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A Note on Ramsey Theorems and Turing Jumps

  • Lorenzo Carlucci
  • Konrad Zdanowski
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)

Abstract

We give a new treatment of the relations between Ramsey’s Theorem, ACA 0 and ACA 0′. First we combine a result by Girard with a colouring used by Loebl and Nešetril for the analysis of the Paris-Harrington principle to obtain a short combinatorial proof of ACA 0 from Ramsey Theorem for triples. We then extend this approach to ACA 0′ using a characterization of this system in terms of preservation of well-orderings due to Marcone and Montalbán. We finally discuss how to apply this method to \(\mathbf{ACA}_0^+\) using an extension of Ramsey’s Theorem for colouring relatively large sets due to Pudlàk and Rödl and independently to Farmaki.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Lorenzo Carlucci
    • 1
  • Konrad Zdanowski
    • 2
  1. 1.Dipartimento di InformaticaUniversità di Roma SapienzaRomaItaly
  2. 2.Uniwersytet Kardynała Stefana Wyszyńskiego w WarszawieWarszawaPoland

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