Almost-Ramsey Sets in c_{0}

Jordi López Abad

We consider a Ramsey-style property of c_{0}, the property of being almost-Ramsey for sets of block subspaces. We give a proof of the following result, announced by W. T. Gowers in [2], namely, that every analytic set of block subspaces of c_{0} which is large, almost-contains all the block subspaces of some block subspace. This result is a natural generalization of the works of N. Hindman [3] on sequences of finite sets of integers and of Gowers [1] on asymptotic sets of c_{0}. We also extend the result to all projective sets under Projective Determinacy.

[1] W. T. Gowers, Lipschitz functions on classical spaces, European Journal of Combinatorics 13 (1992), pp. 141–151.

[2] W. T. Gowers, Analytic sets and games in Banach spaces. Preprint IHES M/94/42.

[3] N. Hindman, Finite sums from sequences within cells of a partition of N, Journal of Combinatorial Theory (A) 17 (1974), pp. 1–11.

Address: Departament de Matematiques. Universitat Autonoma de Barcelona, Spain

E-mail: abad@mat.uab.es