# Gowers’ Ramsey-type theorem in Banach spaces and adversarial Gowers’ games

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## Strategically Ramsey sets and the pigeonhole principle

The aim of this section is to prove a version of Rosendal’s theorem I.13 in the general setting of Gowers spaces. We also introduce the notion of the pigeonhole principle for a Gowers space and see that the last result can be strengthened in the case where this principle holds. This will enable us to see the fundamental dierence between the Mathias{Silver space and the Rosendal space over a eld with at least three elements. We start by introducing Gowers’ game and the asymptotic game in the setting of Gowers spaces, and the notion of a strategically Ramsey set. In this whole section, we x a Gowers space G pP;X;¤;¤; q.
1. Gowers’ game below p, denoted by Gp, is dened in the following way: where the xi’s are elements of X, and the pi’s are elements of P. The rules are the following:
for I: for all i P !, pi ¤ p.
for II: for all i P !, px0; : : : ; xiq  pi.

### The strength of the adversarial Ramsey principle

In section II.1, we proved the adversarial Ramsey property for Borel sets using Borel determinacy, and we saw on the trivial example of the space with only one subspace that, given 􀀀 a suitable class of subsets of Polish spaces, the adversarial Ramsey property for 􀀀-sets implied the determinacy for 􀀀-games on integers. This had two consequences: on one hand, the use of a suciently large fragment of ZFC is necessary to prove the adversarial Ramsey property for Borel sets, and on the other hand, it is not possible to prove it for analytic or coanalytic sets in ZFC. However, the space we used to make this remark is quite articial. Of course, we made the same remark in the introduction of this thesis using the Rosendal space, however we did it by making players play according to the norms  f the vectors, which is quite articial too (we would not do that, for example, in the applications to Banach-space geometry, where we usually restrict our attention to normalized vectors). Therefore, is it natural to ask in which cases using a large fragment of ZFC is necessary to prove the adversarial Ramsey property for Borel sets, or in which cases this property could be provable in ZFC for analytic and coanalytic sets; the aim of this section is to give an answer to this question. We will see, in particular, that Gowers spaces where the pigeonhole principle holds, and those where it does not hold, behave very dierently.
In this section and the next one, we x 􀀀 a suitable class of subsets of Polish spaces. Given a Gowers space G pP;X;¤;¤; q, we denote by AdvGp􀀀q the statement \every 􀀀-subset of X! is adversarially Ramsey », and by StratGp􀀀q the statement \every 􀀀-subset of X! is strategically Ramsey ». We let Advp􀀀q be the statement \for every analytic Gowers space G, AdvGp􀀀q holds », and Stratp􀀀q be the statement \for every analytic Gowers space G, StratGp􀀀q holds ». We proved in the two previous sections the following implications.

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#### Closure properties and limitations for strategically Ramsey sets

In this section, we show the same kind of dierence of behavior between spaces with and without a pigeonhole principle as in the previous section, but this kind for strategically Ramsey sets. We x, in the whole section, a Gowers space G pP;X;¤;¤; q and a suitable class 􀀀 of subsets of Polish spaces. The rst thing to remark is that if G satises the pigeonhole principle, then by corollary II.21, the class of strategically Ramsey sets is closed under taking complements: X  X! is strategically Ramsey if and only if Xc is so. In particular, in ZFC, every 11 subset of X! is strategically Ramsey. In spaces where the pigeonhole principle does not hold, the situation is very dierent; we rstly state the two main results of this section and present their consequences, before proving them.
The rst result is a generalisation of a theorem proved by Lopez-Abad [37] in the context of strategically Ramsey sets in Banach spaces. This result only holds for forgetful Gowers spaces, and to prove it, we need the negation of a slight weakening of the pigeonhole principle. We will say that the forgetful space G satises the weak pigeonhole principle if for every A  X, there exists p P P such that either p  A, or p  Ac (where p  A abusively denotes the fact that for every x P X, if x  p, then x P A). Of course, in most of the concrete spaces we consider, P has a maximum 1 that is isomorphic to every subspace (meaning, here, that for every p0 P P, there are bijections : P ÝÑ tp P P | p ¤ p0u and ‘ : X Ñ tx P X | x  p0u that preserve the relations ¤, ¤ and ); this is, for example, the case of the Mathias{Silver space or of the Rosendal space. In these spaces, the weak pigeonhole principle is equivalent to the pigeonhole principle. Our result is the following: Proposition II.26. Suppose that G is forgetful and does not satisfy the weak pigeonhole principle. Then StratGp􀀀q ñ StratGpD􀀀q.

Remerciements
Courte introduction en francais
Notations and conventions
I Introduction and history
I.1 Determinacy
I.2 Innite-dimensional Ramsey theory
I.3 Gowers’ Ramsey-type theorem in Banach spaces and adversarial Gowers’ games
I.4 Banach-spaces dichotomies and complexity of the isomorphism
I.5 Organisation of the results
II Ramsey theory with and without pigeonhole principle
II.1 Gowers spaces and the aversarial Ramsey property
II.2 Strategically Ramsey sets and the pigeonhole principle
II.3 The strength of the adversarial Ramsey principle
II.4 Closure properties and limitations for strategically Ramsey sets
II.5 The adversarial Ramsey property under large cardinal assumptions
IIIRamsey theory in uncountable spaces
III.1 A counterexample
III.2 Approximate Gowers spaces
III.3 Eliminating the asymptotic game
IV Hilbert-avoiding dichotomies and ergodicity
IV.1 Preliminaries
IV.2 The rst dichotomy
IV.3 The second dichotomy
IV.4 Links with ergodicity and Johnson’s problem
IV.5 A simple proof of Gowers{Maurey’s theorem
Bibliography

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