Universal wide Aronszajn tree

TL;DR

This paper proves the consistency, under certain set-theoretic assumptions, of the existence of a universal wide Aronszajn tree that contains all such trees as subtrees.

Contribution

It demonstrates the relative consistency of a universal wide Aronszajn tree assuming a weakly compact cardinal.

Findings

Existence of a universal wide Aronszajn tree under set-theoretic assumptions.

The result relies on the consistency of a weakly compact cardinal.

Provides a new perspective on the structure of Aronszajn trees.

Abstract

A wide Aronszajn tree is a tree of size $\aleph_1$ with no uncountable branches. Assuming the consistency of the existence of a weakly compact cardinal, we show the consistency of the existence of a wide Aronszajn tree that is \textit{universal} in the sense that it contains an isomorphic copy of every wide Aronszajn tree.