Aronszajn trees and maximality

Abstract

Assuming the consistency of a weakly compact cardinal above a regular uncountable cardinal $\hat{I}¼$, we prove the consistency of the existence of a wide $\hat{I}{¼}^{+}$-Aronszajn tree, i.e. a tree of height and cardinality $\hat{I}{¼}^{+}$ with no branches of length $\hat{I}{¼}^{+}$, into which every wide $\hat{I}{¼}^{+}$-Aronszajn tree can be embedded.