Finite trees and the necessary use of large cardinals

We introduce insertion domains that support the placement of new, higher, vertices into finite trees. We prove that every nonincreasing insertion domain has an element with simple structural properties in the style of classical Ramsey theory. This result is proved using standard large cardinal axioms that go well beyond the usual axioms for mathematics. We also establish that this result cannot be proved without these large cardinal axioms. We also introduce insertion rules that specify the placement of new, higher, vertices into finite trees. We prove that every insertion rule greedily generates a tree with these same structural properties; and every decreasing insertion rule generates (or admits) a tree with these same structural properties. It is also necessary and sufficient to use the same large cardinals (in the precise sense of Corollary D.25). The results suggest new areas of research in discrete mathematics called "Ramsey tree theory" and "greedy Ramsey theory" which demonstrably require more than the usual axioms for mathematics.
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Godel's Unpublished Papers on Foundations of Mathematics.W. W. Tatt - 2001 - Philosophia Mathematica 9 (1):87-126.
The Metamathematics of Ergodic Theory.Jeremy Avigad - 2009 - Annals of Pure and Applied Logic 157 (2-3):64-76.
Subtle Cardinals and Linear Orderings.Harvey M. Friedman - 2000 - Annals of Pure and Applied Logic 107 (1-3):1-34.
Intuition and Its Object.Kai Hauser - 2015 - Axiomathes 25 (3):253-281.
Proof Theory in Philosophy of Mathematics.Andrew Arana - 2010 - Philosophy Compass 5 (4):336-347.

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