Coloring the rationals in reverse mathematics

Emanuele Frittaion (1) , Ludovic Patey (2, 3)

1 Tohoku University [Sendai]
2 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
3 PI.R2 - Design, study and implementation of languages for proofs and programs

Ramsey's theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramsey's theorem for pairs due to Erd˝ os and Rado, which states that every 2-coloring of the pairs of rationals has either an infinite 0-homogeneous set or a 1-homogeneous set of order type η, where η is the order type of the rationals. This theorem is a natural candidate to lie strictly between the arithmetic comprehension axiom and Ramsey's theorem for pairs. This Erd˝ os-Rado theorem, like the tree theorem for pairs, belongs to a family of Ramsey-type statements whose logical strength remains a challenge.

Mots clés: computable reduction; Ramsey; Reverse mathematics

Domaines: Logique [math.LO]

Emanuele Frittaion, Ludovic Patey. Coloring the rationals in reverse mathematics. Computability, 2017, 6 (4), pp.319 - 331. ⟨10.3233/COM-160067⟩. ⟨hal-01888756⟩