Coloring the rationals in reverse mathematics

Emanuele Frittaion 1 Ludovic Patey 2, 3
3 PI.R2 - Design, study and implementation of languages for proofs and programs
Inria de Paris, CNRS - Centre National de la Recherche Scientifique, UPD7 - Université Paris Diderot - Paris 7, PPS - Preuves, Programmes et Systèmes
Abstract : 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.
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Emanuele Frittaion, Ludovic Patey. Coloring the rationals in reverse mathematics. Computability, IOS Press, 2017, 6 (4), pp.319 - 331. ⟨10.3233/COM-160067⟩. ⟨hal-01888756⟩

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