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

Suppose that $(x_s)_{s\in S}$ 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 $\beta$ of $S$ the sequence $(x_s)_{s\in\beta}$ is weakly null, we prove that there exists a subtree $T$ of $S$ such that for any infinite chain $\beta$ of $T$ the sequence $(x_s)_{s\in\beta}$ is nearly (resp., convexly) unconditional. In the case where $(f_s)_{s\in S}$ 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 $\beta$ of $T$, the sequence $(f_s)_{s\in\beta}$ is unconditional. Finally, in the more general setting where for any chain $\beta$, $(x_s)_{s\in\beta}$ is a Schauder basic sequence, we obtain a dichotomy result concerning the semi-boundedly completeness of the sequences $(x_s)_{s\in\beta}$.