The Uncountable Hadwiger Conjecture and Characterizations of Trees Using Graphs

Dávid Uhrik

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arXiv · 2212.02169

The Uncountable Hadwiger Conjecture and Characterizations of Trees Using Graphs

Dávid Uhrik

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Abstract

We prove that the existence of a non-special tree of size $λ$ is equivalent to the existence of an uncountably chromatic graph with no $K_{ω_1}$ minor of size $λ$, 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.

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BibTeX RIS

Dávid Uhrik. 2022-12-05. The Uncountable Hadwiger Conjecture and Characterizations of Trees Using Graphs. https://arxiv.org/abs/2212.02169

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