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For many years I had the idea that if a well-founded tree is both very tall and very narrow, then it must have a cofinal branch. For example, it is a fun exercise to show that any $\omega_1$-tree with all levels finite has a cofinal branch. A similar fact holds also for much taller trees with all levels finite. I used to think that such a phenomenon might hold even more generally, for trees that are much taller than they are wide, although not necessarily finite. But after a conversation this evening with some other set theorists, I am less sure. And so I ask for a counter-example:

Question. Is it (relatively) consistent with ZFC that there is a tree of height $\omega_2$ with all levels countable, with no cofinal branches?

More generally, under what circumstances can we have very tall trees with very small levels, but no cofinal branch? What general theorems about this are available? In what cases are there provable instances where there is a cofinal branch for tall narrow trees?

(Incidentally, I think it would be fine as a warm-up if someone were to post a solution to the fun exercise I mention, that is, that every tree of height $\omega_1$ and all levels finite has a cofinal branch; or some generalization of this. What is the best way to see it?)

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  • $\begingroup$ I think it is a theorem of Kurepa that if $T$ has height $\kappa$ and all levels have size $<\lambda,$ for some $\lambda<\kappa,$ then $T$ has a cofinal branch $\endgroup$ – Mohammad Golshani May 30 '15 at 3:58
  • $\begingroup$ Mohammad, that would fulfill my intuition! But I was somehow convinced to abandon that intuition in the conversation this evening. Can you post a reference? $\endgroup$ – Joel David Hamkins May 30 '15 at 4:01
  • $\begingroup$ Kanamori's book "the higher infinite", proposition 7.9, page 78 $\endgroup$ – Mohammad Golshani May 30 '15 at 4:02
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    $\begingroup$ Mohammad, kindly post that as an answer! $\endgroup$ – Joel David Hamkins May 30 '15 at 4:05
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    $\begingroup$ I would be interested to know what data supported the opposite intuition. $\endgroup$ – Monroe Eskew May 30 '15 at 6:59
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The following theorem of Kurepa answers the question.

Theorem (Kurepa) Suppose that $\kappa$ is regular, $\lambda< \kappa$ , and $T$  is a $\kappa$-tree each of whose levels has cardinality less than $\lambda$. Then $T$  has a cofinal branch.

The theorem is stated in Kanamori's book ``The higher infinite'', as Proposition 7.9 (page 78).

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