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

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