 
Summary: COMBINATORICA
Bolyai Society SpringerVerlag
02099683/110/$6.00 c 2010 J´anos Bolyai Mathematical Society and SpringerVerlag
Combinatorica 30 (4) (2010) 359385
DOI: 10.1007/s0049301025071
A NEW APPROACH TO THE RAMSEYTYPE GAMES
AND THE GOWERS DICHOTOMY IN FSPACES
GEORGE ANDROULAKIS, STEPHEN J. DILWORTH*,
NIGEL J. KALTON
Received May 19, 2008
We give a new approach to the Ramseytype results of Gowers on block bases in Banach
spaces and apply our results to prove the Gowers dichotomy in Fspaces.
1. Introduction
Our aim in this note is to establish the Gowers dichotomy [4] in a gen
eral Fspace (complete metric linear space). We say that an Fspace X is
hereditarily indecomposable if it is impossible to find two separated infinite
dimensional closed subspaces V,W, i.e., such that V W = {0} and V +W
is closed (or equivalently that the natural projection from V +W onto V is
continuous). Our main result is that an Fspace either contains an uncon
ditional basic sequence or an infinitedimensional HI subspace. In order to
