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

Suppose that 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 β of S the sequence is weakly null, we prove that there exists a subtree T of S such that for any infinite chain β of T the sequence is nearly (resp., convexly) unconditional. In the case where 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 β of T, the sequence is unconditional. Finally, in the more general setting where for any chain β, is a Schauder basic sequence, we obtain a dichotomy result concerning the semi‐boundedly completeness of the sequences .