Por Oriola Gjetaj (University of Ghent).

Ramsey’s Theorem has been widely investigated in Computability Theory, Proof Theory and Reverse Mathematics. In this talk we formulate, prove and analyze the  generalizations of Friedman’s Free Set and Thin Set theorems, as well as of the Rainbow Ramsey Theorem to colorings of exactly large sets, i.e. sets such that card(X) = min(X) + 1. The notion of a large set is well-known to proof-theorists for its key role in the famous Paris-Harrington Principle, providing one of the most mathematically natural witnesses of the incompleteness of first-order Peano Arithmetic. The Free Set, Thin Set and Rainbow Ramsey’s Theorem are consequences of the classical Ramsey’s Theorem. The Free Set Theorem for dimension k > 0 states that given a coloring f of k-subsets of N in unboundedly many colors, there exists an infinite set A ⊆N such that for all k-sets s ⊆A, if f (s) ∈A then f (s) ∈s. It is natural to consider the analogous generalizations of the Free Set, Thin Set and Rainbow Ramsey Theorem to colorings of exactly large sets and to inquire into their effective and logical strength.

This is a joint work with Lorenzo Carlucci and Andrea Vivi.

Transmissão via Zoom (pw: 919 4789 5133).