A note on some weak forms of the axiom of choice (Q1187546)

The author shows that in ZF the following assertion implies the axiom of choice for well-ordered families of nonempty sets (and thus is equivalent to it in view of a theorem due to \textit{P. Erdős} and \textit{A. Tarski} [Ann. Math. (2) 44, 315--329 (1943; Zbl 0060.12602)]): If \((T,\leq)\) is a tree of height \(\lambda\) (a well-ordered cardinal) which has fewer than \(\lambda\) branches of height less than \(\lambda\), then one of the following holds for the reverse tree \((T,\geq)\): (i) there is a chain of cardinality \(\lambda\); (ii) there is a strong antichain of cardinality \(\lambda\); (iii) for some \(\kappa<\lambda\) there is no strong antichain of cardinality \(\kappa\). Here a strong antichain is an antichain of \((T,\geq)\) which has at most one element from each level of \((T,\leq)\).