Random reals, the rainbow Ramsey theorem, and arithmetic conservation

Abstract

Abstract. 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 CsimaMileti 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 over1 -sentences. Thus, the set of first-order consequences of 2-RAN is strictly stronger than P − + I Σ 1 and no stronger than P − + B Σ 2 . §1. Introduction. One of the benefits to having a precise formulation of the concept "random infinite binary sequence" is that one can ask and precisely answer questions about what types of objects can be computed from random input and about what sorts of theorems can be proven from the existence of a random sequence.In [3], Csima and Mileti gave an intriguing example of the first type. Their example concerns the Rainbow Ramsey Theorem, which states that if C is a coloring of size-k subsets of N such that there is a uniform finite bound on the number of sets assigned to any particular color, then there is an infinite set X such that C is injective on the size-k subsets of X , i.e. X is a C -rainbow. As is described below, they give a proof of the Rainbow Ramsey Theorem for pairs (k = 2) by showing that if R is a sufficiently random sequence relative to C , then R can be used to compute a C -rainbow. In a sentence, Csima and Mileti show how a random source can be used to produce a solution to a infinitary-combinatorial problem. Further, since they also show that there is a recursive such C with no recursive rainbow, any general method to produce rainbows for colorings must be driven by some such non-recursive data.In this article, we analyze the second question with respect to theorems about finite sets. For this question, we shift the setting from the examination of the computation

Where pigeonhole principles meet Koenig lemmas

“…Let us start by setting up some notation. [7], except that the base theory is weakened from RCA 0 + BΣ 0 2 to RCA 0 . For this improvement, we carefully replace applications of BΣ 0 2 with those of Lemma 4.5.…”

Nonstandard Models in Recursion Theory and Reverse Mathematics

“…As in Theorem 4.12, Conidis and Slaman [21] showed that RRT 2 2 is Π 1 1 -conservative overBΣ 0 2 (this was also proved independently by Wei Wang (unpublished)). In fact, in [21] a stronger combinatorial principle 2-RAN was introduced. It asserts that given an X there is a 2-random set relative to X .…”

Randomness Notions and Reverse Mathematics

“…The statement 2-MLR (i.e., the existence of 2-random sets) is well studied in reverse mathematics. For instance, in the presence of the scheme BΣ 0 2 (i.e., Σ 0 2 -bounding), 2-MLR is equivalent to two formalizations of the dominated convergence theorem [2], and it implies the rainbow Ramsey theorem for pairs and 2-bounded colorings [13,14].…”

“…Therefore M |= RCA 0 + W2R + ¬2-DNR. 2 ) is the statement "for every 2-bounded f : [N] 2 → N, there is a set R that is a rainbow for f." By formalizing work of Csima and Mileti [14], Conidis and Slaman [13] have shown that RCA 0 2-MLR → RRT 2 2 . J. Miller [29], again building on [14], has shown that in fact RCA 0 2-DNR ↔ RRT 2 2 .…”