Some combinatorial principles for trees and applications to tree families in Banach spaces

Suppose that (xs)sS is a normalized family in a Banach space indexed by the dyadic tree S. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree-families. More precisely, assuming that for any infinite chain of S the sequence (xs)s is weakly null, we prove that there exists a subtree T of S such that for any infinite chain of T the sequence (xs)s is nearly (resp., convexly) unconditional. In the case where (fs)sS is a family of continuous functions, under some additional assumptions, we prove the existence of a subtree T of S such that for any infinite chain of T, the sequence (fs)s is unconditional. Finally, in the more general setting where for any chain , (xs)s is a Schauder basic sequence, we obtain a dichotomy result concerning the semi-boundedly completeness of the sequences (xs)s.