One of the most basic questions frequently asked about set-theoretical trees is the question whether they contain any
, a branch that intersects each level of the tree. The fundamental importance of this question has already been realized in the work of Kurepa  and then later in the works of Erdős and Tarski  in their respective attempts to develop the theory of partition calculus and large cardinals. A tree
of height equal to some regular cardinal
may not have a cofinal branch for a very special reason as the following definition indicates.
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