The Uncountable Hadwiger Conjecture and Characterizations of Trees Using Graphs

TL;DR

This paper establishes a deep connection between special trees and uncountably chromatic graphs, introduces a generalized connectedness concept, and characterizes various special trees and large cardinals through graph properties.

Contribution

It proves the equivalence between the existence of non-special trees and uncountably chromatic graphs with no large minors, and introduces a new connectedness notion for characterizing large cardinals.

Findings

Equivalence between non-special trees and uncountably chromatic graphs without $K_{oldsymbol{\omega_1}}$ minors.

Characterizations of Aronszajn, Kurepa, and Suslin trees via graphs.

A new generalized connectedness concept characterizing weakly compact cardinals.

Abstract

We prove that the existence of a non-special tree of size $\lambda$ is equivalent to the existence of an uncountably chromatic graph with no $K_{\omega_1}$ minor of size $\lambda$, establishing a connection between the special tree number and the uncountable Hadwiger conjecture. Also characterizations of Aronszajn, Kurepa and Suslin trees using graphs are deduced. A new generalized notion of connectedness for graphs is introduced using which we are able to characterize weakly compact cardinals.