A new approach to the Ramsey-type games and the Gowers dichotomy in F-spaces

Banach空间Gowers二分法的一个简单组合证明

A simple, combinatorial proof of Gowers Dichotomy for Banach Spaces

本文用纯组合方法，基于无限Ramsey理论，给出了Gowers二分法的简单证明，即每个无限维Banach空间都含无条件基序列或遗传不可分解的子空间。

Parametrized Ramsey theory of infinite block sequences of vectors

We show that the infinite-dimensional versions of Gowers' $\mathrm{FIN}_k$ and $\mathrm{FIN}_{\pm k}$ theorems can be parametrized by an infinite sequence of perfect subsets of $2^ω$. To do so, we use ultra-Ramsey theory to obtain exact and approximate versions of a result which combines elements from both Gowers' theorems and the Hales-Jewett theorem. As a consequence, we obtain a parametrized version of Gowers' $c_0$ theorem.

An exact Ramsey principle for block sequences

We prove an exact, i.e., formulated without $Δ$-expansions, Ramsey principle for infinite block sequences in vector spaces over countable fields, where the two sides of the dichotomic principle are represented by respectively winning strategies in Gowers' block sequence game and winning strategies in the infinite asymptotic game. This allows us to recover Gowers' dichotomy theorem for block sequences in normed vector spaces by a simple application of the basic determinacy theorem for infinite asymptotic games.

Local Ramsey Spaces in Matet Forcing Extensions and Finitely Many Near-Coherence Classes

We introduce Gowers--Matet forcing with a finite sequence of pairwise non-isomorphic Ramsey ultrafilters over $ω$, and with this forcing we settle the long-standing problem of the spectrum of numbers near-coherence classes. We prove that for any finite $n \geq 1$, there is a forcing extension with exactly $n$ near-coherence classes of ultrafilters. For evaluating the new forcing, we prove a strengthening of Gowers's theorem on colourings of ${\rm Fin}_k$.

Determinacy of adversarial Gowers games

We prove a game theoretic dichotomy for $G_{δσ}$ sets of block sequences in vector spaces that extends, on the one hand, the block Ramsey theorem of W. T. Gowers proved for analytic sets of block sequences and, on the other hand, M. Davis' proof of $G_{δσ}$ determinacy.

A discretized approach to W.T. Gowers' game

We give an alternative proof of W. T. Gowers' theorem on block bases by reducing it to a discrete analogue on specific countable nets. We also give a Ramsey type result on k-tuples of block sequences in a normed linear space with a Schauder basis.