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Abstract: Konig's Lemma states that a countably infinite,finitely branching tree has an infinite branch.It is natural to ask whether the analogous result holds for trees of uncountable height.
In 1934, Aronszajn proved that this result fails for trees of height $\aleph_1$. On the other hand, this question for trees of height $\kappa \geq \aleph_2$ is intimately connected with fundamental questions concerning cardinal arithmetic and the existence of large cardinals. I will explain this in this talk, and then I will go over recent results about combinatorial principles and cardinal arithmetic.