Aronszajn trees and maximality

Abstract

Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal $\mu$, we prove the consistency of the existence of a wide $\mu^+$-Aronszajn tree, i.e. a tree of height and cardinality $\mu^+$ with no branches of length $\mu^+$, into which every wide $\mu^+$-Aronszajn tree can be embedded.