Minimal obstructions for normal spanning trees

TL;DR

This paper investigates the minimal obstructions to the existence of normal spanning trees in graphs, showing under certain set-theoretic assumptions that a single forbidden graph suffices, but providing counterexamples under CH to some conjectures.

Contribution

The paper demonstrates that under Martin's Axiom, a single $(eth_1,eth_0)$-graph suffices as an obstruction, and constructs counterexamples to conjectures about minors of $(eth_0,eth_1)$-graphs under CH.

Findings

Under Martin's Axiom, one forbidden graph suffices.

Counterexamples show not all $(eth_0,eth_1)$-graphs are minors of each other.

Certain conjectures about minors of these graphs are false under CH.

Abstract

Diestel and Leader have characterised connected graphs that admit a normal spanning tree via two classes of forbidden minors. One class are Halin's $(\aleph_0,\aleph_1)$-graphs: bipartite graphs with bipartition $(\mathbb{N},B)$ such that $B$ is uncountable and every vertex of $B$ has infinite degree. Our main result is that under Martin's Axiom and the failure of the Continuum Hypothesis, the class of forbidden $(\aleph_0,\aleph_1)$-graphs in Diestel and Leader's result can be replaced by one single instance of such a graph. Under CH, however, the class of $(\aleph_0,\aleph_1)$-graphs contains minor-incomparable elements, namely graphs of binary type, and $\mathcal{U}$-indivisible graphs. Assuming CH, Diestel and Leader asked whether every $(\aleph_0,\aleph_1)$-graph has an $(\aleph_0,\aleph_1)$-minor that is either indivisible or of binary type, and whether any two $\mathcal{U}$-indivisible graphs are necessarily minors of each other. For both questions, we construct examples showing that the answer is in the negative.