RANDOM REALS, THE RAINBOW RAMSEY THEOREM, AND ARITHMETIC CONSERVATION

We investigate the question "To what extent can random reals be used as a tool to establish number theoretic facts?" Let 2-RAN be the principle that for every real X there is a real R which is 2-random relative to X. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory RCA(0) and so RCA(0) + 2-RAN implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is not conservative over RCA(0) for arithmetic sentences. Thus, from the Csima-Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that 2-RAN has non-trivial arithmetic consequences. In Section 4, we show that 2-RAN is conservative over RCA(0) + B Sigma(2) for Pi broken vertical bar-sentences. Thus, the set of first-order consequences of 2-RAN is strictly stronger than P- + I Sigma(1) and no stronger than P- + B Sigma(2).