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 \(K_{\omega1}\) 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.
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Acknowledgement
The author is grateful to Chris Lambie-Hanson for constructive discussions on the topic which greatly improved the exposition of this paper.
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This work has been supported by Charles University Research Center program No. UNCE/SCI/022 and by the Academy of Sciences of the Czech Republic (RVO 67985840).
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Uhrik, D. The uncountable Hadwiger conjecture and characterizations of trees using graphs. Acta Math. Hungar. 172, 19–33 (2024). https://doi.org/10.1007/s10474-024-01399-x
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DOI: https://doi.org/10.1007/s10474-024-01399-x