Open questions about Ramsey-type statements in reverse mathematics

Ludovic Patey 1, 2
2 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 states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey's theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.
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Ludovic Patey. Open questions about Ramsey-type statements in reverse mathematics. Bulletin of Symbolic Logic, Association for Symbolic Logic, 2016, 22 (02), pp.151 - 169. 〈10.1017/bsl.2015.40〉. 〈hal-01888620〉



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