (i) If A is an essentially (respectively, weight) incomparable subset of $\mathcal {T}$ , then every $B\subset A$ is also essentially (respectively, weight) incomparable.

2. (ii) Any subsequence of a pairwise weight incomparable sequence in $\mathcal {T}$ is also pairwise weight incomparable.

3. (iii) Any weight incomparable subset of $\mathcal {T}$ is essentially incomparable.

4. (iv) Let $A=\{(f,1):f\in \{\pm e^*_j:j\in \mathbb {N}\} \}$ . Then A is essentialy incomparable, and, additionally, if $B\subset \mathcal {T}$ is essentially incomparable, then the same holds for $A\cup B$ .

We can finally describe the rule used to define the norming set $W_{(1)}$ of $\mathfrak {X}_{\text {awi}}^{(1)}$ , namely, asymptotically weakly incomparable constraints.

Definition 5.14. Let J be an initial segment of $\mathbb {N}$ or $ J =\mathbb {N}$ . Then a sequence $(f_j)_{j\in J}$ of functionals with successive supports in $V_{(1)}\setminus \{0\}$ is called asymptotically weakly incomparable (AWI) if each $f_j$ admits a tree analysis $(f_{j,\alpha })_{\alpha \in \mathcal {A}_j}$ , $j\in J$ , such that the following are satisfied.

1. (i) There is a partition $\{\bar {f}_j:j\in J\}= C_1^0\cup C_2^0$ , such that $C^0_1$ is essentially incomparable and $C^0_2$ is weight incomparable.

2. (ii) For every $k,j\in J$ with $j\ge k+1$ , there exists a partition

   $$\begin{align*}\{\bar{f}_{j,\alpha}:\alpha\in \mathcal{A}_j\text{ and }|\alpha|=k\}=C^{k}_{1,j}\cup C^{k}_{2,j}, \end{align*}$$
   such that $\cup _{j=k+1}^{\infty } C^k_{1,j}$ is essentially incomparable and $(C^{k}_{2,j})_{j=k+1}^{\infty }$ is pairwise weight incomparable.

Figure 1 The collection of nodes of a fixed level in rectangles across all tree analyses forms an essentially incomparable subset, while circles across a fixed level form a family of pairwise weight incomparable subsets.

Before defining the space $\mathfrak {X}_{\text {awi}}^{(1)}$ , we prove that AWI sequences are stable under taking subsequences and under taking restrictions of functionals to subsets. This fact will imply the unconditionality of the basis of $\mathfrak {X}_{\text {awi}}^{(1)}$ .

Remark 5.15. Let $f\in V_{(1)}$ and $(f_{\alpha })_{\alpha \in \mathcal {A}}$ be a tree analysis of f. Let $\Delta $ be a nonempty subset of $\text {supp}(f)$ , and set $g=f|_{\Delta }$ . First, note that $g\in V_{(1)}$ . Moreover, $(f_{\alpha })_{\alpha \in \mathcal {A}}$ naturally induces a tree analysis $(g_{\alpha })_{\alpha \in \mathcal {B}}$ for g as follows: $\mathcal {B}=\{\alpha \in \mathcal {A}:\text {supp}(f_{\alpha })\cap \Delta \neq \emptyset \}$ and $g_{\alpha }=f_{\alpha }|_{\Delta }$ , $\alpha \in \mathcal {B}$ . Finally, it is easy to see that $w(g)=w(f)$ .

Proposition 5.16. Let J be an initial segment of $\mathbb {N}$ or $ J =\mathbb {N}$ and $(f_j)_{j\in J}$ be an AWI sequence in $V_{(1)}$ .

1. (i) Every subsequence of $(f_j)_{j\in J}$ is also an AWI sequence in $V_{(1)}$ .

2. (ii) If $\Delta _j$ is a nonempty subset of $\text {supp} (f_j)$ and $g_j=f_j|_{\Delta _j}$ , $j\in J$ , then $(g_j)_{j\in J}$ is an AWI sequence in $V_{(1)}$ .

3. (iii) If $(g_j)_{j\in J}$ is a sequence in $V_{(1)}$ , such that $|g_j|=|f_j|$ for all $j\in J$ , then $(g_j)_{j\in J}$ is also AWI.

Proof. Let for every $j\in J$ , $(f_{j,\alpha })_{\alpha \in \mathcal {A}_j}$ be a tree analysis of $f_j$ with

$$\begin{align*}\{ \bar{f}_j:j\in J \}= C^0_1\cup C^0_2 \end{align*}$$

and for every $k,j\in J$ with $j> k$

$$\begin{align*}\{\bar{f}_{j,\alpha}:\alpha\in \mathcal{A}_j\text{ and }|\alpha|=k\}=C^{k}_{1,j}\cup C^{k}_{2,j}, \end{align*}$$

witnessing that $(f_j)_{j\in J}$ is AWI. We will define the desired partitions proving the cases (i)–(iii).

To prove (i), let N be a subset of J and define

$$\begin{align*}F^0_i=\{\bar{f}_j:j\in N\}\cap C^0_i,\quad i=1,2. \end{align*}$$

Then $\{\bar {f}_j:j\in N \} = F^0_1\cup F^0_2$ , where $F^0_1$ is essentially incomparable and $F^0_2$ is weight incomparable. For the remaining part, let $k\in N$ , and note that for $N_k=\{j\in N:j\ge k\}$ , $\cup _{j\in N_k} C^{k}_{1,j}$ is essentially incomparable and $(C^{k}_{2,j})_{j\in N_k}$ is pairwise weight incomparable.

To prove (ii), Remark 5.15 implies that $g_j\in V_{(1)}$ , $w(g_j)=w(f_j)$ , and we let $(g_{j,\alpha })_{\alpha \in \mathcal {B}_j}$ be the tree analysis of g induced by $(f_{j,\alpha })_{\alpha \in \mathcal {A}_j}$ , $j\in J$ . Define

$$\begin{align*}F^0_i=\{\bar{g}_j:j\in J\text{ and }g_j=f_j|_{\Delta_j}\text{ with }\bar{f}_j\in C^0_i \}, \quad i=1,2, \end{align*}$$

and observe that $\{\bar {g}_j:j\in J \}=F^0_1\cup F^0_2$ . Moreover, for $j\in J$ , $\text {supp} (g_j)\subset \text {supp} (f_j)$ and $w(g_j)=w(f_j)$ , and, hence, whenever $g_i\neq g_j$ are in $F^0_1$ with $w(g_i)<_{\mathcal {W}} w(g_j)$ , we have $w(f_i)<_{\mathcal {W}} w(f_j)$ , implying that the generator $t_3\in \mathcal {T}$ of $w(f_j)=w(g_j)$ with $w(t_3)=w(f_i)=w(g_i)$ is such that $f_{t_3}<f_i$ , and thus $f_{t_3}<g_i$ . This yields that $F^0_1$ is essentially incomparable. Clearly, $F^0_2$ is weight incomparable. Next, for $k,j\in J$ with $j>k$ , define

$$\begin{align*}F^k_{i,j}=\{\bar{g}_{j,\alpha}:g_{j,\alpha}=f_{j,\alpha}|_{\Delta_j}\text{ with }\bar{f}_{j,\alpha}\in C^k_{i,j} \}, \quad i=1,2. \end{align*}$$

Note that for each $k\in J$ , $(F^{k}_{2,j})_{j=k+1}^{\infty }$ is pairwise weight incomparable, and the proof that $\cup _{j=k+1}^{\infty } F^k_{1,j}$ is essentially incomparable is identical to that for $F^1_0$ . Finally, the proof of (iii) is similar that of (ii).

Definition 5.17. Let $W_{(1)}$ be the smallest subset of $V_{(1)}$ that is symmetric, contains the singletons and for every $j\in \mathbb {N}$ and every $\mathcal {S}_{n_j}$ -admissible AWI sequence $(f_i)_{i=1}^n$ in $W_{(1)}$ , we have that $m_j^{-1}\sum _{i=1}^nf_i$ is in $W_{(1)}$ . Moreover, let $\mathfrak {X}_{\text {awi}}^{(1)}$ denote the completion of $c_{00}(\mathbb {N})$ with respect to the norm induced by $W_{(1)}$ .

Remark 5.18.

1. (i) The norming set $W_{(1)}$ can be defined as the increasing union of a sequence $(W^n_{(1)})_{n=0}^{\infty }$ , where $W^0_{(1)}=\{\pm e^*_k:k\in \mathbb {N}\}\cup \{0\}$ and

   $$ \begin{align*} W^{n+1}_{(1)}=W^n_{(1)}\cup\Big\{\frac{1}{m_j}\sum_{l=1}^df_l:\;&j,d\in\mathbb{N}\text{ and }(f_l)_{l=1}^d \text{ is an } \mathcal{S}_{n_j}\text{-admissible AWI sequence in }W^n_{(1)}\Big\}. \end{align*} $$

2. (ii) Note that Remark 5.13 (iv) implies that any sequence of singletons is AWI. Hence, we have that

   $$\begin{align*}W^{1}_{(1)}=W^0_{(1)}\cup\Big\{\frac{1}{m_j}\sum_{k\in E}\epsilon_ke^*_k:\;j\in\mathbb{N}, \; E\in\mathcal{S}_{n_j} \text{ and } \epsilon_k\in\{-1,1\} \text{ for }k\in E \Big\}. \end{align*}$$

3. (iii) Proposition 5.16 yields that the standard unit vector basis of $c_{00}(\mathbb {N})$ forms an $1$ -unconditional Schauder basis for $\mathfrak {X}_{\text {awi}}^{(1)}$ .

6 Outline of proof

Although unconditionality of the basis of $\mathfrak {X}_{\text {awi}}^{(1)}$ is almost immediate, it is not, however, straightforward to show that $\mathfrak {X}_{\text {awi}}^{(1)}$ admits $\ell _1$ as an asymptotic model. Indeed, this requires Lemma 7.2, which is based on the combinatorial results concerning measures on well-founded trees of Section 4, which first appeared in [Reference Argyros, Georgiou and Motakis8]. This lemma yields that for any choice of successive families $(F_j)_j$ of normalised blocks in $\mathfrak {X}_{\text {awi}}^{(1)}$ and for any $\varepsilon>0$ , we may pass to a subsequence $(F_j)_{j\in M}$ and find a family $(G_j)_{j\in M}$ of subsets of $W_{(1)}$ , such that for any choice of $x_j\in F_j$ , $j\in M$ , there is a $g_j\in G_j$ with $g_j(x_j)>1-\varepsilon $ so that $(g_j)_{j\in M}$ is AWI. Thus, we are able to prove, employing Lemma 3.4, the aforementioned result.

To prove the nonexistence of Asymptotic $\ell _1$ subspaces in $\mathfrak {X}_{\text {awi}}^{(1)}$ , we start with the notion of exact pairs. This is a key ingredient in the study of Mixed Tsirelson spaces, used for the first time by Schlumprecht [Reference Schlumprecht26].

That is, roughly speaking, for an exact pair $(x,f)$ , the evaluation of a functional g in $W_{(1)}$ , on x, admits an upper bound depending only on the weight of g. In the case of an $m_j$ -strong exact pair $(x,f)$ , any g in $W_{(1)}$ with a tree analysis $(g_{\alpha })_{\alpha \in \mathcal {A}}$ , such that $w(g_{\alpha })\neq m_j$ , has negligible evaluation on x. We will consider certain exact pairs which we call standard exact pairs (SEP) (see Definition 8.7) and which we prove to be strong exact pairs. It is the case that such pairs can be found in any block subspace of $\mathfrak {X}_{\text {awi}}^{(1)}$ , and this is used to prove the reflexivity of $\mathfrak {X}_{\text {awi}}^{(1)}$ as well as the following proposition which yields the nonexistence of Asymptotic $\ell _1$ subspaces.

To this end, we first employ the following lemma that highlights the importance of the asymptotically weakly incomparable constraints.

Then, we consider a sequence of standard exact pairs $(x_1,f_1),\ldots ,(x_n,f_n)$ with $\bar {f}_1<_{\mathcal {T}}\ldots <_{\mathcal {T}} \bar {f}_n$ and fix $0<c<1$ . Pick an $m\in \mathbb {N}$ , such that $3/2^m<c$ . For $f\in W_{(1)}$ , with a tree analysis $(f_{\alpha })_{\alpha \in \mathcal {A}}$ , we consider partitions $f=g+h$ and $g=g_1+g_2$ as follows: First, set

$$\begin{align*}G=\cup\{\text{range} (x_k)\cap \text{range} (f_{\alpha}): k\in\{1,\ldots,n\}\text{ and }\alpha\in\mathcal{A}\text{ with }w(f_{\alpha})=w(f_k)\}, \end{align*}$$

and define $g=f|_G$ and $h=f|_{\mathbb {N}\setminus G}$ (see Figure 2).

Figure 2 The tree analysis of f and the induced tree analyses of g and h. The circled nodes $\alpha $ are such that $w(f_{\alpha })=w(f_k)$ and $\text {supp}(x_k)\cap \text {supp}(f_{\alpha })\neq \emptyset $ for some $k\in \{1,\ldots ,n\}$ .

To define $g_1$ , consider the tree analysis $(g_{\alpha })_{\alpha \in \mathcal {A}_g}$ of g that is induced by $(f_{\alpha })_{\alpha \in \mathcal {A}}$ , that is, $g_{\alpha }=f_{\alpha }|_G$ for $\alpha \in \mathcal {A}_g$ and $\mathcal {A}_g=\{\alpha \in \mathcal {A}:\text {supp}(f_{\alpha })\cap G\neq \emptyset \}$ . Then, we define

$$ \begin{align*} \mathcal{B}^1_k=\{\alpha\in\mathcal{A}_g:&\;|a|\le m,\; w(f_{\alpha})=w(f_k)\text{ and } w(f_{\beta})\neq w(f_k)\text{ for all }\beta<\alpha\text{ in }\mathcal{A}_g\} \end{align*} $$

for $k=1,\ldots ,n$ ,

$$\begin{align*}G_1=\cup_{k=1}^n\cup\left\{\text{supp}(g_{\alpha})\cap \text{supp}(x_k):\alpha\in\mathcal{B}^1_k\right\}, \end{align*}$$

and finally $g_1 = g|_{G_1}$ (see Figure 3). Observe that Lemma 6.3 implies that

(6.1) $$ \begin{align} \#\left\{ k\in\{1,\ldots,n\} : g_1(x_k)\neq 0\right\} \le \ell = e \sum_{k=1}^mk!. \end{align} $$

Moreover, the induced tree analysis $(h_{\alpha })_{\alpha \in \mathcal {A}_h}$ of h is such that $w(h_{\alpha })\neq w(f_k)$ for all $k=1,\ldots ,n$ , and, therefore, the fact that $(x_k,f_k)$ are strong exact pairs yields

(6.2) $$ \begin{align} |h\left(x_k\right)|\le \frac{18}{w(f_k)}, \quad k=1,\ldots,n. \end{align} $$

Considering a further partition of $g_2|_{\text {supp}(x_k)}$ , we show that

(6.3) $$ \begin{align} |g_2\left(x_k\right)|\le \frac{18}{w(f_k)}+\frac{3}{2^m}, \quad k=1,\ldots,n. \end{align} $$

Hence, (6.1), (6.2) and (6.3) imply

$$\begin{align*}\left|f\left( \frac{1}{n} \sum_{k=1}^n x_k \right)\right| \le \frac{36+\ell}{n}+\frac{3}{2^m}. \end{align*}$$

Then, our choice of m yields Proposition 6.2 for sufficiently large n and $w(f_1)$ , where $w(f_1)$ is chosen appropriately to deal with the case where $f\in \{\pm e_j^*:j\in \mathbb {N}\}$ .

Figure 3 We consider the m-th level of the induced tree analysis of g. Nodes $\alpha $ with $w(f_{\alpha })=w(f_k)$ and $|\alpha |\le m$ are used to define $g_1$ , while such nodes of height greater than m define $g_2$ , restricted on each $x_k$ for $k=1,\ldots ,n$ .

Assuming that Y is a C-asymptotic $\ell _1$ block subspace of $\mathfrak {X}^{(1)}_{\text {awi}}$ , we pick a sequence of standard exact pairs $(x_1,f_1),\ldots ,(x_n,f_n)$ in $Y\times W_{(1)}$ , satisfying the conclusion of Proposition 6.2 and derive a contradiction.

The remainder of this part of the paper is organised as follows. In Section 7, we prove that $\mathfrak {X}_{\text {awi}}^{(1)}$ admits $\ell _1$ as a unique asymptotic model. Next, in Section 8, we prove existence and properties of standard exact pairs. The final section of this part contains the results leading up to the proof that $\mathfrak {X}^{(1)}_{\text {awi}}$ does not contain Asymptotic $\ell _1$ subspaces.

7 Asymptotic models generated by block sequences of $\mathfrak {X}_{\text {awi}}^{(1)}$

We show that the space $\mathfrak {X}_{\text {awi}}^{(1)}$ admits a unique asymptotic model, or equivalently, a uniformly unique joint spreading model with respect to $\mathscr {F}_b(\mathfrak {X}_{\text {awi}}^{(1)})$ that is equivalent to the unit vector basis of $\ell _1$ . The key ingredient in the proof is the following lemma concerning bounded positive measures on the tree of initial segments of $\mathcal {T}$ .

Proof. Passing to a subsequence if necessary, we may assume that the (unique) root of ${\widetilde {\mathcal {T}}}$ is not in the support of any $\mu _i$ , $i\in \mathbb {N}$ , $\mathrm {succ}\text {-}\!\lim _i\mu _{i}$ exists and that there exist partitions $\mathrm {supp}(\mu _i)=A_i\cup B_i$ , $i\in \mathbb {N}$ , satisfying the conclusion of Lemma 4.6. Define for each $i\in \mathbb {N}$ , the measures $\mu _i^1$ , $\mu _i^2\in \mathcal {M}_+({\widetilde {\mathcal {T}}})$ given by $\mu _i^1(C) = \mu _i(A_i\cap C)$ and $\mu _i^2(C) = \mu _i(B_i\cap C)$ , and let $\nu = w^*\text {-}\lim _i\mu ^1_i = \mathrm {succ}\text {-}\!\lim _i\mu ^1_i$ . Pick a finite subset F of ${\widetilde {\mathcal {T}}}$ , such that $\nu ({\widetilde {\mathcal {T}}}\setminus F) <\varepsilon /2$ . Then, $\nu = w^*\text {-}\lim _i\mu ^1_i$ implies that $\lim _i\mu _i^1({\widetilde {\mathcal {T}}}) = \nu ({\widetilde {\mathcal {T}}})$ , and, thus, since $\nu = \mathrm {succ}\text {-}\!\lim _i\mu ^1_i$ , we have

$$ \begin{align*} \lim_i\Big|\mu^1_i({\widetilde{\mathcal{T}}}) - \mu^1_i(\cup_{\tilde t\in F}S(\tilde t))\Big| = \Big|\nu({\widetilde{\mathcal{T}}}) - \lim_i\sum_{\tilde t\in F}\mu_i^1(S(\tilde t))\Big| = \nu({\widetilde{\mathcal{T}}}\setminus F)<\frac{\varepsilon}{2}. \end{align*} $$

Hence, we can find $i_0\in \mathbb {N}$ , such that for all $i\geq i_0$ , we have

(7.1) $$ \begin{align} \Big|\mu_i(A_i) - \mu_i\Big(A_i\cap\big(\cup_{\tilde t\in F}S(\tilde t)\big)\Big)\Big| = \Big|\mu^1_i({\widetilde{\mathcal{T}}}) - \mu^1_i(\cup_{\tilde t\in F}S(\tilde t))\Big| <\frac \varepsilon 2. \end{align} $$

We set $\Sigma =\sigma (\{t\in \mathcal {T}:{\tilde {t}}\in F\})$ and

$$ \begin{align*} R = \{ r\in\mathcal{T} : w(r)\in\Sigma \text{, and there is } s\in\mathcal{T} \text{ with }w(s)\in\Sigma\text{, such that }r <_{\mathcal{T}} s\} .\end{align*} $$

Note that $\Sigma $ and R are finite, since F is finite. Thus, using the fact that the sets $\cup \{\text {supp} (f_t):{\tilde {t}}\in \text {supp}(\mu _i)\}$ for $i\in \mathbb {N}$ are disjoint, find $i_1\in \mathbb {N}$ with $i_1\geq i_0$ so that

(7.2) $$ \begin{align} \cup_{r\in R}\text{supp}(f_r) < \text{supp} (f_t), \text{ for all }\: \tilde t\in\cup_{i\geq i_1}\mathrm{supp}(\mu_i^1) .\end{align} $$

For $G_i^1 = A_i\cap (\cup _{{\tilde {t}}\in F}S({\tilde {t}}))$ , (7.1) implies that $|\mu _i(A_i) - \mu _i(G_i^1)|<\varepsilon /2$ , $i\geq i_1$ . We will show that $\{t\in \mathcal {T}:{\tilde {t}}\in \cup _{i\geq i_1}G_i^1\}$ is essentially incomparable, that is, that (iii) is satisfied. To this end, first observe that if ${\tilde {t}}\in \cup _{i\geq i_1}G_i^1$ , then $w(t)\in \Sigma $ . Let ${\tilde {t}}_1,{\tilde {t}}_2\in \cup _{i\geq i_1}G_i^1$ with $w(t_1)<_{\mathcal {W}} w(t_2)$ . It is immediate that if $t_3\in \mathcal {T}$ is the generator of $w(t_2)$ with $w(t_3)=w(t_1)$ , then $t_3\in R$ , and, hence, (7.2) implies that $f_{t_3}<f_{t_1}$ , proving the desired result.

For the remaining part of the proof, recall the root of ${\widetilde {\mathcal {T}}}$ avoids the supports of all $\mu _i^2$ , $i\ge i_1$ . This implies that every ${\tilde {t}}\in \cup _{i\ge i_1} \text {supp}(\mu _i^2)$ is the successor of some node in ${\widetilde {\mathcal {T}}}$ . Then, since for all $i\ge i_1$ , the set $B_i = \text {supp}(\mu _i^2)$ is finite (as a subset of the finite support of $\mu _i$ ), and for each ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , the sequence $(\mu _i^2(S({\tilde {t}})))_{i\ge i_1}$ is eventually zero, we may pass to a subsequence so that for all $i_1\le i<j$ , we have $\{w(t):{\tilde {t}}\in \text {supp}(\mu _i^2)\}\cap \{w(t):{\tilde {t}}\in \text {supp}(\mu _j^2)\} = \emptyset $ . We can, therefore, define the bounded sequence of disjointly supported measures $(\nu _i)_{i\ge i_1}$ on $\widetilde {\mathcal {W}}$ given by $\nu _i(\{{\tilde {w}}\}) = \mu _i^2(\{\tilde t\in {\widetilde {\mathcal {T}}}: \tilde {w}(t) = {\tilde {w}}\})$ . Hence, applying Proposition 4.1 and passing to a subsequence, we obtain a subset $E_i$ of $\text {supp}(\nu _i)$ , such that $\nu _i({\widetilde {\mathcal {W}}}\setminus E_i)<\varepsilon /2$ and the sets $E_i$ , $i\ge i_1$ , are pairwise incomparable. It is easy to verify that if $G^2_i=\{{\tilde {t}}\in B_i:{\tilde {w}}(t)\in E_i \}$ and $F^2_i=\{t\in \mathcal {T}:{\tilde {t}}\in G^2_i\}$ , $i\ge i_1$ , then $(F^2_i)_{i\ge i_1}$ is pairwise weight incomparable and $|\mu _i(B_i) - \mu _i(G_i^2)| = \mu ^2_i({\widetilde {\mathcal {T}}}\setminus G^2_i)<\varepsilon /2$ for every $i\ge i_1$ .

Proof. Let $\mathcal {A}_k$ denote the set of all nodes in $\mathcal {A}$ , such that $|\alpha |=k$ , $1\le k\le h(\mathcal {A})$ . We define g by constructing the tree analysis $(g_{\alpha })_{\alpha \in \mathcal {A}_g}$ . First, define by induction $B_1=\mathcal {A}_1\setminus G_1$ and for $2\le k \le h(\mathcal {A})$ :

$$\begin{align*}B_k = \{ \alpha \in \mathcal{A}_k : \alpha\notin G_{k}\text{ or there is a } \beta\in B_{k-1}\text{, such that } \alpha\in S(\beta) \}. \end{align*}$$

It follows easily that $\alpha \in B_k$ if and only if there exists $\beta \le \alpha $ , such that $\beta \notin G_{|\beta |}$ . Let $\mathcal {C}_g= \mathcal {A}_{h(\mathcal {A})}\setminus {B}_{h(\mathcal {A})}$ . Note that $f_{\alpha }\in \{\pm e_j^*:j\in \mathbb {N}\}$ for every $\alpha \in \mathcal {C}_g$ , and let $\Delta _g=\cup \{\text {supp} (f_{\alpha }):\alpha \in \mathcal {C}_g \}$ . Then $g=f|_{\Delta _g}$ and $(g_{\alpha })_{\alpha \in \mathcal {A}_g}$ is the tree analysis induced by $(f_{\alpha })_{\alpha \in \mathcal {A}}$ .

Observe that, by construction, g satisfies (i) and (iii). To see that it also satisfies (ii), we show by induction that for every $1\le k \le h(\mathcal {A})$

(7.3) $$ \begin{align} \sum_{ \alpha\in\mathcal{A}_k\setminus {B}_k }\frac{f_{\alpha}(x)}{w_f(f_{\alpha})}> f(x) - \sum_{i=1}^k \varepsilon_i. \end{align} $$

This indeed proves (ii), since the left-hand side of (7.3) for $k=h(\mathcal {A})$ is equal to $g(x)$ . We now prove (7.3) by induction. Assume that the inequality holds for some $1\leq k <h(\mathcal {A})$ . Then, for every $\alpha \in \mathcal {A}_{k}\setminus {B}_{k}$ , we have

$$\begin{align*}f_{\alpha}(x)=\sum_{ \beta\in S(\alpha)\cap G_{k+1} }\frac{f_{\beta}(x)}{w(f_{\alpha})}+\sum_{ \beta\in S(\alpha)\setminus G_{k+1} }\frac{f_{\beta}(x)}{w(f_{\alpha})} \end{align*}$$

and

$$\begin{align*}\sum_{\alpha\in\mathcal{A}_k\setminus {B}_k}\sum_{\beta\in S(\alpha)\setminus G_{k+1} }\frac{f_{\beta}(x)}{w_f(f_{\alpha})w(f_{\alpha})} = \sum_{\alpha\in\mathcal{A}_k\setminus {B}_k}\sum_{\beta\in S(\alpha)\setminus G_{k+1} }\frac{f_{\beta}(x)}{w_f(f_{\beta})}<\varepsilon_{k+1}. \end{align*}$$

Hence

$$ \begin{align*} \sum_{\alpha\in\mathcal{A}_k\setminus {B}_k}\sum_{\beta\in S(\alpha)\cap G_{k+1} }\frac{f_{\beta}(x)}{w_f(f_{\alpha})w(f_{\alpha})}&= \sum_{\alpha\in\mathcal{A}_k\setminus {B}_k}\frac{f_{\alpha}(x)}{w_{f}(f_{\alpha})}- \sum_{\alpha\in\mathcal{A}_k\setminus {B}_k}\sum_{\beta\in S(\alpha)\setminus G_{k+1} }\frac{f_{\beta}(x)}{w_f(f_{\beta})}\\ &> ( f(x) - \sum_{i=1}^k \varepsilon_i)-\varepsilon_{k+1} \end{align*} $$

which, along with the previous inequality, proves the desired result since

$$\begin{align*}\{ \beta\in\mathcal{A}:\beta\in S(a)\setminus G_{k+1}\text{ for some }\alpha\in \mathcal{A}_k\setminus {B}_k \} = \mathcal{A}_{k+1}\setminus {B}_{k+1}. \end{align*}$$

Proof. Let $(\varepsilon _k)_{k=0}^{\infty }$ be a sequence of positive reals, such that $\sum _{k=0}^{\infty }\varepsilon _k<\varepsilon /2$ . For every $1\le i\le l$ and $j\in \mathbb {N}$ , pick an $f^i_j\in W_{(1)}$ and a tree analysis $(f^i_{j,\alpha })_{\alpha \in \mathcal {A}^i_j}$ of $f^i_j$ , such that $f^i_j(x^i_j)>1-\varepsilon /2$ and $f^i_{j,\alpha }(x^i_j)> 0$ for every $\alpha \in \mathcal {A}^i_j$ . For $1\le i\le l$ and $j\in \mathbb {N}$ , we set $t_{j}^i=\bar {f}_j^i$ and $t_{j,\alpha }^i=\bar {f}_{j,\alpha }^i$ , $\alpha \in \mathcal {A}_j^i$ . We will choose, by induction, an $L\in [\mathbb {N}]^{\infty }$ and, for every $1\le i\le l$ , $j\in L$ and $k\in \mathbb {N}$ , a subset $G^{k,i}_j$ of $\{ \alpha \in \mathcal {A}^i_j:|\alpha |=k \}$ satisfying the following conditions. For $k\in \mathbb {N}$ , we set $L_{>k}=\{j\in L:j>k\}$ .

1. (i) For every $j\in L$ , there is a partition $\{ t^{i}_j:i=1,\ldots ,l\}=C^{0}_{1,j}\cup C^{0}_{2,j}$ , such that $\cup _{j\in L}C^{0}_{1,j}$ is essentially incomparable and $(C^{0}_{2,j})_{j\in L}$ is pairwise weight incomparable.

2. (ii) For every $1\le i\le l$ , $k\in \mathbb {N}$ and $j\in L_{>k}$ , there is a partition $G^{k,i}_j=G^{k,i}_{1,j}\cup G^{k,i}_{2,j}$ , such that for any choice of $1\le i_j \le l$ , $\cup _{j\in L_{>k} } \{t^{i_j}_{j,\alpha } : \alpha \in G^{k,i_j}_{1,j} \}$ is essentially incomparable and $(\{ t^{i_j}_{j,\alpha } : \alpha \in G^{k,i_j}_{2,j} \})_{j\in L_{>k}}$ is pairwise weight incomparable.

3. (iii) For every $i=1,\ldots ,l$ , $j\in L$ and $k\in \mathbb {N}$ with $k\le h(\mathcal {A}^i_j)$

   $$\begin{align*}\sum_{\alpha\in G^{k,i}_{j}} w_{f^i_j}(f^i_{j,\alpha})^{-1}f^i_{j,\alpha}(x^i_j)>f^i_j(x^i_j)-\varepsilon_k.\end{align*}$$

Observe then that (iii) and an application of the previous lemma yield, for every $1\le i \le l$ and $j\in L$ , a functional $g^i_j\in W_{(1)}$ , such that

$$\begin{align*}g^i_j(x^i_j)>f^i_j(x^i_j)-\sum_{k=1}^{\infty}\varepsilon_k>1-\varepsilon. \end{align*}$$

Fix a choice of $1\le i_j\le l$ , $j\in L$ . Then, (i) implies that $\{ t^{i_j}_j:j\in L\}\cap (\cup _{j\in L}C^0_{1,j})$ is essentially incomparable, and that $\{ t^{i_j}_j:j\in L\}\cap (\cup _{j\in L}C^0_{2,j})$ is weight incomparable. Finally, (ii) and Lemma 7.3 (iii) yield that, for every $1\le i\le l$ , $k\in \mathbb {N}$ and $j\in L_{>k}$ , there is a partition

$$\begin{align*}\{\bar{g}^{i_j}_{j,\alpha}:a\in \mathcal{A}^{i_j}_j\text{ and }|\alpha|=k\}=C^{k,i_j}_{1,j}\cup C^{k,i_j}_{2,j} ,\end{align*}$$

such that $\cup _{j\in L_{>k}}C^{k,i_j}_{1,j}$ is essentially incomparable and $(C^{k,i_j}_{2,j})_{j\in L_{>k}}$ is pairwise weight incomparable. Hence, $g^i_j$ , $1\le i\le l$ and $j\in L$ satisfy the desired conditions.

To obtain L, let us first assume that $\sup _{i,j}h(\mathcal {A}^i_j)=+\infty $ (if $\sup _{i,j}h(\mathcal {A}^i_j)<+\infty $ , then a finite version of the same proof works). Moreover, passing to a subsequence, we may further assume that $\max _i h(\mathcal {A}_j^i)>k$ whenever $j>k$ , for $j,k\in \mathbb {N}$ . Define, for each $j\in \mathbb {N}$ , the measure $\mu ^0_j$ on ${\widetilde {\mathcal {T}}}$ given by

$$\begin{align*}\mu_j^0=\sum_{i=1}^lf^i_j(x^i_j)\delta_{\tilde{t}_{j}^i }. \end{align*}$$

Applying Lemma 7.2, we obtain an $L_0\in [\mathbb {N}]^{\infty }$ , such that, for every $j\in L_0$ , there exist disjoint subsets $G^0_{1,j}$ and $G^0_{2,j}$ of $\text {supp} (\mu ^0_j)$ so that the following hold.

• ( $\alpha _0$ ) $\mu ^0_j( {\widetilde {\mathcal {T}}} \setminus G^0_{1,j}\cup G^0_{2,j} ) < \varepsilon _0$ for every $j\in L_0$ .

• ( $\beta _0$ ) Define $C^0_{1,j}=\{t\in \mathcal {T}:{\tilde {t}}\in G^0_{1,j}\}$ for $j\in L_0$ . Then $\cup _{j\in L_0}C^0_{1,j}$ is essentially incomparable.

• ( $\gamma _0$ ) Define $C^0_{2,j}=\{t\in \mathcal {T}:{\tilde {t}}\in G^0_{2,j}\}$ for $j\in L_0$ . Then the sequence $(C^0_{2,j})_{j\in L_0}$ is pairwise weight incomparable.

Note that ( $\alpha _0$ ) implies that $\text {supp} (\mu ^0_j)=G^0_{1,j}\cup G^0_{2,j}$ since $f^i_j(x^i_j)>1-\varepsilon /2$ , that is, $\{{\tilde {t}}^i_j:i=1,\ldots ,l \}=C^0_{1,j}\cup C^0_{2,j}$ . We proceed by induction on $\mathbb {N}$ . Suppose we have chosen $L_0,\ldots ,L_{k-1}$ and $G^0_{1,j_0},G^0_{2,j_0},\ldots ,G^{k-1}_{1,j_{k-1}},G^{k-1}_{2,j_{k-1}}$ for some $k\in \mathbb {N}$ and every $j_i\in L_i$ , for $i=0,\ldots ,k-1$ . Set $L_k^0=\{j\in L_{k-1}:h(\mathcal {A}^i_j)< k\text { for all }1\le i\le l\}$ . Then, for each $j\in L_{k-1}\setminus L^0_{k}$ , define the following measure on ${\widetilde {\mathcal {T}}}$

$$\begin{align*}\mu^{k}_j=\sum_{i=1}^l\sum_{\substack { \alpha\in\mathcal{A}^i_j \\ |\alpha|=k } } \frac{f^i_{j,\alpha}(x^i_j)}{w_{f^i_j}(f^i_{j,\alpha})}\delta_{\tilde{t}^i_{j,\alpha}}. \end{align*}$$

Again, applying Lemma 7.2 yields an $L^1_{k}\in [L_{k-1}\setminus L^0_k]^{\infty }$ and disjoint subsets $G^{k}_{1,j}$ and $G^{k}_{2,j}$ of $\text {supp} (\mu ^{k}_j)$ , $j\in L^1_k$ , such that

• ( $\alpha _{k}$ ) $\mu ^k_j( {\widetilde {\mathcal {T}}} \setminus G^k_{1,j}\cup G^k_{2,j} ) < \varepsilon _k$ for every $j\in L^1_k$ ,

• ( $\beta _{k}$ ) $\{t\in \mathcal {T}:{\tilde {t}}\in \cup _{j\in {L^1_k}}G^k_{1,j}\}$ is essentially incomparable and

• ( $\gamma _{k}$ ) the sequence $(\{t:{\tilde {t}}\in G^k_{2,j}\})_{j\in {L^1_k}}$ is pairwise weight incomparable.

Then, set $L_k=L_k^0\cup L_k^1$ and $G^k_{i,j}=\{\alpha \in \mathcal {A}_j^i:|\alpha |=k\}$ , for $1\le i\le l$ and $j\in L_{k}^0$ . Finally, choose L to be a diagonalisation of $(L_k)_{k}$ , that is, $L(k)\in L_k$ for $k\in \mathbb {N}$ . Observe that $(\beta _k)$ and $(\gamma _k)$ imply (ii), while $(\alpha _k)$ implies (iii).