Finding paths through narrow and wide trees
  We consider two axioms of second-order arithmetic. These axioms assert, in two different ways, that infinite but narrow binary trees always have infinite paths. We show that both axioms are strictly weaker than Weak K\"onig's Lemma, and incomparable in strength to the dual statement (WWKL) that wide binary trees have paths. 
Stephen Binns and Bj{\o}rn Kjos-Hanssen
math.LO