# A local Ramsey theory for block sequences

Iian B. Smythe  Department of Mathematics, Rutgers, The State University of
New Jersey, Piscataway, NJ, USA 08854<EMAIL_ADDRESS>
(Date: August 20, 2018)

###### Abstract.

We develop local forms of Ramsey-theoretic dichotomies for block sequences in
infinite-dimensional vector spaces, analogous to Mathias’ selective coideal
form of Silver’s theorem for analytic partitions of $[\mathbb{N}]^{\infty}$.
Under large cardinals, these results are extended to partitions in
$\mathbf{L}(\mathbb{R})$ and $\mathbf{L}(\mathbb{R})$-generic filters of block
sequences are characterized. Variants of these results are also established
for block sequences in Banach spaces and for projections in the Calkin
algebra.

###### 2010 Mathematics Subject Classification:

Primary 05D10, 03E05; Secondary 46B20

The author is partially supported by NSERC award PGSD2-453779-2014 and NSF
grant DMS-1600635. He would also like to thank his PhD advisor, Justin Tatch
Moore, for continued guidance and suggesting the problem of characterizing
$\mathbf{L}(\mathbb{R})$-generic filters for the projections in the Calkin
algebra which motivated this work.

##  1\. Introduction

Ramsey-theoretic techniques have a long history of use in Banach space theory,
see e.g., [4]. Most relevant for the present work is Gowers’ dichotomy for
infinite block sequences in Banach spaces:

######  Theorem (Gowers [18] [19]).

Let $B$ be an infinite-dimensional Banach space with a Schauder basis. If
$\mathbb{A}$ is an analytic set of normalized block sequences, then for any
$\Delta>0$, there is a block sequence $Y$ such that either

  1. (i)

every normalized block subsequence of $Y$ is in $\mathbb{A}^{c}$, or

  2. (ii)

II has a strategy in the Gowers game $G^{*}[Y]$ for playing into
$\mathbb{A}_{\Delta}$.

Loosely speaking, this result says that for $\mathbb{A}$ as described, there
is a block sequence $Y$ such that either all of $Y$’s normalized block
subsequences are disjoint from $\mathbb{A}$, or there is a wealth of block
subsequences of $Y$ which are within a small perturbation of $\mathbb{A}$.
This was used, together with work of Komorowski and Tomczak-Jaegerman [23], to
solve (affirmatively) the homogeneous space problem.

In the setting of a discrete countably infinite-dimensional vector space $E$
over a countable field, Rosendal isolated an “exact” version of Gowers’
dichotomy which yields a much simplified proof of the original result:

######  Theorem (Rosendal [35]).

If $\mathbb{A}$ is an analytic set of block sequences in $E$, then there is a
block sequence $Y$ such that either

  1. (i)

I has a strategy in the infinite asymptotic game $F[Y]$ for playing into
$\mathbb{A}^{c}$, or

  2. (ii)

II has a strategy in the Gowers game $G[Y]$ for playing into $\mathbb{A}$.

These dichotomies are analogues, in the Banach space and vector space
settings, respectively, of the following result for partitions of
$[\mathbb{N}]^{\infty}$, the set of infinite subsets of the natural numbers.

######  Theorem (Silver [38]).

If $\mathbb{A}\subseteq[\mathbb{N}]^{\infty}$ is analytic, then there is a
$y\in[\mathbb{N}]^{\infty}$ with either all of its further infinite subsets
disjoint from, or contained in, $\mathbb{A}$.

While the theory of topological Ramsey spaces, in the sense of [40],
encompasses many variations on this result, the dichotomies of Gowers and
Rosendal highlighted above do not fall into this framework.

An important generalization of Silver’s theorem is the following “local”
Ramsey theorem, showing that the witness $y$ in the conclusion can always be
found in a given selective coideal (or “happy family”):

######  Theorem (Mathias [30]).

Let $\mathcal{H}\subseteq[\mathbb{N}]^{\infty}$ be a selective coideal. If
$\mathbb{A}\subseteq[\mathbb{N}]^{\infty}$ is analytic, then there is a
$y\in\mathcal{H}$ with either all of its further infinite subsets disjoint
from, or contained in, $\mathbb{A}$.

By passing to a forcing extension resulting from the Lévy collapse of a Mahlo
cardinal, Mathias extended these results to all partitions $\mathbb{A}$ which
are “reasonably definable”, that is, in the definable closure of the reals
$\mathbf{L}(\mathbb{R})$. Later work of Farah and Todorcevic [14] generalized
this to semiselective coideals and showed that under stronger large cardinal
hypotheses the passage to a forcing extension is not necessary. The extension
of Silver’s theorem to all partitions in $\mathbf{L}(\mathbb{R})$ is due to
Shelah and Woodin [37]. Similar results have been developed recently for
topological Ramsey spaces [31] [12].

The upshot of obtaining these local results is two-fold: We clearly isolate
the combinatorial properties which enable the original dichotomies, and we
obtain greater control over the witnesses to said dichotomies.

This latter point was used by Todorcevic [14] to characterize, under large
cardinal hypotheses, selective ultrafilters as being exactly those which are
generic for $([\mathbb{N}]^{\infty},\subseteq^{*})$ over
$\mathbf{L}(\mathbb{R})$. Such ultrafilters are said to possess “complete
combinatorics”, following Blass and Laflamme [25] who used this phrase to
describe ultrafilters which are generic over $\mathbf{L}(\mathbb{R})$ after
collapsing a Mahlo cardinal. We instead ask for genericity over
$\mathbf{L}(\mathbb{R})$ of the ground model, at the expense of stronger large
cardinal hypotheses.

Using [35] as a starting point, we develop local versions of Gowers’ and
Rosendal’s dichotomies. When $E$ is a countably infinite-dimensional space
with basis $(e_{n})$ over some countable field $F$, we isolate in §2
_$(p^{+})$ -families_ of block sequences, collections of block sequences
closed under certain diagonalizations and witnessing a weak pigeonhole
principle, and in §3 establish our local form of Rosendal’s dichotomy:

######  Theorem 1.1.

Let $\mathcal{H}$ be a $(p^{+})$-family of block sequences in $E$. If
$\mathbb{A}$ is an analytic set of block sequences and $X\in\mathcal{H}$, then
there is a $Y\in\mathcal{H}\upharpoonright X$ such that either

  1. (i)

I has a strategy in $F[Y]$ for playing into $\mathbb{A}^{c}$, or

  2. (ii)

II has a strategy in $G[Y]$ for playing into $\mathbb{A}$.

Stronger properties of families are discussed in §4, notably _strategic_
families. The existence of filters with these properties is considered in §5
and §6, where their existence is proved to be independent of $\mathsf{ZFC}$.

In §7 we show that, under large cardinal hypothesis, strategic
$(p^{+})$-filters have complete combinatorics for infinite block sequences
with the block subsequence ordering, and generalize Theorem 1.1 to partitions
in $\mathbf{L}(\mathbb{R})$ (the corresponding extension of Gowers’ original
result is due to López-Abad [28], see also [6]). This requires an analysis of
a Mathias-like notion of forcing used to build generic block sequences.

######  Theorem 1.2.

Assume that there is a supercompact cardinal. A filter $\mathcal{G}$ of block
sequences in $E$ is $\mathbf{L}(\mathbb{R})$-generic for the partial ordering
of block sequences if and only if it is a strategic $(p^{+})$-filter.

######  Theorem 1.3.

Assume that there is a supercompact cardinal. Let $\mathcal{H}$ be a strategic
$(p^{+})$-family of block sequences in $E$. If $\mathbb{A}$ is a set of block
sequences in $\mathbf{L}(\mathbb{R})$ and $X\in\mathcal{H}$, then there is a
$Y\in\mathcal{H}\upharpoonright X$ such that either

  1. (i)

I has a strategy in $F[Y]$ for playing into $\mathbb{A}^{c}$, or

  2. (ii)

II has a strategy in $G[Y]$ for playing into $\mathbb{A}$.

In §8 we consider normed vector spaces and Banach spaces. For an infinite-
dimensional separable Banach space $B$ with a Schauder basis, we develop the
notion of _spread $(p^{*})$-families_, similar to the $(p^{+})$-families in
§2, and establish the following local form of Gowers’ dichotomy and its
extension to $\mathbf{L}(\mathbb{R})$:

######  Theorem 1.4.

Let $\mathcal{H}$ be a spread $(p^{*})$-family of normalized block sequences
in $B$ which is invariant under small perturbations. If $\mathbb{A}$ is an
analytic set of normalized block sequences and $X\in\mathcal{H}$, then for any
$\Delta>0$, there is a $Y\in\mathcal{H}\upharpoonright X$ such that either

  1. (i)

every normalized block subsequence of $Y$ is in $\mathbb{A}^{c}$, or

  2. (ii)

II has a strategy in $G^{*}[Y]$ for playing into $\mathbb{A}_{\Delta}$.

######  Theorem 1.5.

Assume that there is a supercompact cardinal. Let $\mathcal{H}$ be a strategic
$(p^{*})$-family of normalized block sequences in $B$ which is invariant under
small perturbations. If $\mathbb{A}$ is a set of normalized block sequences in
$\mathbf{L}(\mathbb{R})$ and $X\in\mathcal{H}$, then for any $\Delta>0$, there
is a $Y\in\mathcal{H}\upharpoonright X$ such that either

  1. (i)

every normalized block subsequence of $Y$ is in $\mathbb{A}^{c}$, or

  2. (ii)

II has a strategy in $G^{*}[Y]$ for playing into $\mathbb{A}_{\Delta}$.

It is our hope that Theorem 1.4 will afford new applications of the techniques
introduced by Gowers in [19] to obtain block sequences in Banach spaces with
simultaneous properties, some captured by the target set $\mathbb{A}$, while
others by the family $\mathcal{H}$.

In §9 we apply these results to the study of the projections in the Calkin
algebra, the quotient of the bounded operators $\mathcal{B}(H)$ on a Hilbert
space $H$ by the compact operators. The natural ordering on projections in the
Calkin algebra induces an ordering $\leq_{\mathrm{ess}}$ on
$\mathcal{P}_{\infty}(H)$, the infinite-rank projections in $\mathcal{B}(H)$.
We give a version of Theorem 1.2 for filters in this ordering:

######  Theorem 1.6.

Assume that there is a supercompact cardinal. A filter $\mathcal{G}$ in
$(\mathcal{P}_{\infty}(H),\leq_{\mathrm{ess}})$ is
$\mathbf{L}(\mathbb{R})$-generic if and only if projections onto block
subspaces are $\leq_{\mathrm{ess}}$-dense in $\mathcal{G}$ and the associated
family of block sequences in $H$ is a strategic $(p^{*})$-family.

Generic filters for $(\mathcal{P}_{\infty}(H),\leq_{\mathrm{ess}})$ induce
_pure states_ on $\mathcal{B}(H)$, via the theory of _quantum filters_
introduced by Farah and Weaver [15]. It is known that these generic pure
states are not pure on any atomic maximal abelian self-adjoint subalgebra
(essentially due to Farah and Weaver [15]), and are thus counterexamples to a
conjecture of Anderson [3]. We show that any family satisfying the hypotheses
of Theorem 1.4 and generating a pure state on $\mathcal{B}(H)$ produces such a
counterexample. We caution that our counterexamples remain beyond
$\mathsf{ZFC}$.

######  Theorem 1.7.

A spread $(p^{*})$-family $\mathcal{H}$ of block sequences in $H$ which is
$\leq_{\mathrm{ess}}$-centered induces a singular pure state $\rho$ on
$\mathcal{B}(H)$ which is not pure on any atomic maximal abelian self-adjoint
subalgebra.

§10 concludes the paper with questions for future investigation.

An effort has been made to keep the set-theoretic prerequisites for
understanding this work to a minimum with the hope that the material,
particularly in §3 and §8, may be used for further applications in Banach
space and operator theory. We assume a familiarity with the basic properties
of Polish spaces, Borel sets, and analytic sets (as covered in [22])
throughout. We only make explicit use of the method of forcing and large
cardinal hypotheses in §5 and §7, with occasional reference back to that
material in §8 and §9. The Banach space prerequisites amount to little more
than a familiarity with basic sequences (as covered in the first sections of
[2]).

##  2\. Families of block sequences

Fix a countable field $F$, a countably infinite-dimensional $F$-vector space
$E$, and an Hamel $F$-basis $(e_{n})$ for $E$. Typically we will think of $F$
as a subfield of $\mathbb{C}$, but this is not necessary; $F$ may even be
finite. Given $v\in E$, say with $v=\sum_{n=0}^{N}a_{n}e_{n}$, let
$\mathrm{supp}(v)=\\{n\in\mathbb{N}:a_{n}\neq 0\\}$, the _support_ of $v$. We
write $n<v$ if $n<\min(\mathrm{supp}(v))$ and $v<w$ if
$\max(\mathrm{supp}(v))<\min(\mathrm{supp}(w))$.

We say that a (finite or infinite) sequence $(x_{n})$ of non-zero vectors in
$E$ is a _block sequence_ (with respect to $(e_{n})$) if for all $n$,
$x_{n}<x_{n+1}$. If $\vec{x}=(x_{0},\ldots,x_{n})$ is a finite block sequence,
let $\mathrm{supp}(\vec{x})=\bigcup_{i=0}^{n}\mathrm{supp}(x_{i})$, and for
$X$ any block sequence, let $\langle
X\rangle=\operatorname{span}(X)\setminus\\{0\\}$. We will abuse notation and
write $E$ for $E\setminus\\{0\\}$, and use “vector” to mean non-zero vector.

Let $\mathrm{bb}^{\infty}(E)$ be the collection of all infinite block
sequences in $E$, which we consider as a subspace of $E^{\mathbb{N}}$, where
$E$ has the discrete topology. It is easy to check that
$\mathrm{bb}^{\infty}(E)$ is a $G_{\delta}$ subset of $E^{\mathbb{N}}$, and
thus a Polish space. Let $\mathrm{bb}^{<\infty}(E)$ be the collection of all
finite block sequences in $E$.

For $X=(x_{n})$ and $Y=(y_{n})$ in $\mathrm{bb}^{\infty}(E)$, we write
$X\preceq Y$ if $(x_{n})$ is a block sequence with respect to $(y_{n})$,
sometimes called a _block subsequence_ of $Y$, or equivalently (for _block_
sequences), $\langle X\rangle\subseteq\langle Y\rangle$. We write
$X\preceq^{*}Y$ if for some $m$, $X/m\preceq Y$, where $X/m$ is the tail of
$X$ with supports above $m$. For $\vec{x}\in\mathrm{bb}^{<\infty}(E)$, write
$X/\vec{x}$ for $X/\max(\mathrm{supp}(\vec{x}))$. Note that the orderings
$\preceq$ and $\preceq^{*}$ fail to be antisymmetric, but are reflexive and
transitive.

We will make repeated use of the following order-theoretic notions: A subset
$D$ of a pre-order $(P,\leq)$ (that is, $\leq$ is reflexive and transitive) is
_dense_ if for all $p\in P$, there is a $q\in D$ with $q\leq p$. It is,
moreover, _dense open_ , if whenever $q\leq p\in D$, then $q\in D$. Elements
$p$ and $q$ in $P$ are _compatible_ if they have a common lower bound in $P$,
and _incompatible_ otherwise.

Compatibility in $(\mathrm{bb}^{\infty}(E),\preceq)$ is equivalent to that in
$(\mathrm{bb}^{\infty}(E),\preceq^{*})$ and we write $X\bot Y$ when $X$ and
$Y$ are incompatible. The following observation shows that
$(\mathrm{bb}^{\infty}(E),\preceq)$ can be identified with a dense suborder of
the lattice of all infinite-dimensional subspaces of $E$. In particular, $X$
and $Y$ are compatible if and only if $\langle X\rangle\cap\langle Y\rangle$
is infinite-dimensional.

######  Lemma 2.1.

If $X$ is an infinite-dimensional subspace of $E$, then $X$ contains an
infinite block sequence.

###### Proof.

By taking appropriate linear combinations, one can show that for any $N$, $X$
contains an infinite-dimensional subspace whose supports are above $N$. From
this, it is easy to inductively construct a block sequence in $X$. ∎

Throughout, when we speak of a _family_
$\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$, we mean a non-empty subset
which is closed upwards with respect to $\preceq^{*}$. For $X\in\mathcal{H}$,
we denote by $\mathcal{H}\upharpoonright X=\\{Y\in\mathcal{H}:Y\preceq X\\}$.
A _filter_ $\mathcal{F}\subseteq\mathrm{bb}^{\infty}(E)$ is a family such that
for every $X,Y\in\mathcal{F}$, there is a $Z\in\mathcal{F}$ with $Z\preceq X$
and $Z\preceq Y$.

######  Definition 2.2.

  1. (a)

Given a descending sequence $X_{0}\succeq X_{1}\succeq\cdots$ in
$\mathrm{bb}^{\infty}(E)$, we call $Y\in\mathrm{bb}^{\infty}(E)$ a
_diagonalization_ of $(X_{n})$ if for all $n$, $Y\preceq^{*}X_{n}$.

  2. (b)

Given a sequence $(\mathcal{D}_{n})$ of subsets of $\mathrm{bb}^{\infty}(E)$,
we call $Y$ a _diagonalization_ of $(\mathcal{D}_{n})$ if for each $n$, there
is an $X_{n}\in\mathcal{D}_{n}$ such that $Y\preceq^{*}X_{n}$.

For $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$, a set $\mathcal{D}$ is
_$\preceq$ -dense (open) in $\mathcal{H}$_ if $\mathcal{D}\cap\mathcal{H}$ is.

######  Definition 2.3.

A family $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ is a _$(p)$ -family_,
or has the _$(p)$ -property_, if whenever $X_{0}\succeq X_{1}\succeq\cdots$ is
a decreasing sequence with each $X_{n}\in\mathcal{H}$, there is a
diagonalization $Y\in\mathcal{H}$ of $(X_{n})$.

It is easy to see that $\mathrm{bb}^{\infty}(E)$ itself is a $(p)$-family. We
note that every $(p)$-family $\mathcal{H}$ contains a diagonalization of any
given sequence $(\mathcal{D}_{n})$ of $\preceq$-dense open subsets in
$\mathcal{H}$: build a decreasing sequence $(X_{n})$ in $\mathcal{H}$ with
each $X_{n}\in\mathcal{D}_{n}$, then any diagonalization $Y\in\mathcal{H}$ of
$(X_{n})$ will be a diagonalization of $(\mathcal{D}_{n})$. This can be done
below any given $X\in\mathcal{H}$, so the set of such diagonalizations is
$\preceq$-dense in $\mathcal{H}$. This latter property, which could be called
the _weak $(p)$-property_, will be sufficient for all of the results in §3,
and in particular, for Theorem 1.1.

Recall that $\mathcal{H}\subseteq[\mathbb{N}]^{\infty}$ is a _coideal_ if it
contains all co-finite sets, is closed upwards with respect to $\subseteq$,
and whenever $Y_{0}\cup Y_{1}\in\mathcal{H}$, then one of $Y_{0}$ or $Y_{1}$
is also in $\mathcal{H}$. This last property asserts that $\mathcal{H}$
witnesses the pigeonhole principle. In our setting, provided $|F|>2$,111When
$|F|=2$, such a pigeonhole principle for block subspaces does hold; this is
essentially Hindman’s Theorem [20]. the “obvious” formulation of the
pigeonhole principle is simply false, as the following example shows:

######  Example 2.4.

222The author would like to thank Jordi López-Abad for pointing out this
example which has the advantage of being well-defined at the level of the
spanned subspaces.

Consider the case when $F\subseteq\mathbb{R}$. Similar examples can be
constructed whenever $|F|>2$, cf. Theorem 7 in [26]. For a vector $x\in E$
define the _oscillation_ $\mathrm{osc}(x)$ as the number of times the sign of
the non-zero coefficients of $x$ alternate in its expansion with respect to
$(e_{n})$. So, $\mathrm{osc}(e_{0}-e_{1}+e_{2})=2$,
$\mathrm{osc}(e_{2}+e_{4}-e_{5}+e_{7}-e_{10})=3$, etc.

Define $A_{0}\subseteq E$ (respectively, $A_{1}\subseteq E$) to be the set of
all $x\in E$ such that $\mathrm{osc}(x)$ is even (respectively, odd), and let
$\mathbb{A}_{i}=\\{(x_{n}):x_{0}\in A_{i}\\}$ for $i=0,1$. The
$\mathbb{A}_{i}$ are clopen sets which partition $\mathrm{bb}^{\infty}(E)$.
Moreover, the pair $\mathbb{A}_{0}$, $\mathbb{A}_{1}$ is _asymptotic_ , that
is, for any $X\in\mathrm{bb}^{\infty}(E)$ and $i=0,1$, there is $Y_{i}\preceq
X$ such that $Y_{i}\in\mathbb{A}_{i}$. To see this, suppose that $X=(x_{n})$
is such that $X\in\mathbb{A}_{0}$, so $\mathrm{osc}(x_{0})$ is even. If
$\mathrm{osc}(x_{1})$ is odd, then $(x_{n})_{n\geq 1}\preceq X$ and in
$\mathbb{A}_{1}$. If $\mathrm{osc}(x_{1})$ is even, then let $x=x_{0}-x_{1}$
if the signs of the last non-zero coefficient in $x_{0}$ and the first in
$x_{1}$ agree, and $x=x_{0}+x_{1}$ otherwise. In either case,
$\mathrm{osc}(x)=\mathrm{osc}(x_{0})+\mathrm{osc}(x_{1})+1$, so
$(x,x_{2},x_{3},\ldots)$ is in $\mathbb{A}_{1}$.

The following is a weak analogue of the pigeonhole property of coideals.

######  Definition 2.5.

Let $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ be a family.

  1. (a)

A subset $\mathcal{D}\subseteq\mathrm{bb}^{\infty}(E)$ is _$\mathcal{H}$
-dense_ below some $X\in\mathcal{H}$ if for every
$Y\in\mathcal{H}\upharpoonright X$, there is a $Z\preceq Y$ with
$Z\in\mathcal{D}$. A set $D\subseteq E$ is $\mathcal{H}$-dense below $X$ if
$\\{Z:\langle Z\rangle\subseteq D\\}$ is.

  2. (b)

$\mathcal{H}$ is _full_ if whenever $D\subseteq E$ (not necessarily a
subspace) and $X\in\mathcal{H}$ are such that $D$ is $\mathcal{H}$-dense below
$X$, there is a $Z\in\mathcal{H}\upharpoonright X$ with $\langle
Z\rangle\subseteq D$.

Fullness allows one to upgrade $\\{Z:\langle Z\rangle\subseteq D\\}$ being
$\mathcal{H}$-dense below $X$ to being $\preceq$-dense (open) below $X$ in
$\mathcal{H}$. Obviously $\mathrm{bb}^{\infty}(E)$ itself is a full family. If
the family in question is a filter $\mathcal{F}$, we may simplify the
definition of fullness by replacing $X$ with $(e_{n})$ (or any element of
$\mathcal{F}$). We note that any full filter is maximal; this can be seen by
applying the definition of fullness when $D$ is a block subspace. It is shown
in Proposition 3.6 that fullness is necessary for Theorem 1.1.

######  Definition 2.6.

A family in $\mathrm{bb}^{\infty}(E)$ which is full and has the $(p)$-property
will be called a _$(p^{+})$ -family_. Likewise for _$(p^{+})$ -filter_.

######  Lemma 2.7.

  1. (a)

For $X_{0}\succeq X_{1}\succeq\cdots$ in $\mathrm{bb}^{\infty}(E)$, the set

$\mathcal{D}_{(X_{n})}=\\{Y:Y\text{ is a diagonalization of }(X_{n})\text{ or
}\exists n(Y\bot X_{n})\\}$

is $\preceq$-dense open.

  2. (b)

For $D\subseteq E$ and $X\in\mathrm{bb}^{\infty}(E)$, the set

$\mathcal{D}_{D,X}=\\{Z:\langle Z\rangle\subseteq D\text{ or }\forall V\preceq
X(\langle V\rangle\subseteq D\rightarrow V\bot Z)\\}$

is $\preceq$-dense open below $X$.

###### Proof.

(a) Take $Y\in\mathrm{bb}^{\infty}(E)$ which is compatible with all of the
$X_{n}$. We can build a diagonalization $X=(x_{n})\preceq Y$ by picking
vectors $x_{n}\in\langle X_{n}\rangle\cap\langle Y\rangle$ with
$x_{n}<x_{n+1}$.

(b) Take $Y\preceq X$. If there is no $Z\preceq Y$ such that $\langle
Z\rangle\subseteq D$, then for any $V\preceq X$ with $\langle
V\rangle\subseteq D$, it must be that $V\bot Y$, as otherwise any $Z$
witnessing the compatibility of $V$ and $Y$ would satisfy $\langle
Z\rangle\subseteq D$. ∎

Lemma 2.7 will be used to construct $(p^{+})$-filters in §5. We will see in
Corollary 6.5 that the existence of full filters is independent of
$\mathsf{ZFC}$.

##  3\. Games with vectors and a local Rosendal dichotomy

The _Gowers game_ played below $X\in\mathrm{bb}^{\infty}(E)$, denoted $G[X]$,
is defined as follows: Two players, I and II, alternate with I going first and
playing block sequences $X_{k}\preceq X$, and II responding with vectors
$y_{k}\in\langle X_{k}\rangle$ subject to the constraint $y_{k}<y_{k+1}$. The
block sequence $(y_{k})$ is the _outcome_ of a play of the game. Given
$\vec{x}\in\mathrm{bb}^{<\infty}(E)$ and $X\in\mathrm{bb}^{\infty}(E)$, the
game $G[\vec{x},X]$ is defined exactly as $G[X]$ except that II is restricted
to playing vectors above $\vec{x}$ and the outcome is
$\vec{x}^{\smallfrown}(y_{k})$. This is a discrete version of the game defined
by Gowers in [18] [19]

A _strategy_ for II in $G[\vec{x},X]$ is a function $\alpha$ taking sequences
$(X_{0},\ldots,X_{k})$ of possible prior moves by I to vectors $y\in\langle
X_{k}\rangle$, with $\vec{x}<\alpha(X_{0},\ldots,X_{k-1})<y$, for all $k$.
Given a set $\mathbb{A}\subseteq\mathrm{bb}^{\infty}(E)$, we say that $\alpha$
is a strategy in $G[\vec{x},X]$ for _playing into $\mathbb{A}$_ if whenever II
follows $\alpha$ (that is, at each turn, given as input I’s prior moves, they
play the output of $\alpha$), the resulting outcome lies in $\mathbb{A}$.
These notions are defined likewise for I.

The _infinite asymptotic game_ [34] [35] played below $X$, denoted $F[X]$, is
defined in a similar fashion: Two players, I and II, alternate with I going
first and playing natural numbers $n_{k}$, and II responding with vectors
$y_{k}\in\langle X/n_{k}\rangle$ subject to the constraint $y_{k}<y_{k+1}$.
Again, $(y_{k})$ is the _outcome_ of a play of the game. The game
$F[\vec{x},X]$ is defined as above, as are _strategies_ for I and II, and the
notion of having a strategy for _playing into_ a set.

It is important to note that plays of $F[\vec{x},X]$ can be considered as
plays of $G[\vec{x},X]$ where I is restricted to playing tail block
subsequences of $X$. Consequently, if II has a strategy in $G[\vec{x},X]$ for
playing into a set $\mathbb{A}$, then II has such a strategy in $F[\vec{x},X]$
as well. Similarly, if I has a strategy in $F[\vec{x},X]$ for playing into
$\mathbb{A}$, then they have such a strategy in $G[\vec{x},X]$.

The following generalizes the notion of _strategically Ramsey_ given in [35],
where $\mathcal{H}$ was taken to be all of $\mathrm{bb}^{\infty}(E)$.

######  Definition 3.1.

For $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ a family, we say that a
subset $\mathbb{A}\subseteq\mathrm{bb}^{\infty}(E)$ is _$\mathcal{H}$
-strategically Ramsey_ if for all $\vec{y}\in\mathrm{bb}^{<\infty}(E)$ and
$X\in\mathcal{H}$, there is a $Y\in\mathcal{H}\upharpoonright X$ such that
either

  1. (i)

I has a strategy in $F[\vec{y},Y]$ for playing into $\mathbb{A}^{c}$, or

  2. (ii)

II has a strategy in $G[\vec{y},Y]$ for playing into $\mathbb{A}$.

Note that consequences (i) and (ii) are mutually exclusive by our comments
above. The key fact about $\mathcal{H}$-strategically Ramsey sets is that the
witness, $Y$ in the above definition, can be found in $\mathcal{H}$.

Our goal for the remainder of this section is to outline the proof that, for
any $(p^{+})$-family $\mathcal{H}$, analytic sets are
$\mathcal{H}$-strategically Ramsey, thereby establishing Theorem 1.1. Much of
what follows closely hews to [35], and is a variation on the combinatorial
forcing technique used in [40].

######  Definition 3.2.

Let $\mathcal{H}$ be a family and $\mathbb{A}\subseteq\mathrm{bb}^{\infty}(E)$
be given. For $\vec{y}\in\mathrm{bb}^{<\infty}(E)$ and $Y\in\mathcal{H}$, we
say that

  1. (1)

$(\vec{y},Y)$ is _good (for $\mathbb{A}$)_ if II has a strategy in
$G[\vec{y},Y]$ for playing into $\mathbb{A}$,

  2. (2)

$(\vec{y},Y)$ is _bad (for $\mathbb{A}$)_ if for all
$Z\in\mathcal{H}\upharpoonright Y$, $(\vec{y},Z)$ is not good.

  3. (3)

$(\vec{y},Y)$ is _worse (for $\mathbb{A}$)_ if it is bad and there is an $n$
such that for every $v\in\langle Y/n\rangle$, $(\vec{y}^{\smallfrown}v,Y)$ is
bad.

Reference to $\mathbb{A}$ and $\mathcal{H}$ will be suppressed where
understood.

######  Lemma 3.3.

If $\mathcal{H}$ is a $(p^{+})$-family and
$\mathbb{A}\subseteq\mathrm{bb}^{\infty}(E)$, then for every
$\vec{x}\in\mathrm{bb}^{<\infty}(E)$ and $X\in\mathcal{H}$, there is a
$Y\in\mathcal{H}\upharpoonright X$ such that either

  1. (i)

$(\vec{x},Y)$ is good, or

  2. (ii)

I has a strategy in $F[\vec{x},Y]$ for playing into

$\\{(z_{n}):\forall n(\vec{x}^{\smallfrown}(z_{0},\ldots,z_{n}),Y)\text{ is
worse}\\}.$

###### Proof.

Observe that if $(\vec{y},Y)$ is good/bad/worse and $Z\preceq^{*}Y$ is in
$\mathcal{H}$, then $(\vec{y},Z)$ is also good/bad/worse. It is immediate that
for each $\vec{y}$, the set

$\mathcal{D}_{\vec{y}}=\\{Y\in\mathcal{H}:(\vec{y},Y)\text{ is either good or
bad}\\}$

is $\preceq$-dense open in $\mathcal{H}$.

######  Claim.

If $(\vec{y},Y)$ is bad, then for all $Z\in\mathcal{H}\upharpoonright Y$,
there is a $V\preceq Z$ such that for all $x\in\langle V/\vec{y}\rangle$,
$(\vec{y}^{\smallfrown}x,Y)$ is not good.

###### Proof of claim..

Let $(\vec{y},Y)$ be bad. Towards a contradiction, suppose that there is some
$Z\in\mathcal{H}\upharpoonright Y$ such that for all $V\preceq Z$, there is an
$x\in\langle V/\vec{y}\rangle$ such that $(\vec{y}^{\smallfrown}x,Y)$ is good.
We claim that $(\vec{y},Z)$ is good. If I plays $V\preceq Z$, then by
supposition there is some $x\in\langle V/\vec{y}\rangle$ such that
$(\vec{y}^{\smallfrown}x,Z)$ is good. Let II play that $x$ and from then on
follow the strategy given from $(\vec{y}^{\smallfrown}x,Z)$ being good. This
is contrary to $(\vec{y},Y)$ being bad. ∎

######  Claim.

For each $\vec{y}$, the set

$\mathcal{E}_{\vec{y}}=\\{Z\in\mathcal{H}:(\vec{y},Z)\text{ is either good or
worse}\\}$

is $\preceq$-dense open in $\mathcal{H}$.

###### Proof of claim..

Fix $\vec{y}$ and let $Y\in\mathcal{H}$. Since the sets
$\mathcal{D}_{\vec{x}}$ are dense in $\mathcal{H}$ and there are only
countably many $\vec{x}$, the $(p)$-property allows us to diagonalize all of
them within $\mathcal{H}$ and assume that for all $\vec{x}$, $(\vec{x},Y)$ is
either good or bad. Suppose that $(\vec{y},Y)$ is bad. Let
$D=\\{x:(\vec{y}^{\smallfrown}x,Y)\text{ is not good}\\}$. By the previous
claim, $D$ is $\mathcal{H}$-dense below $Y$. Since $\mathcal{H}$ is full,
there is a $Z\in\mathcal{H}\upharpoonright Y$ such that $\langle
Z\rangle\subseteq D$. If $z\in\langle Z\rangle$, then
$(\vec{y}^{\smallfrown}z,Z)$ is not good, hence bad, by our choice of $Y$.
Thus, $(\vec{y},Z)$ is worse. ∎

We can now prove the lemma. By the previous claim, we have a
$Y\in\mathcal{H}\upharpoonright X$ so that for all $\vec{y}$,
$(\vec{x}^{\smallfrown}\vec{y},Y)$ is either good or worse. If $(\vec{x},Y)$
is good, we’re done, so suppose that $(\vec{x},Y)$ is worse. We will describe
a strategy for I in $F[\vec{x},Y]$: Suppose that at some point in the game
$(z_{0},\ldots,z_{k})$ has been played by II so that
$(\vec{x}^{\smallfrown}(z_{0},\ldots,z_{k}),Y)$ is worse. Then, there is some
$n$ such that for all $z\in\langle Y\rangle$, if $n<z$, then
$(\vec{x}^{\smallfrown}(z_{0},\ldots,z_{k})^{\smallfrown}z,Y)$ is bad, hence
worse. Let I play $n$. ∎

######  Lemma 3.4 (cf. Lemma 2 in [35]).

Let $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ a $(p^{+})$-family. Then,
open sets are $\mathcal{H}$-strategically Ramsey.

###### Proof.

Let $\mathbb{A}\subseteq\mathrm{bb}^{\infty}(E)$ be open. Given
$\vec{x}\in\mathrm{bb}^{<\infty}(E)$ and $X\in\mathcal{H}$, by Lemma 3.3,
there is a $Y\in\mathcal{H}\upharpoonright X$ such that either $(\vec{x},Y)$
is good, in which case we’re done, or I has a strategy in $F[\vec{x},Y]$ to
play $(z_{n})$ such that for all $n$,
$(\vec{x}^{\smallfrown}(z_{0},\ldots,z_{n}),Y)$ is worse. In the latter case,
if I follows this strategy, as II builds $(z_{n})$, for no $m$ can II have a
strategy in $G[\vec{x}^{\smallfrown}(z_{0},\ldots,z_{m}),Y]$ to play in
$\mathbb{A}$. Since $\mathbb{A}$ is open, this means that
$\vec{x}^{\smallfrown}(z_{0},z_{1},\ldots)\notin\mathbb{A}$ and I has a
strategy for playing into $\mathbb{A}^{c}$. ∎

###### Proof (sketch) of Theorem 1.1..

The proof closely follows that of Theorem 5 in [35], where
$\mathcal{H}=\mathrm{bb}^{\infty}(E)$. The idea of the proof is, given a
Souslin scheme $\\{\mathbb{A}_{s}\\}_{s\in\mathbb{N}^{<\infty}}$ for an
analytic set $\mathbb{A}$, we can use Lemma 3.4 and diagonalization to find a
$Y\in\mathrm{bb}^{\infty}(E)$ such that if I does not have a strategy in
$F[Y]$ for playing into $\mathbb{A}^{c}$, then in $G[Y]$, II can build a
sequence $(z_{k})$ such that I continues to have no strategy in
$F[(z_{0},\ldots,z_{k}),Y]$ for playing into $\mathbb{A}_{s}^{c}$, where $s$
is an initial segment of some branch $y$ in $\mathbb{N}^{\mathbb{N}}$. $y$
will witness that II’s strategy has produced an outcome in $\mathbb{A}$. We
omit the details, except to say that the arguments in [35] can be modified for
our result simply by ensuring that the block sequences used are taken in
$\mathcal{H}$. This can be done, in each instance, as a block sequence is
obtained either by applying the result for open sets or by diagonalization. ∎

Theorem 1.1 is consistently sharp and necessarily asymmetric, as there is a
coanalytic counterexample (for $\mathcal{H}=\mathrm{bb}^{\infty}(E)$) in
$\mathbf{L}$ [28].333This counterexample is to Gowers’ theorem, but the
discussion in §5 of [35] shows that this also yields a counterexample to
Rosendal’s dichotomy. In particular, the collection of
$\mathcal{H}$-strategically Ramsey sets may fail to be a $\sigma$-algebra. It
is, however, closed under countable unions. Again, the proof is nearly
identical to that of the corresponding result in [35] and omitted.

######  Theorem 3.5 (cf. Theorem 9 in [35]).

Let $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ be a $(p^{+})$-family. Then,
the collection of $\mathcal{H}$-strategically Ramsey sets is closed under
countable unions.∎

We note that fullness is a necessary assumption for our results:

######  Proposition 3.6.

If $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ is a family for which clopen
sets are $\mathcal{H}$-strategically Ramsey, then $\mathcal{H}$ is full.

###### Proof.

Given $D\subseteq E$, $\mathcal{H}$-dense below some $X\in\mathcal{H}$, let
$\mathbb{D}=\\{(z_{n}):z_{0}\in D\\}$, a clopen subset of
$\mathrm{bb}^{\infty}(E)$. For no $Y\in\mathcal{H}\upharpoonright X$ can II
have a strategy into $\mathbb{D}^{c}$: Consider the round of $G[Y]$ where I
starts by playing some $Z\preceq Y$ with $\langle Z\rangle\subseteq D$. Since
$\mathbb{D}^{c}$ is $\mathcal{H}$-strategically Ramsey, there is a
$Y\in\mathcal{H}\upharpoonright X$ such that I has a strategy $\sigma$ in
$F[Y]$ for playing into $\mathbb{D}$. Let
$Z=Y/\sigma(\emptyset)\in\mathcal{H}$. Since $\sigma$ is a strategy for
playing into $\mathbb{D}$, $\langle Z\rangle\subseteq D$. ∎

##  4\. Stronger properties of families

If an element $Y$ in a family $\mathcal{H}$ witnesses Theorem 1.1, then either
$\mathbb{A}^{c}$ or $\mathbb{A}$ is $\mathcal{H}$-dense below $Y$, depending
on which half of the dichotomy holds. However, it would be desirable to ensure
that $\mathcal{H}$ itself meets whichever one of $\mathbb{A}^{c}$ or
$\mathbb{A}$ the conclusion of the dichotomy provides. To this end, we
consider stronger properties of families, the first of which is based on the
original definition of selectivity (or being “happy”) given in [30].

######  Definition 4.1.

  1. (a)

For $(X_{\vec{x}})_{\vec{x}\in\mathrm{bb}^{<\infty}(E)}$ generating a filter
in $\mathrm{bb}^{\infty}(E)$, we say that $X\in\mathrm{bb}^{\infty}(E)$
_strongly diagonalizes $(X_{\vec{x}})$_ if $X/\vec{x}\preceq X_{\vec{x}}$
whenever $\vec{x}\sqsubseteq X$.

  2. (b)

A family $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ is a _strong
$(p)$-family_, or has the _strong $(p)$-property_, if whenever
$(X_{\vec{x}})_{\vec{x}\in\mathrm{bb}^{<\infty}(E)}$ generates a filter in
$\mathcal{H}$, there is an $Y\in\mathcal{H}$ which strongly diagonalizes
$(X_{\vec{x}})$.

The strong $(p)$-property implies the $(p)$-property: Take $X_{0}\succeq
X_{1}\succeq\cdots$ in $\mathcal{H}$ and define $X_{\vec{x}}=X_{|\vec{x}|}$
for $\vec{x}\in\mathrm{bb}^{<\infty}(E)$. Any $X$ strongly diagonalizing
$(X_{\vec{x}})$ will diagonalize $(X_{n})$.

As in Lemma 2.7, it is useful for constructing families with the strong
$(p)$-property to know that it corresponds to certain $\preceq$-dense sets.

######  Lemma 4.2.

For $(X_{\vec{x}})_{\vec{x}\in\mathrm{bb}^{<\infty}(E)}$ generating a filter
in $\mathrm{bb}^{\infty}(E)$, the set

$\displaystyle\\{Y:Y\text{ is a strong diagonalization of
$(X_{\vec{x}})_{\vec{x}\in\mathrm{bb}^{<\infty}(E)}$, or}$
$\displaystyle\hskip
113.81102pt\\{Y\\}\cup(X_{\vec{x}})_{\vec{x}\in\mathrm{bb}^{<\infty}(E)}\text{
does not generate a filter}\\}$

is $\preceq$-dense.

###### Proof.

Fix $X\in\mathrm{bb}^{\infty}(E)$, and suppose that $\\{X\\}\cup(X_{\vec{x}})$
generates a filter. We build a $Y\preceq X$ which strongly diagonalizes
$(X_{\vec{x}})$: Pick any $y_{0}\in\langle X\rangle\cap\langle
X_{\emptyset}\rangle$. Since $X$, $X_{\emptyset}$ and $X_{(y_{0})}$ generate a
filter, there is a $y_{1}\in\langle X\rangle\cap\langle
X_{\emptyset}\rangle\cap\langle X_{(y_{0})}\rangle$ with $y_{0}<y_{1}$.
Continue in this fashion. ∎

The following result connects the strong $(p)$-property to the infinite
asymptotic game and is based on a characterization of selective ultrafilters
(Theorem 4.5.3 in [7]).

######  Theorem 4.3.

If $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ is a strong $(p)$-family,
then for no $X\in\mathcal{H}$ does I have a strategy in $F[X]$ for playing
into $\mathcal{H}^{c}$.444When $\mathcal{F}$ is a strong $(p)$-filter, one can
improve the conclusion to: for no $X\in\mathcal{F}$ does I have a strategy in
$G_{\mathcal{F}}[X]$ for playing into $\mathcal{F}^{c}$. Here,
$G_{\mathcal{F}}[X]$ is the variant of the Gowers game below $X$ where I is
restricted to playing elements of $\mathcal{F}$. See §3.11 of [39].

###### Proof.

Let $\sigma$ be a strategy for I in $F[X]$ for playing into $\mathcal{H}^{c}$,
where $X\in\mathcal{H}$. Towards a contradiction, suppose that $\mathcal{H}$
has the strong $(p)$-property. Define sets
$\mathcal{A}_{\vec{x}}\subseteq\mathcal{H}$ as follows:
$\mathcal{A}_{\emptyset}=\\{X/\sigma(\emptyset)\\}$ and inductively, for
$\vec{x}=(x_{0},\ldots,x_{n-1})$, $\mathcal{A}_{\vec{x}}$ is the set of all
$X/m$ where $m$ is played by I following $\sigma$ in the first $n$ rounds of
$F[X]$, as II plays $x_{0},x_{1},\ldots,x_{n-1}$. In the case that elements of
a given $\vec{x}$ fail to be valid moves for II against $\sigma$, let
$\mathcal{A}_{\vec{x}}=\mathcal{A}_{\vec{x}^{\prime}}$ where
$\vec{x}^{\prime}$ is the maximal initial segment of $\vec{x}$ consisting of
valid moves. Then, for all $\vec{x}$, $\mathcal{A}_{\vec{x}}$ is finite and
$\mathcal{A}_{\vec{x}}\subseteq\mathcal{A}_{\vec{y}}$ whenever
$\vec{x}\sqsubseteq\vec{y}$.

For each $\vec{x}$, let $M_{\vec{x}}=\max\\{m:X/m\in\mathcal{A}_{\vec{x}}\\}$
and $Y_{\vec{x}}=X/M_{\vec{x}}$. Clearly $(Y_{\vec{x}})$ generates a filter in
$\mathcal{H}$. By the strong $(p)$-property, there is a
$Y=(y_{n})\in\mathcal{H}\upharpoonright X$ such that $Y/\vec{y}\preceq
Y_{\vec{y}}$ for all $\vec{y}\sqsubseteq Y$.

Consider the play of $F[X]$ wherein I follows $\sigma$ and II plays $y_{0}$,
$y_{1}$, and so on. We claim that this is a valid sequence of moves for II.
Note that $y_{0}\in\langle Y/\emptyset\rangle\subseteq\langle
Y_{\emptyset}\rangle\subseteq\langle X/\sigma(\emptyset)\rangle$, so $y_{0}$
is a valid move. Inductively, suppose that $(y_{0},\ldots,y_{k})$ is a valid
sequence of moves. We have $y_{k+1}\in\langle
Y/(y_{0},\ldots,y_{k})\rangle\subseteq\langle
Y_{(y_{0},\ldots,y_{k})}\rangle\subseteq\langle
X/\sigma(y_{0},\ldots,y_{k})\rangle$, where the last containment uses our
induction hypothesis. Thus, $y_{k+1}$ is a valid move. Since the resulting
outcome in this play is in $\mathcal{H}$, we have a contradiction. ∎

Equivalently, Theorem 4.3 says that if $\mathcal{H}$ is a strong $(p)$-family
and $\sigma$ is a strategy for I in $F[X]$, for $X\in\mathcal{H}$, then there
is an outcome of $\sigma$ in $\mathcal{H}$.

######  Lemma 4.4.

If $\mathbb{D}\subseteq\mathrm{bb}^{\infty}(E)$ is $\preceq$-dense open below
$X\in\mathrm{bb}^{\infty}(E)$, then

  1. (a)

II has a strategy in $F[X]$ for playing into $\mathbb{D}$, and

  2. (b)

I has a strategy in $G[X]$ for playing into $\mathbb{D}$.

###### Proof.

For $F[X]$, take $Y\preceq X$ in $\mathbb{D}$ and let II always play vectors
in $Y$. For $G[X]$, take $Y\preceq X$ in $\mathbb{D}$, and let I simply play
$Y$ repeatedly. ∎

It follows from Lemma 4.4, and Theorems 1.1 and 4.3, that whenever
$\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ is a strong $(p^{+})$-family and
$\mathbb{D}$ is a coanalytic $\preceq$-dense open set, then
$\mathcal{H}\cap\mathbb{D}\neq\emptyset$. In particular, strong
$(p^{+})$-families meet all $\preceq$-dense open Borel sets. This is a special
case of Theorem 1.2. The following definition is a counterpart to Theorem 4.3
for II in $G[X]$.

######  Definition 4.5.

A family $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ is _strategic_ if
whenever $X\in\mathcal{H}$ and $\alpha$ is a strategy for II in $G[X]$, there
is an outcome of $\alpha$ which is in $\mathcal{H}$.

As as above, if $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ is a strategic
$(p^{+})$-family and $\mathbb{D}\subseteq\mathrm{bb}^{\infty}(E)$ is an
analytic $\preceq$-dense open set, then
$\mathbb{D}\cap\mathcal{H}\neq\emptyset$. As a consequence for
$(p^{+})$-filters, being strategic subsumes the strong $(p)$-property.

######  Proposition 4.6.

If $\mathcal{F}\subseteq\mathrm{bb}^{\infty}(E)$ is a strategic
$(p^{+})$-filter, then $\mathcal{F}$ is also a strong $(p)$-filter.

###### Proof.

Suppose that $\mathcal{F}$ is as described and
$(X_{\vec{x}})_{\vec{x}\in\mathrm{bb}^{<\infty}(E)}$ is contained
$\mathcal{F}$. Let $\mathcal{D}$ be the set given in Lemma 4.2, so that the
$\preceq$-downwards closure of $\mathcal{D}$ is a $\preceq$-dense open set.
Moreover, $\mathcal{D}$ is easily seen to be Borel and its $\preceq$-downwards
closure analytic. By the comments above, it follows that
$\mathcal{F}\cap\mathcal{D}\neq\emptyset$, and any
$Y\in\mathcal{F}\cap\mathcal{D}$ must be a strong diagonalization of
$(X_{\vec{x}})$. ∎

In §5 we will construct (under set-theoretic hypotheses) strategic
$(p^{+})$-filters. To this end, we again need to know that certain sets are
$\preceq$-dense, but also that there are not “too many” of them. If $\alpha$
is a strategy for II in $G[X]$, then the set of outcomes which result from
$\alpha$, denoted by $[\alpha,X]$, is $\preceq$-dense below $X$. However, as
strategies are functions from finite sequences in $\mathrm{bb}^{\infty}(E)$ to
vectors, there are $2^{2^{\aleph_{0}}}$ many of them.

One way to resolve this is to “finitize” the Gowers game as in [5]: given
$X\in\mathrm{bb}^{\infty}(E)$, the _finite-dimensional Gowers game below $X$_,
denoted by $G_{f}[X]$, is defined as follows: Two players, I and II, alternate
with I going first and playing a non-zero vector $x_{0}^{(0)}\in\langle
X\rangle$. II responds with either a non-zero $y_{0}\in\langle
x_{0}^{(0)}\rangle$ or $0$. If II plays $y_{0}$, then the game “restarts” with
I playing a non-zero vector $x_{0}^{(1)}\in\langle X\rangle$. If II plays $0$,
then I must play a non-zero vector $x_{1}^{(0)}\in\langle
X/x^{(0)}_{0}\rangle$, to which II again responds with either a non-zero
vector $y_{0}\in\langle x_{0}^{(0)},x_{1}^{(0)}\rangle$ or $0$, and so on. The
non-zero plays of II are required to satisfy $y_{n}<y_{n+1}$ and the outcome
is the sequence $(y_{n})$. The notion of strategy for II in $G_{f}[X]$ is
defined in the obvious way (with the added requirement that the outcome must
be infinite) and we denote by $[\alpha,X]_{f}$ the corresponding set of
outcomes.

######  Lemma 4.7.

If $\alpha$ is a strategy for II in $G[X]$, then there is a strategy
$\alpha^{\prime}$ for II in $G_{f}[X]$ such that
$[\alpha^{\prime},X]_{f}\subseteq[\alpha,X]$. Moreover,
$[\alpha^{\prime},X]_{f}$ is still $\preceq$-dense below $X$.

###### Proof.

The proof is identical to the $(\Leftarrow)$ direction of Theorem 1.2 in [5].
∎

It is easy to see that strategies $\alpha$ for II in $G_{f}[X]$ are coded by
reals and $[\alpha,X]_{f}$ is an analytic set. This will suffice for our
constructions in §5.

##  5\. Constructions of filters in $\mathrm{bb}^{\infty}(E)$

In this section we show how to construct filters
$\mathcal{F}\subseteq\mathrm{bb}^{\infty}(E)$ having all of the properties
discussed in §2 and §4. These constructions use either assumptions about
certain “cardinal invariants” (cf. [10]) which hold consistently with
$\mathsf{ZFC}$, or the method of forcing. We will see in Corollary 6.5 that we
cannot hope for a construction in $\mathsf{ZFC}$ alone.

######  Definition 5.1.

  1. (a)

A _tower_ (of _length_ $\kappa$) in $\mathrm{bb}^{\infty}(E)$ is a sequence
$(X_{\alpha})_{\alpha<\kappa}$ such that $\alpha<\beta<\kappa$ implies
$X_{\beta}\preceq^{*}X_{\alpha}$ and there is no $X\in\mathrm{bb}^{\infty}(E)$
with $X\preceq^{*}X_{\alpha}$ for all $\alpha<\kappa$.

  2. (b)

$\mathfrak{t}^{*}$ is the minimum length of a tower in
$\mathrm{bb}^{\infty}(E)$.

$\mathfrak{t}^{*}$ is a regular cardinal and, moreover, uncountable as
$\mathrm{bb}^{\infty}(E)$ has the $(p)$-property. Thus, the Continuum
Hypothesis ($\mathsf{CH}$) implies that $\mathfrak{t}^{*}=2^{\aleph_{0}}$.

We use the following notational conventions for versions of Martin’s Axiom:
for $\kappa<2^{\aleph_{0}}$, $\mathsf{MA}(\kappa)$ is the forcing axiom for
meeting $\kappa$-many dense subsets of a ccc poset, $\mathsf{MA}$ is
$\forall\kappa<2^{\aleph_{0}}(\mathsf{MA}(\kappa))$, and
$\mathsf{MA}(\sigma\text{-centered})$ is $\mathsf{MA}$ restricted to
$\sigma$-centered posets.

######  Lemma 5.2 (Lemma 5 in [17]).

($\mathsf{MA}(\sigma\text{-centered})$) If
$\mathcal{L}\subseteq\mathrm{bb}^{\infty}(E)$ is linearly ordered with respect
to $\preceq^{*}$ and $|\mathcal{L}|<2^{\aleph_{0}}$, then there is a $Y$ such
that $Y\preceq^{*}X$ for all $X\in\mathcal{L}$. In particular,
$\mathfrak{t}^{*}=2^{\aleph_{0}}$.

Consequently, the following theorem holds under $\mathsf{CH}$ or
$\mathsf{MA}(\sigma\text{-centered})$.

######  Theorem 5.3.

$(\mathfrak{t}^{*}=2^{\aleph_{0}})$ There exists a strategic $(p^{+})$-filter
in $\mathrm{bb}^{\infty}(E)$.

###### Proof.

Fix enumerations:

  1. (i)

$\\{X_{\xi}:\xi<2^{\aleph_{0}}\\}=\mathrm{bb}^{\infty}(E)$,

  2. (ii)

$\\{(X_{n}^{\xi}):\xi<2^{\aleph_{0}}\\}$ of all $\preceq^{*}$-decreasing
sequences $(X^{\xi}_{n})$ in $\mathrm{bb}^{\infty}(E)$,

  3. (iii)

$\\{D_{\xi}:\xi<2^{\aleph_{0}}\\}$ of all subsets $D_{\xi}$ of $E$,

  4. (iv)

$\\{[\alpha_{\xi},X_{\xi}]_{f}:\xi<2^{\aleph_{0}}\\}$ of all sets
$[\alpha,X]_{f}$ of outcomes of $\alpha$, where $\alpha$ is a strategy for II
in $G_{f}[X]$.

This can be done in (i) and (ii) since
$|\mathrm{bb}^{\infty}(E)|=2^{\aleph_{0}}$, in (iii) since $E$ is countable,
and in (iv) since the strategies $\alpha$ are coded by reals.

Define sets, for $\xi,\gamma<2^{\aleph_{0}}$, with $\langle\cdot,\cdot\rangle$
a bijection $2^{\aleph_{0}}\times 2^{\aleph_{0}}\to 2^{\aleph_{0}}$,

$\displaystyle\mathcal{D}_{\xi}$ $\displaystyle=\\{Y:Y\text{ is a
diagonalization of }(X^{\xi}_{n})\text{ or }\exists n(Y\bot X^{\xi}_{n})\\}$
$\displaystyle\mathcal{F}_{\langle\xi,\gamma\rangle}$
$\displaystyle=\\{Y:\langle Y\rangle\subseteq D_{\xi}\text{ or }\forall
V\preceq X_{\gamma}(\langle V\rangle\subseteq D_{\xi}\Rightarrow V\perp Y)\\}$
$\displaystyle\mathcal{S}_{\xi}$
$\displaystyle=\\{Y:Y\in[\alpha_{\xi},X_{\xi}]_{f}\text{ or }Y\perp
X_{\xi}\\}.$

Note that the first two sets above are $\preceq$-dense in
$\mathrm{bb}^{\infty}(E)$ by Lemma 2.7, and the third is $\preceq$-dense by
Lemma 4.7.

We construct a $\preceq^{*}$-descending chain $(Y_{\eta})$ of length
$2^{\aleph_{0}}$ in $\mathrm{bb}^{\infty}(E)$ by transfinite induction on
$\eta$. For $\eta=0$, pick $Y_{0}$ below conditions in each of
$\mathcal{D}_{0}$, $\mathcal{F}_{0}$, and $\mathcal{S}_{0}$. If we have
already defined $Y_{\beta}$ for all $\beta<\eta$, pick $Y_{\eta}$ below each
$Y_{\beta}$ for $\beta<\eta$ and below conditions in each of
$\mathcal{D}_{\eta}$, $\mathcal{F}_{\eta}$, and $\mathcal{S}_{\eta}$. This is
possible since $t^{*}>\eta$.

Let $\mathcal{F}$ be the filter generated by
$\\{Y_{\eta}:\eta<2^{\aleph_{0}}\\}$ in $\mathrm{bb}^{\infty}(E)$. To see that
$\mathcal{F}$ is a $(p)$-filter, suppose that $(X_{n}^{\xi})$ is a
$\preceq^{*}$-decreasing sequence in $\mathcal{F}$. Let
$Y\in\mathcal{F}\cap\mathcal{D}_{\xi}$. It cannot be the case that $Y\bot
X_{n}^{\xi}$ for any $n$, as $\mathcal{F}$ is a filter, so $Y$ must be a
diagonalization of $(X^{\xi}_{n})$. Similarly, using the sets
$\mathcal{S}_{\xi}$, $\mathcal{F}$ is strategic.

To see that $\mathcal{F}$ is full, suppose $D_{\xi}\subseteq E$ and
$X_{\gamma}\in\mathcal{F}$ are such that $D_{\xi}$ is $\mathcal{F}$-dense
below $X_{\gamma}$. Take
$Z\in\mathcal{F}\cap\mathcal{F}_{\langle\xi,\gamma\rangle}\neq\emptyset$. By
assumption, there is a $Y^{\prime}$ below both $Y$ and $X_{\gamma}$ such that
$\langle Y^{\prime}\rangle\subseteq D_{\xi}$, but obviously it cannot be that
$Y^{\prime}\perp Y$. Thus, it must be that $\langle Y\rangle\subseteq
D_{\xi}$. ∎

The next result allows us to obtain $(p^{+})$-filters generically by forcing
with $(\mathrm{bb}^{\infty}(E),\preceq^{*})$. Since the dense sets involved
are all definable in a simple way from real parameters, they are contained in
$\mathbf{L}(\mathbb{R})$. In particular, this establishes (without any large
cardinals) the $(\Rightarrow)$ direction of Theorem 1.2.

######  Lemma 5.4.

For $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ a $(p^{+})$-family, forcing
with $(\mathcal{H},\preceq^{*})$ adds no new reals and if
$\mathcal{G}\subseteq\mathcal{H}$ is $\mathbf{L}(\mathbb{R})$-generic for
$(\mathcal{H},\preceq^{*})$, then $\mathcal{G}$ will be a $(p^{+})$-filter. If
$\mathcal{H}$ is strategic (has the strong $(p)$-property, respectively), then
$\mathcal{G}$ will also be strategic (have the strong $(p)$-property,
respectively).

###### Proof.

$\mathcal{H}$ being a $(p)$-family implies that $(\mathcal{H},\preceq^{*})$ is
$\sigma$-closed, and thus adds no new reals. We use this fact implicitly in
what follows. Let $\mathcal{G}$ be as described. To see that $\mathcal{G}$ is
full, let $D\subseteq E$ be $\mathcal{G}$-dense below some $X\in\mathcal{G}$.
Translating this into the forcing language, there must be an
$X^{\prime}\in\mathcal{G}$, which we may assume is below $X$, with

$X^{\prime}\mathrel{\Vdash}_{\mathcal{H}}\forall
Y\in\dot{\mathcal{G}}\upharpoonright X\exists Z\preceq Y(\langle
Z\rangle\subseteq\check{D}).$

We claim that the set $\mathcal{D}=\\{Z:\langle Z\rangle\subseteq D\\}$ is
$\preceq$-dense below $X^{\prime}$ in $\mathcal{H}$. If not, then by fullness
of $\mathcal{H}$, $D$ must fail to be $\mathcal{H}$-dense below $X^{\prime}$.
That is, there is some $Y\in\mathcal{H}\upharpoonright X^{\prime}$ with no
$Z\preceq Y$ such that $\langle Z\rangle\subseteq D$. Then, $Y$ fails to force
the statement in the displayed line above, contrary to $Y\preceq X^{\prime}$.
Since $X^{\prime}\in\mathcal{G}$ and $\mathcal{D}$ is $\preceq$-dense below
$X^{\prime}$ in $\mathcal{H}$, $\mathcal{G}\cap\mathcal{D}\neq\emptyset$,
showing that $\mathcal{G}$ is full. The remainder of the proof consists of
observing that the relevant $\preceq$-dense sets in Lemmas 2.7, 4.2, and 4.7
are $\preceq$-dense in $\mathcal{H}$ under these hypotheses. ∎

##  6\. Connections to filters on a countable set

We would like to relate the filters discussed thus far to filters of subsets
of a countable set. In our case, the countable set will be
$E\setminus\\{0\\}$, but we will call these filters on $E$.

######  Definition 6.1.

A filter $\mathcal{F}$ on $E$ is a _block filter_ if it has a base consisting
of sets of the form $\langle X\rangle$, for $X\in\mathrm{bb}^{\infty}(E)$.

It is tempting to define a _block ultrafilter_ on $E$ to be a block filter on
$E$ which is also an ultrafilter. However, unless $|F|=2$, such objects do not
exist: Let $\mathcal{F}$ be a block filter on $E$. For $A_{0},A_{1}\subseteq
E$ given in Example 2.4, note that $E=A_{0}\cup A_{0}$. But, for every
$X\in\mathrm{bb}^{\infty}(E)$, $\langle X\rangle\cap A_{0}\neq\emptyset$ and
$\langle X\rangle\cap A_{1}\neq\emptyset$, so neither set can be in
$\mathcal{F}$.

Let $\mathrm{FIN}$ be the set of nonempty finite subsets of $\mathbb{N}$. An
ultrafilter $\mathcal{U}$ on $\mathrm{FIN}$ is said to be an _ordered union
ultrafilter_ [11] if it has a base consisting of sets of the form $\langle
X\rangle=\\{x_{n_{0}}\cup\cdots\cup x_{n_{k}}:n_{0}<\cdots<n_{k}\\}$, where
$X=(x_{n})$ is a block sequence in $\mathrm{FIN}$ (that is, for all $n$,
$\max(x_{n})<\min(x_{n+1})$). The set of infinite block sequences in
$\mathrm{FIN}$ is denoted by $\mathrm{FIN}^{[\infty]}$. We have, perhaps,
overloaded the notation $\langle X\rangle$, but its intended interpretation
should be clear from context. If $X=(x_{n})\in\mathrm{bb}^{\infty}(E)$, denote
by $\widetilde{X}=(\mathrm{supp}(x_{n}))\in\mathrm{FIN}^{[\infty]}$.

If $|F|=2$, then $E\setminus\\{0\\}$ can be identified with $\mathrm{FIN}$ via
each vector’s support. Sums of vectors in block position corresponds to unions
of their supports. As a consequence of Hindman’s Theorem (Corollary 3.3 in
[20]), one can construct (under hypotheses such as $\mathsf{CH}$ or
$\mathsf{MA}$) ordered union ultrafilters on $\mathrm{FIN}$; these will
correspond to block ultrafilters on $E$.

For the remainder of this section we will consider a general countable field
$F$. The map which takes a vector to its support will provide the connection
between this general setting and $\mathrm{FIN}$.

######  Definition 6.2.

Let $\mathcal{F}$ be a block filter on $E$.

  1. (a)

A subset $D\subseteq E$ is _$\mathcal{F}$ -dense_ if for every $\langle
X\rangle\in\mathcal{F}$, there is a $Z\preceq X$ with $\langle
Z\rangle\subseteq D$.

  2. (b)

$\mathcal{F}$ is _full_ if whenever $D\subseteq E$ is $\mathcal{F}$-dense, we
have that $D\in\mathcal{F}$.

As in the case for filters in $\mathrm{bb}^{\infty}(E)$, every full block
filter on $E$ is maximal with respect to containment amongst block filters.

The map $s:X\mapsto\langle X\rangle$ takes block sequences to subsets of $E$.
It is straightforward to show that the image of a (full) filter in
$\mathrm{bb}^{\infty}(E)$ under $s$ generates a (full) block filter on $E$ and
that the inverse image of a (full) block filter on $E$ is a (full) filter in
$\mathrm{bb}^{\infty}(E)$. By Theorem 5.3 (or Lemma 5.4), it is consistent
that such filters exist.

######  Theorem 6.3.

Suppose that $\mathcal{F}$ is a full block filter on $E$, and let

$\mathrm{supp}(\mathcal{F})=\\{A\subseteq\mathrm{FIN}:\exists
F\in\mathcal{F}(A\supseteq\\{\mathrm{supp}(v):v\in F\\})\\}.$

Then, $\mathrm{supp}(\mathcal{F})$ is an ordered union ultrafilter on
$\mathrm{FIN}$.

###### Proof.

Let $A,B\in\mathrm{supp}(\mathcal{F})$, say with
$A\supseteq\\{\mathrm{supp}(v):v\in F\\}$ and
$B\supseteq\\{\mathrm{supp}(v):v\in G\\}$, for $F,G\in\mathcal{F}$. Then,

$\displaystyle A\cap B$ $\displaystyle\supseteq\\{s:\exists v\in F\exists w\in
G(s=\mathrm{supp}(v)=\mathrm{supp}(w))\\}$
$\displaystyle\supseteq\\{\mathrm{supp}(v):v\in F\cap G\\},$

which is in $\mathrm{supp}(\mathcal{F})$, as $F\cap G\in\mathcal{F}$. Since
$\mathrm{supp}(\mathcal{F})$ is upwards closed by definition, we have that
$\mathrm{supp}(\mathcal{F})$ is a filter on $\mathrm{FIN}$. As $\mathcal{F}$
is a block filter, it follows that $\mathrm{supp}(\mathcal{F})$ has a base
consisting of sets $\langle\widetilde{X}\rangle$ for
$X\in\mathrm{bb}^{\infty}(E)$.

It remains to show that $\mathrm{supp}(\mathcal{F})$ is an ultrafilter. Take
$A\subseteq\mathrm{FIN}$ such that for all $B\in\mathrm{supp}(\mathcal{F})$,
$A\cap B\neq\emptyset$. Let

$\displaystyle D_{0}$ $\displaystyle=\\{v\in E:\mathrm{supp}(v)\in A\\}$
$\displaystyle D_{1}$ $\displaystyle=\\{v\in E:\mathrm{supp}(v)\notin A\\}.$

Towards a contradiction, suppose that for all $\langle
X\rangle\in\mathcal{F}$, there is a $\langle Z\rangle\subseteq\langle
X\rangle$ with $\langle Z\rangle\subseteq D_{1}$. Since $\mathcal{F}$ is full,
there is a $\langle Z\rangle\in\mathcal{F}$ with $\langle Z\rangle\subseteq
D_{1}$. Then $\langle\widetilde{Z}\rangle\in\mathrm{supp}(\mathcal{F})$, but
$A\cap\langle\widetilde{Z}\rangle=\emptyset$, a contradiction.

Thus, there is some $\langle X\rangle\in\mathcal{F}$ such that for no $\langle
Z\rangle\subseteq\langle X\rangle$ is $\langle Z\rangle\subseteq D_{1}$. Take
$\langle Y\rangle\in\mathcal{F}\upharpoonright\langle X\rangle$. By Hindman’s
Theorem applied to $\langle\widetilde{Y}\rangle$, there is a
$\widetilde{Z}\in\mathrm{FIN}^{[\infty]}$ such that
$\langle\widetilde{Z}\rangle\subseteq\langle\widetilde{Y}\rangle$ and either
(i) $\langle\widetilde{Z}\rangle\subseteq A$, or (ii)
$\langle\widetilde{Z}\rangle\subseteq\langle\widetilde{Y}\rangle\setminus A$.

Take any $Z\preceq Y$ in $\mathrm{bb}^{\infty}(E)$ whose supports agree with
$\widetilde{Z}$, then if (ii) holds, $\langle Z\rangle\subseteq D_{1}$,
contrary to what we know about $\langle X\rangle$. Thus,
$\langle\widetilde{Z}\rangle\subseteq A$ and $\langle Z\rangle\subseteq
D_{0}$. Since $\langle Y\rangle\in\mathcal{F}\upharpoonright\langle X\rangle$
was arbitrary, we have that $D_{0}$ is $\mathcal{F}$-dense. As $\mathcal{F}$
is full, we can find a $\langle Z\rangle\in\mathcal{F}$ with $\langle
Z\rangle\subseteq D_{0}$. Then,
$\langle\widetilde{Z}\rangle\in\mathrm{supp}(\mathcal{F})$ and
$\langle\widetilde{Z}\rangle\subseteq A$, so $A\in\mathrm{supp}(\mathcal{F})$.
∎

As a consequence of Theorem 6.3 and the Corollary on p. 87 of [11] we have:

######  Corollary 6.4.

If $\mathcal{F}$ is a full filter on $E$, then

$\displaystyle\min(\mathcal{F})$
$\displaystyle=\\{\\{n=\min(\mathrm{supp}(v)):v\in F\\}:F\in\mathcal{F}\\},$
$\displaystyle\max(\mathcal{F})$
$\displaystyle=\\{\\{n=\max(\mathrm{supp}(v)):v\in F\\}:F\in\mathcal{F}\\}$

are selective ultrafilters on $\mathbb{N}$.

As it is consistent that there are no selective ultrafilters [24], we have:

######  Corollary 6.5.

The existence of full block filters on $E$, and thus full filters in
$\mathrm{bb}^{\infty}(E)$, is independent of $\mathsf{ZFC}$.

An ordered union ultrafilter $\mathcal{U}$ on $\mathrm{FIN}$ is _stable_ [9]
if whenever $(\langle X_{n}\rangle)_{n\in\mathbb{N}}$ is contained in
$\mathcal{U}$, for $X_{n}\in\mathrm{FIN}^{[\infty]}$, there is an $\langle
X\rangle\in\mathcal{U}$ with $\langle X\rangle\subseteq^{*}\langle
X_{n}\rangle$ for all $n$. Much as selective ultrafilters on $\mathbb{N}$
provide local witnesses to Silver’s theorem, selective ultrafilters on
$\mathrm{FIN}$ witness a theorem of Milliken [32] on analytic partitions of
$\mathrm{FIN}^{[\infty]}$. It is easy to see, given Theorem 6.3, that
$(p^{+})$-filters in $\mathrm{bb}^{\infty}(E)$ induce stable ordered union
ultrafilters on $\mathrm{FIN}$. See [12], [31], and [42] for (equivalent)
alternate definitions of “selective ultrafilter” on $\mathrm{FIN}$.

##  7\. Extending to universally Baire sets and $\mathbf{L}(\mathbb{R})$

In this section, we show that under additional set-theoretic hypotheses,
Theorem 1.1 can be extended beyond the analytic sets to obtain Theorems 1.2
and 1.3, provided the families involved are strategic. We begin by noting the
following result:

######  Theorem 7.1 (Rosendal [35]).

($\mathsf{MA}(\aleph_{1})$) A union of $\aleph_{1}$-many strategically Ramsey
sets is strategically Ramsey.

The above theorem, plus existing results in the literature, yields:

######  Theorem 7.2.

Assume that there is a supercompact cardinal.555Throughout this section, the
assumption of supercompactness can be weakened to the existence of a proper
class of Woodin cardinals, see [27]. We use supercompactness due to its
central role in the literature and verbal brevity. Every subset of
$\mathrm{bb}^{\infty}(E)$ in $\mathbf{L}(\mathbb{R})$ is strategically
Ramsey.666Noé de Rancourt has announced a different proof of this result using
methods inspired by determinacy considerations.

###### Proof.

We follow the proof of Theorem 4 in [28]. The existence of a supercompact
cardinal implies that $\mathbf{L}(\mathbb{R})$ is a _Solovay model_ in the
sense of [13], and Lemma 4.4 of the same reference shows that every set of
reals in such a model is a union of $\aleph_{1}$-many analytic sets. By
Theorem 7.1, under $\mathsf{MA}(\aleph_{1})$ a union of $\aleph_{1}$-many
strategically Ramsey sets is again strategically Ramsey. Since
supercompactness implies [37] that $\mathbf{L}(\mathbb{R})^{\mathbf{V}[G]}$ is
elementarily equivalent to $\mathbf{L}(\mathbb{R})$ for any set-forcing
extension $\mathbf{V}[G]$, and one can force $\mathsf{MA}(\aleph_{1})$ in a
way which preserves $\aleph_{1}$, the same is true in
$\mathbf{L}(\mathbb{R})$. As analytic sets are strategically Ramsey by Theorem
1.1, every set in $\mathbf{L}(\mathbb{R})$ is as well. ∎

Following [33], given a notion of forcing $Q$ and a complete metric space
$(X,d)$, we say that a $Q$-name $\dot{x}$ is a _nice $Q$-name for an element
of $\dot{X}$_ if there is a countable collection $\mathcal{D}$ of dense
subsets of $Q$ such that $\dot{x}(G)$ (the interpretation of $\dot{x}$ by $G$)
is an element of $X$ whenever $G$ is a $\mathcal{D}$-generic filter for $Q$.
One can show that if $\dot{y}$ is a $Q$-name and
$p\mathrel{\Vdash}_{Q}\dot{y}\in\dot{X}$, then there is a nice $Q$-name
$\dot{x}$ for an element of $\dot{X}$ such that
$p\mathrel{\Vdash}_{Q}\dot{y}=\dot{x}$.

A subset $A\subseteq X$ is _universally Baire_ if whenever $Q$ is a notion of
forcing, there is a $Q$-name $\dot{A}$ such that for every nice $Q$-name
$\dot{x}$ for an element of $\dot{X}$, there is a countable collection
$\mathcal{D}$ of dense subsets of $Q$ such that

  1. (1)

$\\{q\in Q:q\text{ decides }\dot{x}\in\dot{A}\\}$ is in $\mathcal{D}$,

  2. (2)

whenever $G$ is $\mathcal{D}$-generic for $Q$, $\dot{x}(G)$ is in $X$ and
$\dot{x}(G)$ is in $A$ if and only if there is a $q\in G$ such that
$q\mathrel{\Vdash}\dot{x}\in\dot{A}$.

The following result will be the main tool for going beyond the analytic sets.

######  Theorem 7.3 (Feng–Magidor–Woodin [16]).

Assume that there is a supercompact cardinal. Every set of reals in
$\mathbf{L}(\mathbb{R})$ is universally Baire.

Consider the following variant of the infinite asymptotic game: If $A\subseteq
E$ is an infinite dimensional subspace of $E$, we define $F[A]$ to be the game
in which I plays natural numbers $n_{k}$, which we assume are increasing, and
II plays vectors $y_{k}\in A$ subject to the constraint $n_{k}<y_{k}<y_{k+1}$.
By Lemma 2.1, this is well-defined. One can define _outcome_ , _strategies_ ,
and the game $F[\vec{x},A]$ exactly as in §3. Note that the game
$F[\vec{x},\langle X\rangle]$ in this sense, where
$X\in\mathrm{bb}^{\infty}(E)$, coincides with $F[\vec{x},X]$ from §3, and we
will denote it as such.

Suppose that $\sigma$ is a strategy for I in $F[A]$ and $\tau$ a strategy for
I in $F[B]$, where $B\subseteq A$ are infinite-dimensional subspaces. We write
$\tau\geq\sigma$ if for all $\vec{y}$ in the domain of $\tau$,
$\tau(\vec{y})\geq\sigma(\vec{y})$ ($\sigma(\vec{y})$ is well-defined by
induction). Observe that if $\tau\geq\sigma$, then whenever $(y_{n})$ is an
outcome of $F[B]$ where I follows $\tau$, then it is also an outcome of $F[A]$
where I follows $\sigma$. In particular, if $\sigma$ is a strategy for playing
into a set $\mathbb{A}$, then so is $\tau$.

If $\sigma$ is a strategy for I in $F[A]$ and $B\subseteq A$ as above, then
denote by $\sigma\upharpoonright B$ the restriction of $\sigma$ to the part of
its domain contained in $B$, a strategy for I in $F[B]$. Clearly,
$\sigma\upharpoonright B\geq\sigma$. Let $\varepsilon$ be the strategy in
$F[E]$ where I plays $n$ on the $n$th move. Then, for all $A$ and strategies
$\sigma$ for $I$ in $F[A]$, we have that $\sigma\geq\varepsilon\upharpoonright
A$.

######  Definition 7.4.

Let $\mathbb{P}$ be the set of all triples $(\vec{x},A,\sigma)$, where
$\vec{x}\in\mathrm{bb}^{<\infty}(E)$, $A$ is an infinite-dimensional subspace
of $E$ and $\sigma$ is a strategy for I in $F[\vec{x},A]$. We say that
$(\vec{y},B,\tau)\leq(\vec{x},A,\sigma)$ if

  1. (i)

$\vec{y}=\vec{x}^{\smallfrown}(y_{0},\ldots,y_{k-1})$ where
$y_{0},\ldots,y_{k-1}$ are the first $k$ moves by II in a round of
$F[\vec{x},A]$ where I follows $\sigma$,

  2. (ii)

$B\subseteq A$,

  3. (iii)

$\tau(\cdot)\geq\sigma((y_{0},\ldots,y_{k})^{\smallfrown}\,\cdot\,)$.

The ordering $\leq$ on $\mathbb{P}$ is reflexive and transitive, though fails
to be antisymmetric. We treat $\mathbb{P}$ as a notion of forcing. Note that
$\mathbb{P}$ has a maximal element, namely $(\emptyset,E,\varepsilon)$. If
$X\in\mathrm{bb}^{\infty}(E)$, we write $(\vec{x},X,\sigma)$ for
$(\vec{x},\langle X\rangle,\sigma)$. If
$\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ is a family, let

$\mathbb{P}(\mathcal{H})=\\{(\vec{x},A,\sigma)\in\mathbb{P}:\exists
X\in\mathcal{H}(\langle X\rangle\subseteq A)\\},$

a suborder of $\mathbb{P}$. Note that if
$\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ is a family, then the set of
conditions $(\vec{x},X,\sigma)$ where $X\in\mathcal{H}$ is dense in
$\mathbb{P}(\mathcal{H})$.

For $(\vec{x},A,\sigma)\in\mathbb{P}$, let

$[\vec{x},A,\sigma]=\\{Y\in\mathrm{bb}^{\infty}(E):Y\text{ is an outcome of
$F[\vec{x},A]$ where I follows $\sigma$}\\}.$

We collect some basic properties of $\mathbb{P}$ in the following lemma:

######  Lemma 7.5.

  1. (a)

If $(\vec{y},B,\tau)\leq(\vec{x},A,\sigma)$ in $\mathbb{P}$, then
$[\vec{y},B,\tau]\subseteq[\vec{x},A,\sigma]$. Conversely, if
$[\vec{y},B,\tau]\subseteq[\vec{x},A,\sigma]$, then $(\vec{y},B,\tau)$ is
below $(\vec{x},A,\sigma)$ in the separative quotient of $\mathbb{P}$.

  2. (b)

If $(\vec{x},A,\sigma)\in\mathbb{P}$, then the set $[\vec{x},A,\sigma]$ is
(topologically) closed.

  3. (c)

If $\mathcal{F}\subseteq\mathrm{bb}^{\infty}(E)$ is a filter, then
$\mathbb{P}(\mathcal{F})$ is $\sigma$-centered.

###### Proof.

(a) The first part follows from our observations about the ordering on
strategies for I. For the converse, suppose that
$[\vec{y},B,\tau]\subseteq[\vec{x},A,\sigma]$. Then, every outcome of
$F[\vec{y},B]$ where I follows $\tau$ is an outcome of $F[\vec{x},A]$ where I
follows $\sigma$. In particular,
$\vec{y}=\vec{x}^{\smallfrown}(y_{0},\ldots,y_{k-1})$ where
$y_{0},\ldots,y_{k-1}$ are the first $k$ moves by II in a round of
$F[\vec{x},A]$ where I follows $\sigma$.

We claim that $B/m\subseteq A$, where
$m=\max\\{\mathrm{supp}(\vec{y}),\tau(\emptyset)\\}$ and $B/m=\\{y\in
B:y>m\\}$. To see this, note that for any $y\in B/m$, there is an outcome
$\vec{y}^{\smallfrown}y^{\smallfrown}Z\in[\vec{y},B,\tau]$ and thus in
$[\vec{x},A,\sigma]$. In particular, $y\in A$.

By our choice of $m$, $\tau\upharpoonright B/m=\tau$. So,
$(\vec{y},B/m,\tau)\leq(\vec{x},A,\sigma)$ and the sets of extensions of
$(\vec{y},B/m,\tau)$ and $(\vec{y},B,\tau)$ coincide. Thus, their images in
the separative quotient of $\mathbb{P}$ coincide.

(b) If $Y=(y_{n})\notin[\vec{x},A,\sigma]$, then either
$\vec{x}\not\sqsubseteq Y$, or there is some least $n$ such that $y_{n}$ is
not a valid response to $\sigma(y_{0},\ldots,y_{n-1})$, i.e., $y_{n}\notin A$
or $y_{n}\not>\sigma(y_{0},\ldots,y_{n-1})$. As $E$ is discrete, these are
open conditions.

(c) Suppose that $(\vec{x},A,\sigma)$ and $(\vec{x},B,\tau)$ are both in
$\mathbb{P}(\mathcal{F})$. There are $X,Y\in\mathcal{F}$ with $\langle
X\rangle\subseteq A$ and $\langle Y\rangle\subseteq B$. Since $\mathcal{F}$ is
a filter, there is a $Z\in\mathcal{F}$ below both. Let $\rho$ be the strategy
for I in $F[Z]$ given by
$\rho(\vec{z})=\max\\{\sigma(\vec{z}),\tau(\vec{z})\\}$. Then,
$(\vec{x},Z,\rho)\in\mathbb{P}(\mathcal{F})$ and extends both
$(\vec{x},A,\sigma)$ and $(\vec{x},B,\tau)$. Since there are only countably
many such $\vec{x}$, this shows that $\mathbb{P}(\mathcal{F})$ is
$\sigma$-centered. ∎

Given a family $\mathcal{H}\subseteq\mathrm{bb}^{\infty}(E)$ and a
sufficiently generic filter $G$ for $\mathbb{P}(\mathcal{H})$, we denote by
$X_{\text{gen}}(G)$ the _generic block sequence_ determined by $G$,

$\displaystyle X_{\text{gen}}(G)$
$\displaystyle=\bigcup\\{\vec{x}:\exists(\vec{x},A,\sigma)\in G\\}.$

In what follows, $G$ will be $\mathcal{D}$-generic for some countable
collection of dense sets $\mathcal{D}$ coming from the definition of
universally Baire, and so $G$ can be taken to be in $\mathbf{V}$. Any such
$\mathcal{D}$ will ensure that $X_{\text{gen}}(G)$ is infinite. We write
$\dot{X}_{\text{gen}}$ to be a nice (as defined above)
$\mathbb{P}(\mathcal{H})$-name for this block sequence.

######  Lemma 7.6.

Let $\mathcal{F}\subseteq\mathrm{bb}^{\infty}(E)$ be a filter, $\mathcal{D}$ a
collection of dense subsets of $\mathbb{P}(\mathcal{F})$, and $G$ a
$\mathcal{D}$-generic filter for $\mathbb{P}(\mathcal{F})$. For
$X=X_{\text{gen}}(G)$, the set

$G(X)=\\{(\vec{x},A,\sigma)\in\mathbb{P}(\mathcal{F}):X\in[\vec{x},A,\sigma]\\}$

is a $\mathcal{D}$-generic filter for $\mathbb{P}(\mathcal{F})$ which contains
$G$ and $X_{\text{gen}}(G(X))=X$.

###### Proof.

By Lemma 7.5(a), $G(X)$ is closed upwards. If $(\vec{x},A,\sigma)\in G$, then
one can build a decreasing sequence $(\vec{x}_{n},A_{n},\sigma_{n})$ in $G$
with $(\vec{x}_{0},A_{0},\sigma_{0})=(\vec{x},A,\sigma)$,
$|\vec{x}_{n}|\to\infty$ as $n\to\infty$, and $X$ the union of the
$\vec{x}_{n}$. By construction, $X$ must be in $[\vec{x},A,\sigma]$. This
shows that $G\subseteq G(X)$, and consequently the latter is
$\mathcal{D}$-generic. It remains to show that $G(X)$ is a filter. Take
$(\vec{x},A,\sigma),(\vec{y},B,\tau)\in G(X)$. As $X$ has both $\vec{x}$ and
$\vec{y}$ as an initial segment, one must be an initial segment of the other,
say $\vec{x}\sqsubseteq\vec{y}$, and the part of $\vec{y}$ above $\vec{x}$ is
a sequence of moves by II against $\sigma$. As $\mathcal{F}$ is a filter,
$A\cap B$ is infinite-dimensional. Let $\rho$ be the strategy for I in
$F[A\cap B]$ given by $\rho(\vec{v})=\max\\{\sigma(\vec{v}),\tau(\vec{v})\\}$,
for $\vec{v}$ in its domain. Then, $(\vec{y},A\cap B,\rho)$ is below both
$(\vec{x},A,\sigma)$ and $(\vec{y},B,\tau)$. Moreover, $X\in[\vec{y},A\cap
B,\rho]$, and so $(\vec{y},A\cap B,\rho)\in G(X)$. That
$X_{\text{gen}}(G(X))=X$ is clear. ∎

A consequence of Lemma 7.6 is that if $G$ is generic for
$\mathbb{P}(\mathcal{F})$ over a model of a sufficient fragment of
$\mathsf{ZFC}$, then $G(X)=G$, though we will not make use of this here.

######  Lemma 7.7.

Let $\mathcal{F}\subseteq\mathrm{bb}^{\infty}(E)$ be a filter and
$\mathcal{D}$ a countable collection of dense open subsets of
$\mathbb{P}(\mathcal{F})$.

  1. (a)

For any $(\vec{x},A,\sigma)\in\mathbb{P}(\mathcal{F})$, the set

$\mathbb{G}_{\mathcal{D},(\vec{x},A,\sigma)}=\\{X_{\text{gen}}(G):G\text{ a
$\mathcal{D}$-generic filter for $\mathbb{P}(\mathcal{F})$ with
$(\vec{x},A,\sigma)\in G$}\\}$

is an $F_{\sigma\delta}$ subset of $\mathrm{bb}^{\infty}(E)$.

  2. (b)

If $X\in\mathcal{F}$, then for no $Y\in\mathcal{F}\upharpoonright X$ does I
have a strategy in $F[\vec{x},Y]$ for playing into
$(\mathbb{G}_{\mathcal{D},(\vec{x},X,\sigma)})^{c}$.

  3. (c)

If $\mathcal{F}$ is a $(p^{+})$-filter and $X\in\mathcal{F}$, then there is a
$Y\in\mathcal{F}\upharpoonright X$ for which II has a strategy in
$G[\vec{x},Y]$ for playing into $\mathbb{G}_{\mathcal{D},(\vec{x},X,\sigma)}$.

###### Proof.

(a) Enumerate $\mathcal{D}=\\{D_{n}:n\in\mathbb{N}\\}$. Since
$\mathbb{P}(\mathcal{F})$ is ccc by Lemma 7.5(c), each $D_{n}$ contains a
countable maximal antichain $A_{n}$ below $(\vec{x},A,\sigma)$. We claim that

$\mathbb{G}_{\mathcal{D},(\vec{x},A,\sigma)}=\bigcap_{n\in\mathbb{N}}\bigcup\\{[\vec{y},B,\tau]:(\vec{y},B,\tau)\in
A_{n}\\},$

which is $F_{\sigma\delta}$, as each set $[\vec{y},Y,\tau]$ is closed by Lemma
7.5(b).

If $X=X_{\text{gen}}(G)$ where $G$ is a $\mathcal{D}$-generic filter with
$(\vec{x},A,\sigma)\in G$, then for each $n$, $G\cap A_{n}\neq\emptyset$, say
with $(\vec{y}_{n},B_{n},\tau_{n})\in G\cap A_{n}$. By Lemma 7.6, for each
$n$, $X\in[\vec{y}_{n},B_{n},\tau_{n}]$, and so $X$ is in the set on right
hand side of the above displayed line. For the reverse inclusion, suppose that
$X$ is in set on the right hand side. Then, by Lemma 7.6, $G(X)$ is a
$\mathcal{D}$-generic filter containing $(\vec{x},A,\sigma)$ for which
$X_{\text{gen}}(G(X))=X$, and so
$X\in\mathbb{G}_{\mathcal{D},(\vec{x},A,\sigma)}$.

(b) Let $X\in\mathcal{F}$ and $Y\in\mathcal{F}\upharpoonright X$ be given.
Towards a contradiction, suppose that $\rho$ is a strategy for I in
$F[\vec{x},Y]$ for playing into
$(\mathbb{G}_{\mathcal{D},(\vec{x},X,\sigma)})^{c}$. We may assume
$\rho\geq\sigma\upharpoonright Y$. Consider the following play of
$F[\vec{x},Y]$: I plays $\rho(\emptyset)=n_{0}$. Pick

$p_{0}=(\vec{x}^{\smallfrown}(y_{0}^{0},\ldots,y_{k_{0}}^{0}),B^{0},\rho^{0})\leq(\vec{x},Y,\rho)\leq(\vec{x},X,\sigma)$

in $D_{0}$ and let II play $y_{0}=y^{0}_{0}$. Note that this is a valid move
by definition of $\leq$ in $\mathbb{P}(\mathcal{F})$. Next, I plays
$\rho(y_{0})=n_{1}$. Pick

$p_{1}=(\vec{x}^{\smallfrown}(y_{0}^{0},\ldots,y_{k_{0}}^{0})^{\smallfrown}(y_{0}^{1},\ldots,y_{k_{1}}^{1}),B^{1},\rho^{1})\leq(\vec{x}^{\smallfrown}(y_{0}^{0},\ldots,y_{k_{0}}^{0}),B^{0},\rho^{0})$

in $D_{1}$ and let II play $y_{1}=y^{0}_{1}$ if $k_{0}\geq 1$, and
$y_{1}=y^{1}_{0}$ otherwise. Continuing in this fashion, we build an outcome
$(y_{n})$. Observe that $(y_{n})$ must be in
$\mathbb{G}_{\mathcal{D},(\vec{x},X,\sigma)}$: the conditions $p_{n}$ picked
in $D_{n}$ above form a $\mathcal{D}$-generic chain in
$\mathbb{P}(\mathcal{F})$ below $(\vec{x},X,\sigma)$, thus generate a
$\mathcal{D}$-generic filter $G$ with $X_{\text{gen}}(G)=(y_{n})$ and
$(\vec{x},X,\sigma)\in G$. This contradicts our choice of $\rho$.

(c) follows from (a) and (b) by an application of Theorem 1.1. ∎

######  Lemma 7.8.

Let $\mathcal{F}\subseteq\mathrm{bb}^{\infty}(E)$ be a $(p^{+})$-filter. If
$\mathbb{A}\subseteq\mathrm{bb}^{\infty}(E)$ is universally Baire, then for
any $\vec{x}\in\mathrm{bb}^{<\infty}(E)$ and $X\in\mathcal{F}$, there is a
$Y\in\mathcal{F}\upharpoonright X$ such that II has a strategy in
$G[\vec{x},Y]$ for playing into one of $\mathbb{A}$ or $\mathbb{A}^{c}$.

###### Proof.

Let $X\in\mathcal{F}$ be given. We may assume that $\vec{x}$ is $\emptyset$.
Recall, for $\vec{y}\in\mathrm{bb}^{<\infty}(E)$ and $Y\in\mathcal{F}$,
Definition 3.2 of $(\vec{y},Y)$ being good/bad/worse (for the set
$\mathbb{A}$). By Lemma 3.3, there is a $Y\in\mathcal{F}\upharpoonright X$
such that either $(\emptyset,Y)$ is good or I has a strategy $\sigma$ in
$F[Y]$ to play into the set

$\\{(z_{n}):\forall n(z_{0},\ldots,z_{n},Y)\text{ is worse}\\}.$

In the former case we’re done, so we assume the latter.

Since $\mathbb{A}$ is universally Baire, we may let $\dot{\mathbb{A}}$ be a
$\mathbb{P}(\mathcal{F})$-name for $\mathbb{A}$ and $\mathcal{D}$ countable
collection of dense open subsets of $\mathbb{P}(\mathcal{F})$ such that

  1. (i)

$\\{q\in\mathbb{P}(\mathcal{F}):q\text{ decides
}\dot{X}_{\text{gen}}\in\dot{\mathbb{A}}\\}$ is in $\mathcal{D}$, and

  2. (ii)

whenever $G$ is $\mathcal{D}$-generic in $\mathbb{P}(\mathcal{F})$,
$X_{\text{gen}}(G)$ is in $\mathrm{bb}^{\infty}(E)$ and $X_{\text{gen}}(G)$ is
in $\mathbb{A}$ if and only if there is a $q\in G$ such that
$q\mathrel{\Vdash}_{\mathbb{P}(\mathcal{F})}\dot{X}_{\text{gen}}\in\dot{\mathbb{A}}$.

Thus, if $G$ is $\mathcal{D}$-generic for $\mathbb{P}(\mathcal{F})$, contains
$(\emptyset,Y,\sigma)$, and
$(\emptyset,Y,\sigma)\mathrel{\Vdash}_{\mathbb{P}(\mathcal{F})}\dot{X}_{\text{gen}}\notin\dot{\mathbb{A}}$,
then $X_{\text{gen}}(G)\notin\mathbb{A}$. We claim that
$(\emptyset,Y,\sigma)\mathrel{\Vdash}_{\mathbb{P}(\mathcal{F})}\dot{X}_{\text{gen}}\notin\dot{\mathbb{A}}$.

Suppose not, then there is a $(\vec{y},Z,\tau)\leq(\emptyset,Y,\sigma)$, with
$Z\in\mathcal{F}$, such that
$(\vec{y},Z,\tau)\mathrel{\Vdash}_{\mathbb{P}(\mathcal{F})}\dot{X}_{\text{gen}}\in\dot{\mathbb{A}}$.
Applying Lemma 7.7(c), take $W\in\mathcal{F}\upharpoonright Z$ such that II
has a strategy $\alpha$ in $G[\vec{y},W]$ for playing into
$\mathbb{G}_{\mathcal{D},(\vec{y},Z,\tau)}$. We claim that
$\mathbb{G}_{\mathcal{D},(\vec{y},Z,\tau)}\subseteq\mathbb{A}$. Let $(z_{n})$
be in $\mathbb{G}_{\mathcal{D},(\vec{y},Z,\tau)}$. Take $G$ a
$\mathcal{D}$-generic filter for which $(z_{n})=X_{\text{gen}}(G)$ and
$(\vec{y},Z,\tau)\in G$. Since
$(\vec{y},Z,\tau)\mathrel{\Vdash}_{\mathbb{P}(\mathcal{F})}\dot{X}_{\text{gen}}\in\dot{\mathbb{A}}$,
we have that $(z_{n})\in\mathbb{A}$. Thus, $\alpha$ is a strategy for II in
$G[\vec{y},W]$ for playing into $\mathbb{A}$. This, however, contradicts the
fact that $\sigma$ ensures that $(\vec{y},Z)$ is bad.

Thus,
$(\emptyset,Y,\sigma)\mathrel{\Vdash}_{\mathbb{P}(\mathcal{F})}\dot{X}_{\text{gen}}\notin\dot{\mathbb{A}}$.
Then, exactly as in the preceding paragraph, we may find
$W\in\mathcal{F}\upharpoonright Y$ such that II has a strategy in $G[W]$ for
playing into $\mathbb{G}_{\mathcal{D},(\emptyset,Y,\sigma)}$, and thus into
$\mathbb{A}^{c}$. ∎

While the symmetric result in Lemma 7.8 is appealing on its own, and applies
to all analytic sets (being universally Baire [16]) in $\mathsf{ZFC}$, it is
not a true “dichotomy” as II can easily have strategies for playing into both
$\mathbb{A}$ and $\mathbb{A}^{c}$.

One consequence of Lemma 7.7 and the proof of Lemma 7.8 is that, given
$(p^{+})$-filter $\mathcal{F}$ and a universally Baire set
$\mathbb{A}\subseteq\mathrm{bb}^{\infty}(E)$, there is always an
$X\in\mathcal{F}$ such that one of $\mathbb{A}$ or $\mathbb{A}^{c}$ contains
an $F_{\sigma\delta}$ set $\preceq$-dense below $X$.

We can now complete the proofs of Theorems 1.2 and 1.3.

###### Proof of Theorem 1.2..

We have already proven the $(\Rightarrow)$ direction in Lemma 5.4. For the
remaining direction, let $\mathbb{D}\subseteq\mathrm{bb}^{\infty}(E)$ be a
$\preceq$-dense open set which is in $\mathbf{L}(\mathbb{R})$, and thus
universally Baire by Theorem 7.3. By Lemma 7.8, there is an $X\in\mathcal{F}$
such that II has a strategy in $G[X]$ for playing into either $\mathbb{D}$ or
$\mathbb{D}^{c}$. By Lemma 4.4, the latter can never occur. Thus, II has a
strategy in $G[X]$ for playing into $\mathbb{D}$. Since $\mathcal{F}$ is
strategic, there is a play by this strategy, say $Z$, with
$Z\in\mathbb{D}\cap\mathcal{F}\neq\emptyset$. ∎

######  Lemma 7.9.

Assume that there is a supercompact cardinal. Let
$\mathcal{F}\subseteq\mathrm{bb}^{\infty}(E)$ be a strategic $(p^{+})$-filter.
Every subset of $\mathrm{bb}^{\infty}(E)$ in $\mathbf{L}(\mathbb{R})$ is
$\mathcal{F}$-strategically Ramsey.

###### Proof.

Let $\mathbb{A}\subseteq\mathrm{bb}^{\infty}(E)$ be in
$\mathbf{L}(\mathbb{R})$, and fix $\vec{x}\in\mathrm{bb}^{<\infty}(E)$ and
$X\in\mathcal{F}$. By Theorem 7.2, the set of all $Y\preceq X$ witnessing that
$\mathbb{A}$ is strategically Ramsey is $\preceq$-dense below $X$, and is
clearly in $\mathbf{L}(\mathbb{R})$. Since $\mathcal{F}$ is
$\mathbf{L}(\mathbb{R})$-generic, $\mathcal{F}$ must contain such a $Y$. ∎

###### Proof of Theorem 1.3..

Let $\mathbb{A}\subseteq\mathrm{bb}^{\infty}(E)$ be in
$\mathbf{L}(\mathbb{R})$, and fix $\vec{x}\in\mathrm{bb}^{<\infty}(E)$ and
$X\in\mathcal{H}$. Let $\mathcal{G}$ be $\mathbf{V}$-generic for
$(\mathcal{H},\preceq^{*})$ and contain $X$. By Lemma 5.4, $\mathcal{G}$ is a
strategic $(p^{+})$-filter in $\mathbf{V}[\mathcal{G}]$. By Lemma 7.9, there
is a $Y\in\mathcal{G}\upharpoonright X$ witnessing that $\mathbb{A}$ is
strategically Ramsey in $\mathbf{V}[\mathcal{G}]$. Since forcing with
$(\mathcal{H},\preceq^{*})$ adds no new reals, $Y$ witnesses that $\mathbb{A}$
is $\mathcal{H}$-strategically Ramsey in $\mathbf{V}$. ∎

##  8\. Normed spaces and a local Gowers dichotomy

We now consider the case when $E$ is a countably infinite-dimensional normed
vector space, with _normalized_ basis $(e_{n})$ (that is, $\|e_{n}\|=1$ for
all $n$), over a countable subfield $F$ of $\mathbb{C}$ so that the norm takes
values in $F$. If $V$ is a subspace of $E$, let $S(V)=\\{x\in V:\|x\|=1\\}$.

Let $\mathrm{bb}^{\infty}_{1}(E)=\\{(x_{n})\in\mathrm{bb}^{\infty}(E):\forall
n(\|x_{n}\|=1)\\}$ and
$\mathrm{bb}^{<\infty}_{1}(E)=\\{\vec{x}\in\mathrm{bb}^{<\infty}(E):\forall
n<|\vec{x}|(\|x_{n}\|=1)\\}$. For $X\in\mathrm{bb}^{\infty}(E)$, let
$[X]=\\{Y\in\mathrm{bb}^{\infty}_{1}(E):Y\preceq X\\}$. Taking $E$ discrete,
$\mathrm{bb}^{\infty}_{1}(E)$ is a closed subset of the Polish space
$\mathrm{bb}^{\infty}(E)$, thus itself Polish.

For $X=(x_{n}),Y=(y_{n})\in\mathrm{bb}^{\infty}_{1}(E)$ and
$\Delta=(\delta_{n})$ a sequence of positive real numbers, written $\Delta>0$,
we write $d(X,Y)\leq\Delta$ if for all $n$, $\|x_{n}-y_{n}\|\leq\delta_{n}$.
Given $\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ and $\Delta>0$, let

$\mathbb{A}_{\Delta}=\\{Y\in\mathrm{bb}^{\infty}_{1}(E):\exists
X\in\mathbb{A}(d(X,Y)\leq\Delta)\\},$

the _$\Delta$ -expansion_ of $\mathbb{A}$. We collect a few useful properties
of $\Delta$-expansions in a lemma which will be used tacitly in what follows.
The proof is left to the reader.

######  Lemma 8.1.

Let $\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ and $\Delta>0$.

  1. (a)

If $\mathbb{A}=\bigcup_{i\in I}\mathbb{A}_{i}$, then
$\mathbb{A}_{\Delta}=\bigcup_{i\in I}(\mathbb{A}_{i})_{\Delta}$.

  2. (b)

If $\mathbb{A}$ is analytic, then so is $\mathbb{A}_{\Delta}$.

  3. (c)

$(\mathbb{A}_{\Delta})^{c}\subseteq((\mathbb{A}_{\Delta})^{c})_{\Delta}\subseteq\mathbb{A}^{c}$.

  4. (d)

If $0<\Gamma\leq\Delta/2$, then
$((\mathbb{A}_{\Delta})^{c})_{\Gamma}\subseteq(\mathbb{A}_{\Gamma})^{c}$.

∎

The notions of _family_ , _filter_ , _fullness_ , _$(p)$ -property_, etc, in
$\mathrm{bb}^{\infty}_{1}(E)$, are defined exactly as for
$\mathrm{bb}^{\infty}(E)$ in §2. Moreover, all of the results established in
the previous sections could have been carried out in
$\mathrm{bb}^{\infty}_{1}(E)$ in the event that $E$ is normed. The only
necessary modification is that in the games $G[\vec{x},X]$ and $F[\vec{x},X]$,
the two players must play normalized block sequences and vectors,
respectively. This will be assumed in what follows.

For $D\subseteq S(E)$, let

$D_{\epsilon}=\\{x\in S(E):\exists y\in D(\|x-y\|\leq\epsilon)\\}.$

We weaken the notion of fullness to the following approximate version.777While
this hampers our ability to reuse results of §3 and §7, we hope that it will
enable further applications. An elementary proof of Proposition 8.21, without
the hypothesis of being “strategic”, would greatly simplify the situation in
the cases of interest.

######  Definition 8.2.

A family $\mathcal{H}\subseteq\mathrm{bb}_{1}^{\infty}(E)$ is _almost full_ if
whenever $D\subseteq S(E)$ and $X\in\mathcal{H}$ are such that $D$ is
$\mathcal{H}$-dense below $X$ (that is, for all
$Y\in\mathcal{H}\upharpoonright X$, there is a $Z\preceq Y$ with $S(\langle
Z\rangle)\subseteq D$), then for any $\epsilon>0$, there is a
$Z\in\mathcal{H}\upharpoonright X$ with $S(\langle Z\rangle)\subseteq
D_{\epsilon}$.

######  Definition 8.3.

If a family has the $(p)$-property and is almost full we call it a _$(p^{*})$
-family_. Likewise for _$(p^{*})$ -filter_, _strategic $(p^{*})$-family_, etc.

The following is a discrete version of Gowers weakly Ramsey property [19],
relativized to a family $\mathcal{H}$.

######  Definition 8.4.

Given a family $\mathcal{H}\subseteq\mathrm{bb}^{\infty}_{1}(E)$, a set
$\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ is _$\mathcal{H}$ -weakly
Ramsey_ if for every $\Delta>0$ and $X\in\mathcal{H}$, there is a
$Y\in\mathcal{H}\upharpoonright X$ such that either

  1. (i)

$[Y]\subseteq\mathbb{A}^{c}$, or

  2. (ii)

II has a strategy in $G[Y]$ for playing into $\mathbb{A}_{\Delta}$.

The first goal of this section is to show that for certain $(p^{*})$-families
$\mathcal{H}$, analytic sets in $\mathrm{bb}_{1}^{\infty}(E)$ are
$\mathcal{H}$-weakly Ramsey. We begin with variants of Lemmas 3.3 and 3.4, and
Theorem 1.1, for $(p^{*})$-families. Since dealing with both families and
$\Delta$-expansions requires some care, we include proofs of these results. As
in §3, they are very similar to those in [35].

######  Definition 8.5.

Given a family $\mathcal{H}\subseteq\mathrm{bb}^{\infty}_{1}(E)$,
$\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(H)$ and $\Delta>0$, for
$\vec{y}\in\mathrm{bb}_{1}^{<\infty}(E)$ and $Y\in\mathcal{H}$, we say the
pair $(\vec{y},Y)$ is _$\Delta$ -good_/_$\Delta$ -bad_/_$\Delta$ -worse_ if it
is good/bad/worse for the set $\mathbb{A}_{\Delta}$ (in the sense of
Definition 3.2). Further:

  1. (1)

$(\vec{y},Y)$ is _$\Delta^{*}$ -good_ if it is $\Delta(|\vec{y}|)$-good,

  2. (2)

$(\vec{y},Y)$ is _$\Delta^{*}$ -bad_ if it is $\Delta(|\vec{y}|)$-bad,

  3. (3)

$(\vec{y},Y)$ is _$\Delta^{*}$ -worse_ if it is $\Delta^{*}$-bad and there is
a $n$ such that for all $v\in S(\langle Y/n\rangle)$,
$(\vec{y}^{\smallfrown}v,Y)$ is $\Delta^{*}$-bad.

Here,
$\Delta(m)=(\delta_{0}/2,\delta_{1}/2,\ldots,\delta_{m-1}/2,\delta_{m},\delta_{m+1},\ldots)$.

Note that $\Delta^{*}$-good implies $\Delta$-good and $\Delta^{*}$-bad implies
$\Delta/2$-bad.

######  Lemma 8.6.

If $\mathcal{H}$ is a $(p^{*})$-family and
$\mathbb{A}\subseteq\mathrm{bb}^{\infty}(E)$, then for every
$\vec{x}\in\mathrm{bb}^{<\infty}(E)$, $X\in\mathcal{H}$ and $\Delta>0$, there
is a $Y\in\mathcal{H}\upharpoonright X$ such that either

  1. (i)

$(\vec{x},Y)$ is $\Delta$-good, or

  2. (ii)

I has a strategy in $F[\vec{x},Y]$ for playing into

$\\{(z_{n}):\forall n(\vec{x}^{\smallfrown}(z_{0},\ldots,z_{n}),Y)\text{ is
$\Delta/2$-bad}\\}.$

###### Proof.

Let $\mathcal{H}$, $\mathbb{A}$, $X\in\mathcal{H}$ and $\Delta>0$ be given. As
in the proof of Lemma 3.3, for any $\vec{y}$ and $\Gamma>0$, the set

$\mathcal{D}_{\vec{y}}^{\Gamma}=\\{Y:(\vec{y},Y)\text{ is $\Gamma$-good or
$\Gamma$-bad}\\}$

is $\preceq$-dense open in $\mathcal{H}$, and if $(\vec{y},Y)$ is
$\Gamma$-bad, then for every $V\in\mathcal{H}\upharpoonright Y$, there is a
$Z\preceq V$ such that for all $x\in S(\langle Z\rangle)$,
$(\vec{y}^{\smallfrown}x,Y)$ is not $\Gamma$-good.

######  Claim.

For any $\vec{y}\in\mathrm{bb}^{<\infty}_{1}(E)$, the set

$\mathcal{E}_{\vec{y}}=\\{Y:(\vec{y},Y)\text{ is $\Delta^{*}$-good or
$\Delta^{*}$-worse}\\}$

is $\preceq$-dense open in $\mathcal{H}$.

###### Proof of claim..

Let $Y\in\mathcal{H}$. By diagonalizing over the sets
$\mathcal{D}_{\vec{z}}^{\Delta(|\vec{z}|)}$, we may assume that for all
$\vec{z}$, $(\vec{z},Y)$ is $\Delta^{*}$-good or $\Delta^{*}$-bad. Assume that
$(\vec{y},Y)$ is $\Delta^{*}$-bad. Let

$D=\\{x\in S(E):(\vec{y}^{\smallfrown}x,Y)\text{ is not
$\Delta(|\vec{y}|)$-good}\\}.$

Take $\epsilon=\delta_{|\vec{y}|}/2$. By almost fullness, there is a
$Z\in\mathcal{H}\upharpoonright Y$ such that $S(\langle Z\rangle)\subseteq
D_{\epsilon}$. Given $z\in S(\langle Z\rangle)$, pick $z^{\prime}\in D$ with
$\|z-z^{\prime}\|<\epsilon$. If $(\vec{y}^{\smallfrown}z,Z)$ is
$\Delta^{*}$-good, then there is a strategy $\alpha$ for II in
$G[\vec{y}^{\smallfrown}z,Z]$ for playing into
$\mathbb{A}_{\Delta(|\vec{y}|+1)}$. We may assume that all plays according to
$\alpha$ are above $z$ and $z^{\prime}$, so we can treat $\alpha$ as a
strategy $\alpha^{\prime}$ for II in $G[\vec{y}^{\smallfrown}z^{\prime},Z]$.
If $\vec{y}^{\smallfrown}{z^{\prime}}^{\smallfrown}W$ is an outcome of
$\alpha^{\prime}$, then $\vec{y}^{\smallfrown}z^{\smallfrown}W$ is an outcome
of $\alpha$, and thus in $\mathbb{A}_{\Delta(|\vec{y}|+1)}$. By our choice of
$\epsilon$, it follows that $\vec{y}^{\smallfrown}{z^{\prime}}^{\smallfrown}W$
is in $\mathbb{A}_{\Delta(|\vec{y}|)}$. Then,
$(\vec{y}^{\smallfrown}z^{\prime},Z)$ is $\Delta(|\vec{y}|)$-good,
contradicting that $z^{\prime}\in D$. Thus, $(\vec{y}^{\smallfrown}z,Z)$ is
$\Delta^{*}$-bad, and $(\vec{y},Z)$ is $\Delta^{*}$-worse. ∎

Returning to the proof of the lemma, assume that $\vec{x}=\emptyset$. By the
claim, we can find $Y\in\mathcal{H}\upharpoonright X$ such that for all
$\vec{y}$, $(\vec{y},Y)$ is either $\Delta^{*}$-good or $\Delta^{*}$-worse. If
$(\emptyset,Y)$ is $\Delta^{*}$-good, we’re done, so assume that it is
$\Delta^{*}$-worse. In this case, we define a strategy for I in $F[Y]$ for
playing into $\\{(z_{n}):\forall n(z_{0},\ldots,z_{n},Y)\text{ is
$\Delta^{*}$-worse}\\}$ exactly as in the proof of Lemma 3.3. ∎

######  Lemma 8.7 (cf. Lemma 2 in [35]).

Let $\mathcal{H}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ be a $(p^{*})$-family.
Given $\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ open,
$x\in\mathrm{bb}^{<\infty}_{1}(E)$, $X\in\mathcal{H}$, and $\Delta>0$, there
is a $Y\in\mathcal{H}\upharpoonright Y$ such that either

  1. (i)

I has a strategy in $F[\vec{x},Y]$ for playing into
$(\mathbb{A}_{\Delta/2})^{c}$, or

  2. (ii)

II has a strategy in $G[\vec{x},Y]$ for playing into $\mathbb{A}_{\Delta}$.

###### Proof.

The proof is similar to Lemma 3.4, using Lemma 8.6. ∎

######  Lemma 8.8 (cf. Lemma 4 in [35]).

Let $\mathcal{H}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ be a $(p^{*})$-family.
Suppose that $\mathbb{A}=\bigcup_{n\in\mathbb{N}}\mathbb{A}_{n}$, each
$\mathbb{A}_{n}\subseteq\mathrm{bb}^{\infty}_{1}(E)$. Let $\vec{x}$,
$X\in\mathcal{H}$, and $\Delta>0$ be given. Then, there is a
$Y\in\mathcal{H}\upharpoonright X$ such that either

  1. (i)

I has a strategy in $F[\vec{x},Y]$ for playing into
$(\mathbb{A}_{\Delta/2})^{c}$, or

  2. (ii)

II has a strategy in $G[Y]$ for playing into

$\displaystyle\\{(z_{k}):\exists n\forall V\in\mathcal{H}\upharpoonright Y($ I
has no strategy in $F[\vec{x}^{\smallfrown}(z_{0},\ldots,z_{n}),V]$
$\displaystyle\text{for playing into $((\mathbb{A}_{n})_{\Delta})^{c}$})\\}.$

###### Proof.

For $Y\in\mathcal{H}$, $\vec{y}\in\mathrm{bb}^{<\infty}(E)$, and
$n\in\mathbb{N}$, we say $(\vec{y},n)$ _$\Gamma$ -accepts_ $Y$ if I has a
strategy in $F[\vec{y},Y]$ for playing into $((\mathbb{A}_{n})_{\Gamma})^{c}$
and $(\vec{y},n)$ _$\Gamma$ -rejects_ $Y$ if for all
$Z\in\mathcal{H}\upharpoonright Y$, $(\vec{y},n)$ does not $\Gamma$-accept
$Z$. Both acceptance and rejection are $\preceq^{*}$-hereditary in
$\mathcal{H}$, and the sets

$\mathcal{D}^{\Gamma}_{\vec{y},n}=\\{Y:(\vec{y},n)\text{ $\Gamma$-accepts or
$\Gamma$-rejects }Y\\}$

are clearly $\preceq$-dense open in $\mathcal{H}$. By the $(p)$-property, we
can find $Y\in\mathcal{H}\upharpoonright X$ such that for all $\vec{y}$ and
$n$, $(\vec{y},n)$ either $\Delta/2$-accepts or $\Delta/2$-rejects $Y$. Put

$R=\\{(z_{k}):\exists n(\vec{x}^{\smallfrown}(z_{0},\ldots,z_{n}),n)\text{
$\Delta/2$-rejects }Y\\},$

and notice that $R$ is open in $\mathrm{bb}^{\infty}_{1}(E)$. By Lemma 8.7,
there is $Y^{\prime}\in\mathcal{H}\upharpoonright Y$ such that either II has a
strategy in $G[Y^{\prime}]$ for playing into $R_{\Delta/2}$, or I has a
strategy in $F[Y^{\prime}]$ for playing into $(R_{\Delta/4})^{c}\subseteq
R^{c}$. In the first case, suppose that $(z_{k})$ is an outcome of II’s
strategy. Then, there is $(z_{k}^{\prime})$ with
$\|z_{k}-z_{k}^{\prime}\|\leq\delta_{k}/2$ for all $k$, and an $n$ such that
$(\vec{x}^{\smallfrown}(z_{0}^{\prime},\ldots,z_{n}^{\prime}),n)$
$\Delta/2$-rejects $Y$. We claim
$(\vec{x}^{\smallfrown}(z_{0},\ldots,z_{n}),n)$ $\Delta$-rejects $Y$. If not,
then for some $Z\in\mathcal{H}\upharpoonright Y$, I has a strategy in
$F[\vec{x}^{\smallfrown}(z_{0},\ldots,z_{n}),Z]$ for playing into
$((\mathbb{A}_{n})_{\Delta})^{c}$. This yields a strategy for I in
$F[\vec{x}^{\smallfrown}(z_{0}^{\prime},\ldots,z_{n}^{\prime}),Z]$ for playing
into $(((\mathbb{A}_{n})_{\Delta})^{c})_{\Delta/2}$. By Lemma 8.1(d),
$(((\mathbb{A}_{n})_{\Delta})^{c})_{\Delta/2}\subseteq((\mathbb{A}_{n})_{\Delta/2})^{c}$,
and so $(\vec{x}^{\smallfrown}(z_{0}^{\prime},\ldots,z_{n}^{\prime}),n)$ fails
to $\Delta/2$-reject $Y$, a contradiction. Thus, $(z_{k})$ is as desired for
(ii).

Suppose that I has a strategy $\sigma$ in $F[Y]$ for playing into
$(R_{\Delta/4})^{c}\subseteq R^{c}$. In particular, I plays $(z_{k})$ such
that for all $n$, I has a strategy $\sigma_{(z_{0},\ldots,z_{n})}$ in
$F[\vec{x}^{\smallfrown}(z_{0},\ldots,z_{n}),Y]$ to play into
$((\mathbb{A}_{n})_{\Delta/2})^{c}$. As in the proof of Lemma 4 in [35], we
successively put more strategies for I into play, and obtain a strategy for
playing into
$\bigcap_{n}((\mathbb{A}_{n})_{\Delta/2})^{c}=(\mathbb{A}_{\Delta/2})^{c}$. ∎

######  Theorem 8.9 (cf. Theorem 5 in [35]).

Let $\mathcal{H}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ be a $(p^{*})$-family.
If $\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ is analytic, $\Delta>0$,
$\vec{x}\in\mathrm{bb}^{<\infty}_{1}(E)$, and $X\in\mathcal{H}$, then there is
a $Y\in\mathcal{H}\upharpoonright Y$ such that either

  1. (i)

I has a strategy in $F[\vec{x},Y]$ for playing into
$(\mathbb{A}_{\Delta/2})^{c}$, or

  2. (ii)

II has a strategy in $G[\vec{x},Y]$ for playing into $\mathbb{A}_{\Delta}$.

###### Proof.

We consider the case when $\vec{x}=\emptyset$. Let
$F:\mathbb{N}^{\mathbb{N}}\to\mathbb{A}$ be a continuous surjection and for
each $s\in\mathbb{N}^{<\mathbb{N}}$, let
$\mathbb{A}_{s}=F^{\prime\prime}(N_{s})$ where
$N_{s}=\\{\alpha\in\mathbb{N}^{\mathbb{N}}:s\subseteq\alpha\\}$. Note that
$\mathbb{A}_{s}=\bigcup_{n}\mathbb{A}_{s^{\smallfrown}n}$.

Let $R(s,\vec{x},Y)$ (for $Y\in\mathcal{H}$) be the set of all $(z_{k})$ for
which there is an $n$ such that for all $Z\in\mathcal{H}\upharpoonright Y$, I
has no strategy in $F[\vec{x}^{\smallfrown}(z_{0},\ldots,z_{n}),Z]$ for
playing into $((\mathbb{A}_{s^{\smallfrown}n})_{\Delta})^{c}$. By Lemma 8.8
and the $(p)$-property, there is an $Y\in\mathcal{H}\upharpoonright X$ such
that for all $\vec{x}$ and $s\in\mathbb{N}^{<\mathbb{N}}$, either

  1. (i)

I has a strategy in $F[\vec{x},Y]$ for playing into
$((\mathbb{A}_{s})_{\Delta/2})^{c}$, or

  2. (ii)

II has a strategy in $G[Y]$ for playing into $R(s,\vec{x},X)$.

Suppose I has no strategy in $F[Y]$ for playing into
$(\mathbb{A}_{\Delta/2})^{c}=((\mathbb{A}_{\emptyset})_{\Delta/2})^{c}$. We
will describe a strategy for II in $G[Y]$ for playing into
$\mathbb{A}_{\Delta}$: As II has a strategy in $G[Y]$ for playing into
$R(\emptyset,\emptyset,Y)$, they follow this strategy until
$(z_{0},\ldots,z_{n_{0}})$ has been played such that I has no strategy in
$F[(z_{0},\ldots,z_{n_{0}}),Y]$ for playing into
$((\mathbb{A}_{s^{\smallfrown}n_{0}})_{\Delta})^{c}$. By the assumption on
$Y$, II must have a strategy in $G[Y]$ to play in
$R((n_{0}),(z_{0},\ldots,z_{n_{0}}),Y)$. II follows this until a further
$(z_{n_{0}+1},\ldots,z_{n_{0}+n_{1}+1})$ has been played so that I has no
strategy in $F[(z_{0},\ldots,z_{n_{0}},\ldots,z_{n_{0}+n_{1}+1}),Y]$ for
playing into
$((\mathbb{A}_{{s}^{\smallfrown}{n_{0}}^{\smallfrown}n_{1}})_{\Delta})^{c}$.

We continue in this fashion, exactly as in the proof of Theorem 5 in [35], so
that the outcome $Z=(z_{n})$ satisfies that for all $k$, with
$m_{k}=(\sum_{j\leq k}n_{k})+k$, there is some
$Z^{k}\sqsupseteq(z_{0},\ldots,z_{m_{k}})$ in
$(\mathbb{A}_{(n_{0},\ldots,n_{k})})_{\Delta}=(F^{\prime\prime}(N_{(n_{0},\ldots,n_{k})}))_{\Delta}$.
Continuity of $F$ ensures that, for $\alpha=(n_{0},n_{1},\ldots)$,
$d(F(\alpha),Z)\leq\Delta$. ∎

The following result provides the link between strategically Ramsey sets and
weakly Ramsey sets.

######  Theorem 8.10 (Rosendal [34] [35]).

Suppose that, for some $X\in\mathrm{bb}^{\infty}_{1}(E)$, I has a strategy in
$F[X]$ to play into some set $\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(E)$.
Then, for any $\Delta>0$, there is a sequence of finite intervals
$I_{0}<I_{1}<\cdots$ in $\mathbb{N}$ such whenever $Y=(y_{n})\preceq X$ and
$\forall n\exists m(I_{0}<y_{n}<I_{m}<y_{n+1})$, we have that
$Y\in\mathbb{A}_{\Delta}$.

Inspired by this theorem, we define:

######  Definition 8.11.

A family $\mathcal{H}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ is _spread_ if
whenever $X=(x_{n})\in\mathcal{H}$ and $I_{0}<I_{1}<\cdots$ is a sequence of
intervals in $\mathbb{N}$, there is a $Y=(y_{n})\in\mathcal{H}\upharpoonright
X$ such that $\forall n\exists m(I_{0}<y_{n}<I_{m}<y_{n+1})$.

This property is analogous to the “(q)-property” (see Lemma 7.4 of [40]) for
coideals on $\mathbb{N}$: One can show that a coideal $\mathcal{H}$ on
$\mathbb{N}$ has the (q)-property if and only if for every $x\in\mathcal{H}$
and sequence of finite intervals $I_{0}<I_{1}<\cdots$, there is a
$y\in\mathcal{H}\upharpoonright x$ such that $\forall n\exists
m(I_{0}<y_{n}<I_{m}<y_{n+1})$.

By appropriately thinning down a block sequence, we see the following:

######  Lemma 8.12.

Given a sequence of intervals $I_{0}<I_{1}<\cdots$ in $\mathbb{N}$, the set

$\\{(y_{n}):\forall n\exists m(I_{0}<y_{n}<I_{m}<y_{n+1})\\}.$

is $\preceq$-dense open in $\mathrm{bb}^{\infty}_{1}(E)$.∎

Clearly, $\mathrm{bb}^{\infty}_{1}(E)$ itself is spread. As in §5, one can
build spread filters (which are full, almost full, strategic, etc) under
additional set-theoretic hypotheses or by forcing. We note that the strong
$(p)$-property suffices:

######  Lemma 8.13.

If $\mathcal{H}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ is a strong $(p)$-family,
then it is spread. In particular, strategic families are spread.

###### Proof.

Fix $X\in\mathcal{H}$, and let $I_{0}<I_{1}<\cdots$ be an increasing sequence
of intervals in $\mathbb{N}$. Consider the following strategy $\sigma$ for I
in $F[X]$: $\sigma(\emptyset)=\max(I_{0})$. If II responds with some
$y_{0}>\sigma(\emptyset)$, then let $\sigma(y_{0})=\max(I_{m})$, where $I_{m}$
is the first interval entirely above $\mathrm{supp}(y_{0})$. Continue in this
fashion. Any outcome $(y_{n})$ will satisfy $\forall n\exists
m(I_{0}<y_{n}<I_{m}<y_{n+1})$. Since $\mathcal{H}$ is a strong $(p)$-family,
Theorem 4.3 implies that some outcome is in $\mathcal{H}$. ∎

######  Theorem 8.14.

Let $\mathcal{H}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ be a spread
$(p^{*})$-family. Then, every analytic set is $\mathcal{H}$-weakly Ramsey.

###### Proof.

Let $\mathbb{A}\subseteq\mathrm{bb}_{1}^{\infty}(E)$ be analytic. Fix
$X\in\mathcal{H}$ and $\Delta>0$. By Theorem 8.9, there is
$Y\in\mathcal{H}\upharpoonright X$ such that either I has a strategy in $F[Y]$
for playing into $(\mathbb{A}_{\Delta/2})^{c}$, or II has a strategy in $G[Y]$
for playing into $\mathbb{A}_{\Delta}$. In the latter case, we’re done, so
assume the former. Theorem 8.10 and $\mathcal{H}$ being spread implies that
there is some $Z\in\mathcal{H}\upharpoonright Y$ with
$[Z]\subseteq((\mathbb{A}_{\Delta/2})^{c})_{\Delta/2}\subseteq\mathbb{A}^{c}$.
∎

In order to extend to sets in $\mathbf{L}(\mathbb{R})$, we will use the
following analogue of Lemma 7.8.

######  Lemma 8.15.

Let $\mathcal{F}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ be a $(p^{*})$-filter.
If $\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ is such that continuous
images of $\mathbb{A}$ are universally Baire, then for any $X\in\mathcal{F}$
and $\Delta>0$, there is a $Y\in\mathcal{F}\upharpoonright X$ for which II has
a strategy in $G[Y]$ for playing into one of $(\mathbb{A}_{\Delta/8})^{c}$ or
$\mathbb{A}_{\Delta}$.

###### Proof.

Let $X\in\mathcal{F}$ and $\Delta>0$. By Lemma 8.6, there is a
$Y\in\mathcal{F}\upharpoonright X$ such that either $(\emptyset,Y)$ is
$\Delta$-good or I has a strategy $\sigma$ in $F[Y]$ for playing into

$\\{(z_{n}):\forall n(z_{0},\ldots,z_{n},Y)\text{ is $\Delta/2$-bad}\\}.$

In the former case, we’re done, so assume the latter.

By hypothesis, $\mathbb{A}_{\Gamma}$ is universally Baire for all $\Gamma$. In
particular, we may let $\dot{\mathbb{A}}_{\Delta/4}$ be a
$\mathbb{P}(\mathcal{F})$-name for $\mathbb{A}_{\Delta/4}$ and $\mathcal{D}$ a
countable collection of dense open subsets of $\mathbb{P}(\mathcal{F})$ such
that

  1. (i)

$\\{q\in\mathbb{P}(\mathcal{F}):q\text{ decides
}\dot{X}_{\text{gen}}\in\dot{A}\\}$ is in $\mathcal{D}$, and

  2. (ii)

whenever $G$ is $\mathcal{D}$-generic in $\mathbb{P}(\mathcal{F})$,
$\dot{X}_{\text{gen}}$ is in $\mathrm{bb}^{\infty}_{1}(E)$ and
$\dot{X}_{\text{gen}}(G)$ is in $\mathbb{A}_{\Delta/4}$ if and only if there
is a $q\in G$ such that
$q\mathrel{\Vdash}_{\mathbb{P}(\mathcal{F})}\dot{X}_{\text{gen}}\in\dot{\mathbb{A}}_{\Delta/4}$.

We claim that
$(\emptyset,Y,\sigma)\mathrel{\Vdash}_{\mathbb{P}(\mathcal{F})}\dot{X}_{\text{gen}}\notin\dot{\mathbb{A}}_{\Delta/4}$.

Suppose not, then there is a $(\vec{y},Z,\tau)\leq(\emptyset,Y,\sigma)$, with
$Z\in\mathcal{F}$, such that
$(\dot{y},Z,\tau)\mathrel{\Vdash}_{\mathbb{P}(\mathcal{F})}\dot{X}_{\text{gen}}\in\dot{\mathbb{A}}_{\Delta/4}$.
Applying Lemma 7.7(b) and Theorem 8.9, there is a
$W\in\mathcal{F}\upharpoonright Z$ such that II has a strategy $\alpha$ in
$G[\vec{y},W]$ for playing into
$(\mathbb{G}_{\mathcal{D},(\vec{y},Z,\tau)})_{\Delta/4}$. As in the proof of
Lemma 7.8,
$\mathbb{G}_{\mathcal{D},(\vec{y},Z,\tau)}\subseteq\mathbb{A}_{\Delta/4}$, so
$\alpha$ is a strategy for II in $G[\vec{y},W]$ for playing into
$\mathbb{A}_{\Delta/2}$. This, however, contradicts the fact that $\sigma$
ensures $(\vec{y},Z)$ is $\Delta/2$-bad.

Thus,
$(\emptyset,Y,\sigma)\mathrel{\Vdash}_{\mathbb{P}(\mathcal{F})}\dot{X}_{\text{gen}}\notin\dot{\mathbb{A}}_{\Delta/4}$.
But then, exactly as in the preceding paragraph, we may find
$W\in\mathcal{F}\upharpoonright Y$ such that II has a strategy in $G[W]$ for
playing into $(\mathbb{G}_{\mathcal{D},(\emptyset,Y,\sigma)})_{\Delta/8}$, and
thus into
$((\mathbb{A}_{\Delta/4})^{c})_{\Delta/8}\subseteq(\mathbb{A}_{\Delta/8})^{c}$,
where the last containment follows from Lemma 8.1(d). ∎

In what follows, we strengthen the hypotheses on the basis $(e_{n})$,
asserting that there is some $K>0$ such that for all $m\leq n$ and scalars
$(a_{k})$,

$\|\sum_{k\leq m}a_{k}e_{k}\|\leq K\|\sum_{k\leq n}a_{k}e_{k}\|.$

This is equivalent to $(e_{n})$ being a Schauder basis of the completion
$\overline{E}$ of $E$, cf. Proposition 1.1.9 [2]. The infimum of all such $K$
as above is called the _basis constant_ of $(e_{n})$. The following Lemma
about perturbations of blocks sequences appears to be well-known.

######  Lemma 8.16.

For any $\Delta>0$, there is a $\Gamma>0$ such that whenever
$X=(x_{n}),X^{\prime}=(x^{\prime}_{n})\in\mathrm{bb}^{\infty}_{1}(E)$ satisfy
$d(X^{\prime},X)\leq\Gamma$, then $[X^{\prime}]\subseteq[X]_{\Delta}$. In
fact, if $Y^{\prime}\in[X^{\prime}]$, then $\widetilde{Y}\in[X]$ and
$d(Y^{\prime},\widetilde{Y})\leq\Delta$, where $\widetilde{Y}$ is the
normalization of the image of $Y^{\prime}$ under the linear map extending
$x^{\prime}_{n}\mapsto x_{n}$.

###### Proof.

Let $\Delta>0$. If $K$ is the basis constant of $(e_{n})$, then by Lemma 1.3.5
in [2], the basis constant of $X$ is $\leq K$. Pick $\Gamma>0$ with
$\sum_{n\geq m}\gamma_{n}\leq\min\\{1/6K,\delta_{m}/8K\\}$. For
$X^{\prime}=(x_{n}^{\prime})$ with $d(X^{\prime},X)\leq\Gamma$, consider the
map on the completions $T:\overline{\langle X\rangle}\to\overline{\langle
X^{\prime}\rangle}$ extending $x_{n}\mapsto x_{n}^{\prime}$. $T$ is a bounded
linear isomorphism, as whenever $v=\sum a_{n}x_{n}\in\overline{\langle
X\rangle}$,

$\displaystyle\|Tv\|-\|v\|\leq\|Tv-v\|$ $\displaystyle\leq\|\sum
a_{n}x_{n}^{\prime}-\sum
a_{n}x_{n}\|\leq\sup_{n}|a_{n}|\sum\|x_{n}^{\prime}-x_{n}\|$
$\displaystyle\leq 2K\|v\|\sum\|x_{n}^{\prime}-x_{n}\|\leq 1/3\|v\|,$

and so $\|T\|\leq 4/3$. Using $1/\|T^{-1}\|=\inf_{\|v\|=1}\|Tv\|$, we have
$\|T^{-1}\|\leq 3/2$.

As the basis constant for $X^{\prime}$ is also $\leq K$, for
$v^{\prime}=\sum_{n\geq m}a_{n}x^{\prime}_{n}\in\overline{\langle
X^{\prime}\rangle}$, we have that
$\|T^{-1}v^{\prime}-v^{\prime}\|\leq\delta_{m}/4\|v^{\prime}\|$ by a similar
argument as above.

If $v^{\prime}$ is a unit vector, then we also have that

$|1-\frac{1}{\|T^{-1}v^{\prime}\|}|\leq\|T\|\|T^{-1}v^{\prime}-v^{\prime}\|\leq(4/3)(\delta_{m}/4)\leq\delta_{m}/3.$

For $Y^{\prime}=(y_{m}^{\prime})\in[X^{\prime}]$, we claim
$d(Y^{\prime},\widetilde{Y})\leq\Delta$, where $\widetilde{Y}$ is the
normalization of $Y=(y_{m})=(T^{-1}(y_{m}^{\prime}))$. Observe that

$\displaystyle\|y_{m}-\frac{1}{\|y_{m}\|}y_{m}\|\leq\left|1-\frac{1}{\|T^{-1}(y^{\prime}_{m})\|}\right|\|T^{-1}(y^{\prime}_{m})\|\leq(\delta_{m}/3)(3/2)=\delta_{m}/2.$

Thus, for all $m$,

$\displaystyle\|y^{\prime}_{m}-\frac{1}{\|y_{m}\|}y_{m}\|$
$\displaystyle=\|y_{m}^{\prime}-y_{m}\|+\|y_{m}-\frac{1}{\|y_{m}\|}y_{m}\|\leq\delta_{m}.$

∎

The following lemma expresses the uniform continuity of the games $F[X]$ and
$G[X]$.

######  Lemma 8.17.

Let $\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ and $\Delta>0$. There is
a $\Gamma>0$ such that whenever $X\in\mathrm{bb}^{\infty}_{1}(E)$ is such that
I (II, respectively) has a strategy in $F[X]$ ($G[X]$, respectively) for
playing into $\mathbb{A}$ and $d(X,X^{\prime})\leq\Gamma$, then I (II,
respectively) has a strategy in $F[X^{\prime}]$ ($G[X^{\prime}]$,
respectively) for playing into $\mathbb{A}_{\Delta}$.

###### Proof.

Take $\Gamma>0$ as in Lemma 8.16. Suppose I has a strategy $\sigma$ in $F[X]$
for playing into $\mathbb{A}$ and $d(X,X^{\prime})\leq\Gamma$. We define a
strategy $\sigma^{\prime}$ for I in $F[X^{\prime}]$. Let
$\sigma^{\prime}(\emptyset)=\sigma(\emptyset)$. Inductively, suppose that
$\sigma^{\prime}(y^{\prime}_{0},\ldots,y^{\prime}_{k})$ has been defined and
is equal to $\sigma(y_{0},\ldots,y_{k})$, where $y_{0},\ldots,y_{k}$ is a
valid play by II in $F[X]$ against $\sigma$, and
$\|y^{\prime}_{i}-y_{i}\|\leq\gamma_{i}$ for $0\leq i\leq k$. Suppose that
$y^{\prime}_{k+1}>\sigma^{\prime}(y^{\prime}_{0},\ldots,y^{\prime}_{k})$ in
$S(\langle X^{\prime}\rangle)$. By our choice of $\Gamma$, there is a
$y_{k+1}>\sigma^{\prime}(y^{\prime}_{0},\ldots,y^{\prime}_{k})=\sigma(y_{0},\ldots,y_{k})$
in $S(\langle X\rangle)$ with $\|y^{\prime}_{k+1}-y_{k+1}\|\leq\gamma_{k+1}$.
Let
$\sigma^{\prime}(y^{\prime}_{0},\ldots,y^{\prime}_{k},y^{\prime}_{k+1})=\sigma(y_{0},\ldots,y_{k},y_{k+1})$.
It follows that $\sigma^{\prime}$ is a strategy for playing into
$\mathbb{A}_{\Delta}$.

Suppose that II has a strategy $\alpha$ in $G[X]$ for playing into
$\mathbb{A}$, and $d(X,X^{\prime})\leq\Gamma$. Let $T:\langle
X\rangle\to\langle X^{\prime}\rangle$ be as in the proof of Lemma 8.16. We
define a strategy $\alpha^{\prime}$ for II in $G[X^{\prime}]$. Suppose that I
begins by playing $Y^{\prime}_{0}\in[X^{\prime}]$. Let
$\alpha^{\prime}(Y^{\prime}_{0})=\widetilde{T}(\alpha(\widetilde{T}^{-1}(Y^{\prime}_{0})))$,
where $\widetilde{T}$ and $\widetilde{T^{-1}}$ indicate taking normalizations.
Continue in this fashion. Then, $\alpha$ is a strategy for playing into
$\mathbb{A}_{\Delta}$. ∎

######  Theorem 8.18.

Assume that there is a supercompact cardinal. Let
$\mathcal{F}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ be a strategic
$(p^{*})$-filter. Then, every set
$\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ in $\mathbf{L}(\mathbb{R})$
is $\mathcal{F}$-weakly Ramsey.

###### Proof.

Let $\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ be in
$\mathbf{L}(\mathbb{R})$, $X\in\mathcal{F}$, and $\Delta>0$. By Theorem 7.2,
the set $\mathbb{D}$ of all $Y\preceq X$ such that either I has a strategy in
$F[Y]$ for playing into $(\mathbb{A}_{\Delta/2})^{c}$, or II has a strategy in
$G[Y]$ for playing into $\mathbb{A}_{\Delta/2}$, is $\preceq$-dense open, and
is clearly in $\mathbf{L}(\mathbb{R})$. By Lemmas 4.4 and 8.15, there is a
$Y\in\mathcal{F}\upharpoonright X$ such that II has a strategy for playing
into $\mathbb{D}_{\Gamma}$, where $\Gamma$ is as in Lemma 8.17, applied to
$\Delta/4$. Since $\mathcal{F}$ is strategic, there is a
$Z\in\mathcal{F}\upharpoonright Y$ which is in $\mathbb{D}_{\Gamma}$. By our
choice of $\Gamma$, then either I has a strategy in $F[Z]$ for playing into
$((\mathbb{A}_{\Delta/2})^{c})_{\Delta/4}\subseteq(\mathbb{A}_{\Delta/2})^{c}$,
or II has a strategy in $G[Z]$ for playing into $\mathbb{A}_{\Delta}$. In the
latter case, we’re done, and in the former case, we need only apply Theorem
8.10 and Lemma 8.13. ∎

We will use the following analogue of Lemma 5.4, whose proof is similar and
left to the reader.

######  Lemma 8.19.

For $\mathcal{H}\subseteq\mathrm{bb}_{1}^{\infty}(E)$ a $(p^{*})$-family,
forcing with $(\mathcal{H},\preceq^{*})$ adds no new reals and if
$\mathcal{G}\subseteq\mathcal{H}$ is $\mathbf{L}(\mathbb{R})$-generic for
$(\mathcal{H},\preceq^{*})$, $\mathcal{G}$ will be a $(p^{*})$-filter. If
$\mathcal{H}$ is strategic (spread, respectively), then $\mathcal{G}$ will
also be strategic (spread, respectively).

######  Theorem 8.20.

Assume that there is a supercompact cardinal. Let
$\mathcal{H}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ be a strategic
$(p^{*})$-family. Then, every set
$\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ in $\mathbf{L}(\mathbb{R})$
is $\mathcal{H}$-weakly Ramsey.

###### Proof.

The proof is similar to that of Theorem 1.3, using Lemma 8.19 and Theorem
8.18. ∎

Some of the above can be simplified in the case when the family $\mathcal{H}$
in question is _invariant under small perturbations_ , that is, there is some
$\Delta>0$ so that $\mathcal{H}_{\Delta}=\mathcal{H}$. The reason lies in the
following fact:

######  Proposition 8.21.

If $\mathcal{H}$ is a strategic $(p^{*})$-family which is invariant under
small perturbations, then $\mathcal{H}$ is a $(p^{+})$-family as well.

###### Proof.

888We suspect that an elementary proof of this result can be found and that
“strategic” can be relaxed to “spread”.

Let $D\subseteq S(E)$ be $\mathcal{H}$-dense below some $X\in\mathcal{H}$ and
put $\mathbb{D}=\\{Y\preceq X:S(\langle Y\rangle)\subseteq D\\}$. Take
$\Delta>0$ so that $\mathcal{H}_{\Delta}=\mathcal{H}$. Note that $\mathbb{D}$
is closed and thus it and its continuous images are universally Baire. Let
$\mathcal{G}$ be a $\mathbf{V}$-generic filter for $(\mathcal{H},\preceq^{*})$
which contains $X$, so that by Lemma 8.19, $\mathcal{G}$ is a strategic
$(p^{*})$-filter in $\mathbf{V}[\mathcal{G}]$. By Lemma 8.15 in
$\mathbf{V}[\mathcal{G}]$, there is a $Y\in\mathcal{G}\upharpoonright X$ so
that II has a strategy in $G[Y]$ for playing into one of
$(\mathbb{D}_{\Delta/8})^{c}$ or $\mathbb{D}_{\Delta}$. However, as I has a
strategy in $G[Y]$ for playing into $\mathbb{D}$, and
$(\mathbb{D}_{\Delta/8})^{c}\subseteq\mathbb{D}^{c}$ by Lemma 8.1(c), II’s
strategy must be for playing into $\mathbb{D}_{\Delta}$. Since forcing with
$(\mathcal{H},\preceq^{*})$ added no new reals, such a strategy must exist in
$\mathbf{V}$ (we are using Lemma 4.7 implicitly here). As $\mathcal{H}$ is
strategic and $\mathcal{H}_{\Delta}=\mathcal{H}$, we have that
$\mathcal{H}\cap\mathbb{D}\neq\emptyset$, showing that $\mathcal{H}$ is full.
∎

We now extend these principles to Banach spaces. In what follows, $B$ is a
(separable) Banach space with normalized Schauder basis $(e_{n})$. We say that
a countable field $F$ is _suitable_ if the norm on $E_{F}$, the $F$-span of
$E$, takes values in $F$. Let $\langle X\rangle_{F}$ the $F$-span of
$X\in\mathrm{bb}^{\infty}(E_{F})$. If $V$ is a subspace of $B$, let
$S(V)=\\{v\in V:\|v\|=1\\}$.

Let $\mathrm{bb}^{\infty}_{1}(B)$ be the set of all infinite block sequences
(with respect to $(e_{n})$) in $B$, which we endow with the Polish topology
inherited from $B^{\mathbb{N}}$. The relations $\preceq$ and $\preceq^{*}$
extend to $\mathrm{bb}^{\infty}_{1}(B)$. For
$Y\in\mathrm{bb}^{\infty}_{1}(B)$, let
$[Y]^{*}=\\{Z\in\mathrm{bb}^{\infty}_{1}(B):Z\preceq Y\\}$. We denote by
$G^{*}[Y]$ the _Gowers game_ defined as before, except that the players may
now play real (complex) block sequences and block vectors. The notions of
_family_ , _$(p)$ -family_, _spread_ , and _strategic_ are defined as before,
with appropriate modifications for real (complex) scalars.

Strategic families in $\mathrm{bb}^{\infty}_{1}(B)$ arise naturally from
strategic families in $\mathrm{bb}^{\infty}_{1}(E_{F})$: Given a strategic
$\mathcal{H}\subseteq\mathrm{bb}^{\infty}_{1}(E_{F})$, if
$\widehat{\mathcal{H}}$ is invariant under small perturbations and equal to
the $\preceq$-upwards closure of $\mathcal{H}_{\Delta}$ (taken in
$\mathrm{bb}^{\infty}_{1}(B)$) for some small $\Delta>0$, then
$\widehat{\mathcal{H}}$ is strategic. This follows from the fact that Lemma
8.16 and the proof of Lemma 8.17 can be carried out in $B$.

######  Definition 8.22.

We say that $\mathcal{H}$ is _almost full_ if whenever $D\subseteq S(B)$ is
closed and $\mathcal{H}$-dense below some $X\in\mathcal{H}$ (that is, for all
$Y\in\mathcal{H}\upharpoonright X$, there is a $Z\preceq Y$ with $S(\langle
Z\rangle)\subseteq D$), then for any $\epsilon>0$, there is a
$Y\in\mathcal{H}\upharpoonright X$ with $S(\overline{\langle
Y\rangle})\subseteq D_{\epsilon}$.

######  Definition 8.23.

An almost full $(p)$-family in $\mathrm{bb}_{1}^{\infty}(B)$ is called a
_$(p^{*})$ -family_.

While we have reused this terminology, the meaning should be clear from
context. The following is the relativized version of Gowers weakly Ramsey
property [19].

######  Definition 8.24.

Given a family $\mathcal{H}\subseteq\mathrm{bb}^{\infty}_{1}(B)$, a set
$\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(B)$ is _$\mathcal{H}$ -weakly
Ramsey_ if for every $\Delta>0$ and $X\in\mathcal{H}$, there is a
$Y\in\mathcal{H}\upharpoonright X$ such that either

  1. (i)

$[Y]^{*}\subseteq\mathbb{A}^{c}$, or

  2. (ii)

II has a strategy in $G^{*}[Y]$ for playing into $\mathbb{A}_{\Delta}$.

Proving Theorems 1.4 and 1.5 amounts to showing that for spread (strategic)
$(p^{*})$-families $\mathcal{H}\subseteq\mathrm{bb}_{1}^{\infty}(B)$ which are
invariant under small perturbations, analytic ($\mathbf{L}(\mathbb{R})$) sets
are $\mathcal{H}$-weakly Ramsey.

######  Lemma 8.25.

Let $F$ be suitable. If $X_{0}\succeq X_{1}\succeq X_{2}\succeq\cdots$ is a
$\preceq$-decreasing sequence in $\mathrm{bb}^{\infty}_{1}(E_{F})$,
$X\in\mathrm{bb}^{\infty}_{1}(B)$ is such that $X\preceq X_{n}$ for all $n$,
and $\Delta>0$, then there is an
$X^{\prime}\in\mathrm{bb}^{\infty}_{1}(E_{F})$ with
$X^{\prime}\in[X]_{\Delta}$, and $X^{\prime}\preceq^{*}X_{n}$ for all $n$

###### Proof.

Let $(X_{n})$, $X$ and $\Delta>0$ be as described, say with $X=(x_{n})$. We
construct $X^{\prime}=(x_{n}^{\prime})$ as follows: There is an
$M_{0}\in\mathbb{N}$ so that $\overline{\langle
X/M_{0}\rangle_{F}}\subseteq\overline{\langle X_{0}\rangle_{F}}$. Let
$x_{n_{0}}$ be the first entry of $X/M_{0}$. Pick a unit vector
$x_{0}^{\prime}\in\langle X_{0}\rangle_{F}$ such that
$d(x_{n_{0}},x_{0}^{\prime})\leq\delta_{0}$. Continue inductively. At stage
$k$, we have chosen $M_{0}<\cdots<M_{k}$ and
$x_{0}^{\prime}<\cdots<x_{k}^{\prime}$ so that if $x_{n_{i}}$ is the first
entry of $X/M_{i}$, then $x_{i}^{\prime}\in\langle X_{i}\rangle_{F}$ and
$d(x_{n_{i}},x_{i}^{\prime})\leq\delta_{i}$, for $i\leq k$. By construction,
$X^{\prime}/n\preceq X_{n}$ for all $n$, and $X^{\prime}\in[X]_{\Delta}$. ∎

######  Lemma 8.26.

If $\mathcal{H}\subseteq\mathrm{bb}^{\infty}_{1}(B)$ is a $(p^{*})$-family
which is invariant under small perturbations, then
$\mathcal{H}\cap\mathrm{bb}^{\infty}_{1}(E_{F})$ is a $(p^{*})$-family for any
suitable subfield $F$ of $\mathbb{R}$ (or $\mathbb{C}$). If $\mathcal{H}$ is
spread (strategic, respectively), then so is
$\mathcal{H}\cap\mathrm{bb}^{\infty}_{1}(E_{F})$.

###### Proof.

Let $\mathcal{H}$ and $F$ be as described and put
$\widetilde{\mathcal{H}}=\mathcal{H}\cap\mathrm{bb}^{\infty}_{1}(E_{F})$.
Lemma 8.25 implies that $\widetilde{\mathcal{H}}$ is a $(p)$-family. To see
that $\widetilde{\mathcal{H}}$ is almost full, let $D\subseteq S(E_{F})$ be
$\widetilde{\mathcal{H}}$-dense below $X\in\widetilde{\mathcal{H}}$, and take
$\epsilon>0$. Consider $\overline{D_{\epsilon/3}}\subseteq S(B)$. For
$\Delta=(\epsilon/3,\epsilon/3,\ldots)$, let $\Gamma$ be as in Lemma 8.16. For
any $Y\in\mathcal{H}\upharpoonright X$, there is a
$Y^{\prime}\in\widetilde{\mathcal{H}}\upharpoonright X$, with
$d(Y,Y^{\prime})\leq\Gamma$ and $Z^{\prime}\preceq Y^{\prime}$ with $S(\langle
Z\rangle)\subseteq D$. By our choice of $\Gamma$, there is a $Z\in[Y]^{*}$
with $S(\langle Z\rangle)\subseteq D_{\epsilon/3}$, and so
$S(\overline{\langle Z\rangle})\subseteq\overline{D_{\epsilon/3}}$. Thus,
$\overline{D_{\epsilon/3}}$ is $\mathcal{H}$-dense below $X$. By almost
fullness of $\mathcal{H}$, there is a $W\in\mathcal{H}\upharpoonright X$ with
$S(\overline{\langle
W\rangle})\subseteq(\overline{D_{\epsilon/3}})_{\epsilon/3}$. Then, one can
find a $W^{\prime}\in\widetilde{\mathcal{H}}\upharpoonright X$ with $S(\langle
W^{\prime}\rangle)\subseteq D_{\epsilon}$, showing that
$\widetilde{\mathcal{H}}$ is almost full.

To see that $\mathcal{H}$ being strategic implies that $\widehat{\mathcal{H}}$
is strategic, let $\alpha$ be a strategy for II in $G[X]$, with
$X\in\widehat{\mathcal{H}}$. Define a strategy $\alpha^{\prime}$ in $G^{*}[X]$
which is equal to $\alpha$ on their shared domain, and otherwise plays so that
the outcomes are sufficiently small (using Lemma 8.16 and our assumption about
$\mathcal{H}$) perturbations of outcomes of $\alpha$. Then, if any outcome of
$\alpha^{\prime}$ is in $\mathcal{H}$, an outcome of $\alpha$ must be in
$\widehat{\mathcal{H}}$. The proof for being spread is left to the reader. ∎

###### Proof of Theorem 1.4..

Suppose that $\mathbb{A}\subseteq\mathrm{bb}^{\infty}_{1}(B)$ is analytic,
$\Delta>0$, and $X\in\mathcal{H}$ is such that for no
$Y\in\mathcal{H}\upharpoonright X$ is $[Y]^{*}\subseteq\mathbb{A}^{c}$. Let
$F$ be a suitable field for $(e_{n})$. Let
$\widetilde{\mathcal{H}}=\mathcal{H}\cap\mathrm{bb}^{\infty}_{1}(E_{F})$. If
there was some $Y\in\widetilde{\mathcal{H}}\upharpoonright X$ with
$[Y]\subseteq(\mathbb{A}_{\Delta/3})^{c}\cap\mathrm{bb}^{\infty}_{1}(E_{F})$,
then
$[Y]^{*}\subseteq((\mathbb{A}_{\Delta/3})^{c})_{\Delta/3}\subseteq\mathbb{A}^{c}$,
contrary to our assumption. Thus, by Lemma 8.26 and Theorem 8.14, there is a
$Y\in\widetilde{\mathcal{H}}\upharpoonright X$ such that II has a strategy in
$G[Y]$ for playing into
$\mathbb{A}_{\Delta/2}\cap\mathrm{bb}^{\infty}_{1}(E_{F})$. Easy perturbation
arguments show that II has a strategy in $G^{*}[Y]$ for playing into
$\mathbb{A}_{\Delta}$. ∎

###### Proof of Theorem 1.5..

The proof is similar to that of Theorem 1.4, using Theorem 8.20, or
alternatively, Proposition 8.21 and Theorem 1.3. ∎

The following is an analytical example of a strategic $(p^{*})$-family, which,
though trivial in the sense that is $\preceq$-downwards closed, we hope
suggests further applications:

######  Example 8.27.

Given $B$ as above, suppose that $B$ contains a normalized block sequence $X$
equivalent to the standard basis of $c_{0}$ or $\ell^{p}$ for $1\leq
p<\infty$. Let $\mathcal{H}$ be the set of all block sequences in $B$ which
have a further block subsequence equivalent to $X$. Then, $\mathcal{H}$ is a
strategic $(p^{*})$-family which is invariant under small perturbations. These
facts follows from the block homogeneity characterization of the standard
bases of $c_{0}$ and $\ell^{p}$, Lemma 2.1.1 in [2].

##  9\. Projections in the Calkin algebra

Given a Banach space with a Schauder basis, one might wish to develop a notion
of forcing with block sequences “modulo small perturbation” and then prove an
analog of Theorem 1.2, characterizing $\mathbf{L}(\mathbb{R})$-generic
filters.999There are obstacles to this being a meaningful endeavor in general,
e.g., in a hereditarily indecomposable Banach space, the collection of _all_
infinite-dimensional subspaces modulo small perturbations forms a filter, cf.
(iii) on p. 820 of [19], and is thus trivial as a forcing notion. We focus on
a particular variant of this which is of significant interest.

Let $H$ be a complex infinite-dimensional separable Hilbert space with
orthonormal basis $(e_{n})$. Note that any normalized block sequence (with
respect to $(e_{n})$) is necessarily orthonormal. Throughout, $E$ will denote
the $\overline{\mathbb{Q}}$-linear span of $(e_{n})$ in $H$,
$\mathrm{bb}^{\infty}_{1}(E)$ the space of infinite normalized block sequences
in $E$, and for $X\in\mathrm{bb}^{\infty}_{1}(E)$, $\langle X\rangle$ is the
$\overline{\mathbb{Q}}$-span of $X$.

For $X\in\mathrm{bb}^{\infty}_{1}(E)$, let $P_{X}$ be the orthogonal
projection onto $\overline{\langle X\rangle}$. Note that, for
$X,Y\in\mathrm{bb}^{\infty}_{1}(E)$, $X\preceq Y$ if and only if $P_{X}\leq
P_{Y}$ in the usual ordering of projections (that is, $P\leq Q$ if
$\operatorname{ran}(P)\subseteq\operatorname{ran}(Q)$, or equivalently
$PQ=P$). We call such projections _block projections_.

Let $\mathcal{B}(H)$ be the C*-algebra of bounded operators on $H$ and
$\mathcal{K}(H)$ the ideal of compact operators on $H$. The quotient
$\mathcal{C}(H)=\mathcal{B}(H)/\mathcal{K}(H)$ is also a C*-algebra, called
the _Calkin algebra_. We write $\pi:\mathcal{B}(H)\to\mathcal{C}(H)$ for the
quotient map.

Denote by $\mathcal{P}(H)$ ($\mathcal{P}_{\infty}(H)$, respectively) the set
(infinite-rank, respectively) projections in $\mathcal{B}(H)$, and
$\mathcal{P}(\mathcal{C}(H))$ ($\mathcal{P}(\mathcal{C}(H))^{+}$,
respectively) the set of (non-zero, respectively) projections, i.e., self-
adjoint idempotents, in $\mathcal{C}(H)$. By Proposition 3.1 in [41],
$\mathcal{P}(\mathcal{C}(H))=\pi(\mathcal{P}(H))$. The ordering $\leq$ on
$\mathcal{P}(\mathcal{C}(H))$ is inherited from the ordering on
$\mathcal{P}(H)$.

######  Definition 9.1.

  1. (a)

For projections $P,Q\in\mathcal{P}(H)$, we write $P\leq_{\mathrm{ess}}Q$ if
$\pi(P)\leq\pi(Q)$ in $\mathcal{P}(\mathcal{C}(H))$ and
$P\equiv_{\mathrm{ess}}Q$ if $\pi(P)=\pi(Q)$.

  2. (b)

For $X,Y\in\mathrm{bb}^{\infty}_{1}(E)$, we write $X\leq_{\mathrm{ess}}Y$ if
$P_{X}\leq_{\mathrm{ess}}P_{Y}$ and $X\equiv_{\mathrm{ess}}Y$ if
$P_{X}\equiv_{\mathrm{ess}}P_{Y}$.

The last sentence of the following lemma requires a slight modification of the
original proof and is left to the reader.

######  Lemma 9.2 (Proposition 3.3 in [41]).

For $P$ and $Q$ projections on $H$, the following are equivalent:

  1. (i)

$P\leq_{\mathrm{ess}}Q$.

  2. (ii)

For every $\epsilon>0$, there is a finite-codimensional subspace $V$ of
$\operatorname{ran}(P)$ such that every unit vector $v\in V$ satisfies
$d(v,\operatorname{ran}(Q))\leq\epsilon$.

In the event that $P=P_{X}$ and $Q=P_{Y}$ for
$X,Y\in\mathrm{bb}^{\infty}_{1}(E)$, one can replace “finite-codimensional
subspace” in (ii) with “tail subspace”.

The following lemma is well-known:

######  Lemma 9.3.

Suppose that $\Delta=(\delta_{n})>0$ is summable and $P$ and $Q$ are
projections on $H$ whose ranges have orthonormal bases $(x_{n})$ and $(y_{n})$
respectively. If for all $n$, $\|x_{n}-y_{n}\|\leq\delta_{n}$, then
$P\equiv_{\mathrm{ess}}Q$.

###### Proof.

Assuming that for all $n$, $\|x_{n}-y_{n}\|\leq\delta_{n}$, we will show that
$P\leq_{\mathrm{ess}}Q$. The result follows by symmetry. Let $\epsilon>0$ and
choose an $N$ such that $\sum_{n\geq N}\delta_{n}\leq\epsilon$. Let
$V=\overline{\langle(x_{n})_{n\geq N}\rangle}$, a finite-codimensional
subspace of $\operatorname{ran}(P)$. If $v\in V$ is a unit vector, say with
$v=\sum_{n\geq N}a_{n}x_{n}$, then for $y=\sum_{n\geq
N}a_{n}y_{n}\in\operatorname{ran}(Q)$, we have

$\|v-y\|=\|\sum_{n\geq N}a_{n}(x_{n}-y_{n})\|\leq\sum_{n\geq
N}\|x_{n}-y_{n}\|\leq\epsilon.$

The claim follows by Lemma 9.2. ∎

In particular, $\equiv_{\mathrm{ess}}$-invariant families in
$\mathrm{bb}^{\infty}_{1}(E)$ or $\mathrm{bb}^{\infty}_{1}(H)$ are invariant
under small perturbations. The following observation can be proved using Lemma
9.3 and standard manipulations with basic sequences (cf. Proposition 1.3.10 in
[2]).

######  Lemma 9.4.

The set of block projections is dense in
$(\mathcal{P}_{\infty}(H),\leq_{\mathrm{ess}})$.

It follows that $(\mathcal{P}(\mathcal{C}(H))^{+},\leq)$,
$(\mathcal{P}_{\infty}(H),\leq_{\mathrm{ess}})$, and
$(\mathrm{bb}^{\infty}_{1}(E),\leq_{\mathrm{ess}})$ are equivalent as notions
of forcing. It is for this reason that we focus on
$(\mathrm{bb}^{\infty}_{1}(E),\leq_{\mathrm{ess}})$.

######  Lemma 9.5.

If $X_{0}\succeq X_{1}\succeq X_{2}\succeq\cdots$ is a $\preceq$-decreasing
sequence in $\mathrm{bb}^{\infty}_{1}(E)$ and
$X\in\mathrm{bb}^{\infty}_{1}(E)$ is such that $X\leq_{\mathrm{ess}}X_{n}$ for
all $n$, then there is an $X^{\prime}\leq_{\mathrm{ess}}X$ such that
$X^{\prime}\preceq^{*}X_{n}$ for all $n$.

###### Proof.

This can be proved using Lemmas 9.2 and 9.3 in a way similar to Lemma 8.25. ∎

Clearly, any $\preceq$-dense subset of $\mathrm{bb}^{\infty}_{1}(E)$ is also
$\leq_{\mathrm{ess}}$-dense. The following lemma is a converse to this.

######  Lemma 9.6.

If $\mathcal{D}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ is
$\leq_{\mathrm{ess}}$-dense open, then it is $\preceq$-dense open.

###### Proof.

Suppose $\mathcal{D}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ is
$\leq_{\mathrm{ess}}$-dense open. Given any $X\in\mathrm{bb}^{\infty}_{1}(E)$,
there is a $Y\in\mathcal{D}$ with $Y\leq_{\mathrm{ess}}X$. Applying Lemma 9.5
(with $X_{n}=X$ for all $n$), there is a $Y^{\prime}\leq_{\mathrm{ess}}Y$ with
$Y^{\prime}\preceq X$. Then, $Y^{\prime}\in\mathcal{D}$. ∎

We can now establish Theorem 1.6, an analog of Theorem 1.2 for projections in
the Calkin algebra. We first prove a more general result.

######  Theorem 9.7.

  1. (a)

If $\mathcal{G}$ is an $\mathbf{L}(\mathbb{R})$-generic filter for
$(\mathrm{bb}^{\infty}_{1}(E),\leq_{\mathrm{ess}})$, then $\mathcal{G}$ is a
strategic $(p^{+})$-family.

  2. (b)

Assume that there is a supercompact cardinal. If
$\mathcal{G}\subseteq\mathrm{bb}^{\infty}_{1}(E)$ is a strategic
$(p^{*})$-family which is also a $\leq_{\mathrm{ess}}$-filter, then
$\mathcal{G}$ is $\mathbf{L}(\mathbb{R})$-generic for
$(\mathrm{bb}_{1}^{\infty}(E),\leq_{\mathrm{ess}})$.

###### Proof.

(a) Let $\mathcal{G}$ be as described. Clearly, it is a family. To see that it
is full, suppose that $D\subseteq S(E)$ is $\mathcal{G}$-dense below some
$X\in\mathcal{G}$. Let

$\mathcal{D}_{0}=\\{Z:\langle Z\rangle\subseteq D\text{ or }\forall V\preceq
X(\langle V\rangle\subseteq D\rightarrow V\bot Z)\\},$

where $\bot$ denotes incompatibility with respect to $\preceq$.
$\mathcal{D}_{0}$ is $\preceq$-dense open by Lemma 2.7, thus
$\leq_{\mathrm{ess}}$-dense as well, and clearly in $\mathbf{L}(\mathbb{R})$,
so there is a $Z\in\mathcal{D}_{0}\cap(\mathcal{G}\upharpoonright X)$. Then,
there is a $Z^{\prime}\preceq Z\preceq X$ with $S(\langle
Z^{\prime}\rangle)\subseteq D$, so we have that $\langle Z\rangle\subseteq D$,
showing that $\mathcal{G}$ is full.

To see that $\mathcal{G}$ is a $(p)$-family, let $X_{0}\succeq X_{1}\succeq
X_{2}\succeq\cdots$ in $\mathcal{G}$. Let

$\mathcal{D}_{1}=\\{Y:\forall n(Y\preceq^{*}X_{n})\text{ or }\exists
n(Y\bot_{\mathrm{ess}}X_{n})\\},$

where $\bot_{\mathrm{ess}}$ denotes incompatibility with respect to
$\leq_{\mathrm{ess}}$. We want to show that $\mathcal{D}_{1}$ is
$\leq_{\mathrm{ess}}$-dense. The set

$\mathcal{D}_{1}^{\prime}=\\{Y:\forall n(Y\leq_{\mathrm{ess}}X_{n})\text{ or
}\exists n(Y\bot_{\mathrm{ess}}X_{n})\\}$

is $\leq_{\mathrm{ess}}$-dense open. Then, given any $X$, we can find a
$Y\in\mathcal{D}_{1}^{\prime}$ below $X$. If $Y\bot_{\mathrm{ess}}X_{n}$ for
some $n$, we’re done. Otherwise, $Y\leq_{\mathrm{ess}}X_{n}$ for all $n$, and
we can apply Lemma 9.5 to find a $Y^{\prime}\leq_{\mathrm{ess}}Y$ with
$Y\preceq^{*}X_{n}$ for all $n$. Such a $Y^{\prime}$ is in $\mathcal{D}_{1}$,
verifying that this set is $\leq_{\mathrm{ess}}$-dense. As $\mathcal{D}_{1}$
is in $\mathbf{L}(\mathbb{R})$, $\mathcal{G}\cap\mathcal{D}_{1}\neq\emptyset$,
and anything in this intersection must be a diagonalization of $(X_{n})$. It
is likewise easy to see that $\mathcal{G}$ must be strategic.

(b) Let $\mathcal{D}\subseteq\mathrm{bb}^{\infty}_{1}(H)$ be
$\leq_{\mathrm{ess}}$-dense open and in $\mathbf{L}(\mathbb{R})$. By Lemma
9.6, $\mathcal{D}$ is also $\preceq$-dense open. For $\Delta>0$ summable,
$\mathcal{D}_{\Delta}=\mathcal{D}$ by Lemma 9.3. Thus, by Theorem 8.14, there
is an $X\in\mathcal{H}$ such II has a strategy for playing into $\mathcal{D}$.
Since $\mathcal{G}$ is strategic, it follows that
$\mathcal{G}\cap\mathcal{D}\neq\emptyset$. ∎

###### Proof of Theorem 1.6.

The ($\Rightarrow$) direction is proved by a straightforward verification of
the relevant sets being $\preceq$-dense open, thus $\leq_{\mathrm{ess}}$-dense
by Lemma 9.6. The ($\Leftarrow$) direction follows from Theorem 9.7(b) or
Theorem 1.5. ∎

We conclude this section by describing a hoped-for application of our
machinery and its limitations. A _state_ $\tau$ on $\mathcal{B}(H)$ is a
linear functional on $\mathcal{B}(H)$ which is _positive_ , that is,
$\tau(T^{*}T)\geq 0$ for all $T$, and satisfies $\tau(I)=1$, where $I$ is the
identity operator. The set of states forms a weak*-compact convex subset of
the dual of $\mathcal{B}(H)$ and thus has extreme points, called _pure
states_. These definitions generalize to any unital C*-algebra, including
$\mathcal{C}(H)$.

A state on $\mathcal{B}(H)$ is _singular_ if it vanishes on $\mathcal{K}(H)$.
Composing with the quotient map $\pi:\mathcal{B}(H)\to\mathcal{C}(H)$ yields a
bijective correspondence between singular pure states on $\mathcal{B}(H)$ and
pure states on $\mathcal{C}(H)$.

For any choice of orthonormal basis $(f_{k})$ for $H$, and any ultrafilter
$\mathcal{U}$ on $\mathbb{N}$, the functional defined by