A ramsey theorem for trees,with an application to Banach spaces

LetS be the binary tree of all sequences of 0’s and 1’s. A chain ofS is any infinite linearly ordered subset. Letℋ be an analytic set of chains, we show that there exists a binary subtreeS’ ofS such that either all chains ofS’ lie inℋ or no chain ofS’ lies inℋ. As an application, we prove the following result on Banach spaces: If (x _s) _sɛs is a bounded sequence of elements in a Banach spaceE, there exists a subtreeS’ ofS such that for any chainβ ofS’ the sequence (x _s ) _{s ∈β} is either a weak Cauchy sequence or equivalent to the usuall ¹ basis.