References

In [1] W. T. Gowers has formulated and proved a Ramsey-type result which lies at the heart of his famous dichotomy for Banach spaces. He defines a family G of weakly Ramsey sets of block sequences and shows that every analytic set of block sequences belongs to G, though his dichotomy can be deduced from the fact that every Gδ set of block sequences, i.e. countable intersection of open sets, belongs to G. We show that G is not closed under taking complements and that the full generality really appears at the Gδ level. More precisely, we supply a rather direct proof of Gowers ’ result that G contains all analytic sets as a direct consequence of the fact that Gδ sets of block sequences belong to G. This fact can explain why the only known applications of this technique are based on very low-ranked Borel sets (open, closed, Fσ or Gδ). We also show, answering a question of Gowers ([1]), that under a suitable large-cardinal assumption every definable set of block sequences belongs to G