A Note on Ramsey Theorems and Turing Jumps

We give a new treatment of the relations between Ramsey's Theorem, ACA0 and ACA0′. First we combine a result by Girard with a colouring used by Loebl and Nesetril for the analysis of the Paris-Harrington principle to obtain a short combinatorial proof of ACA0 from Ramsey Theorem for triples. We then extend this approach to ACA0′ using a characterization of this system in terms of preservation of well-orderings due to Marcone and Montalban. 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 Pudlak and Rodl and independently to Farmaki.