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Explore generalizations of the classic finite Ramsey theorem in this 54-minute conference talk that substitutes traditional cardinality constraints with sophisticated notions of "large sets." Delve into extensions of the Paris-Harrington theorem from 1977, examining how concepts of α-largeness for ordinals below ε_0 can be further expanded using barriers from better quasi-order theory. Discover how these barrier-based largeness notions can be applied throughout generalized Ramsey statements while maintaining only the number of colors as a cardinality constraint. Learn about the surprising achievement of obtaining tight bounds on these "generalized Ramsey numbers," contrasting sharply with the classic finite case where such precise bounds remain elusive except for very specific small-number scenarios. Gain insights into this collaborative research with Antonio Montalban and Andrea Volpi that advances our understanding of combinatorial mathematics and reverse mathematics paradigms.
