Aronszajn trees

In set theory, an Aronszajn tree is an uncountable tree with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal κ, a κ-Aronszajn tree is a tree of height κ such that all levels have size less than κ and all branches have height less than κ (so Aronszajn trees are the same as ${\displaystyle \aleph _{1}}$-Aronszajn trees). They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by Kurepa (1935).

A cardinal κ for which no κ-Aronszajn trees exist is said to have the tree property. (sometimes the condition that κ is regular and uncountable is included.)

König's lemma states that ${\displaystyle \aleph _{0}}$-Aronszajn trees do not exist.

The existence of Aronszajn trees (${\displaystyle =\aleph _{1}}$-Aronszajn trees) was proven by Nachman Aronszajn, and implies that the analogue of König's lemma does not hold for uncountable trees.

...
Wikipedia