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Abstract

We state a combinatorial problem for trees, and provide a sharp answer for a particular case. We introduce an ordinal index which characterizes weak compactness of operators between Banach spaces. As an application of the solution to the combinatorial problem, we prove that certain classes of weakly compact operators determined by this index form operator ideals. We also discuss the distinctness of these classes, as well as the descriptive set-theoretic properties of this index.