A computable analysis of variable words theorems

Abstract

The Carlson-Simpson lemma is a combinatorial statement occurring in the proof of the Dual Ramsey theorem. Formulated in terms of variable words, it informally asserts that given any finite coloring of the strings, there is an infinite sequence with infinitely many variables such that for every valuation, some specific set of initial segments is homogeneous. Friedman, Simpson, and Montalban asked about its reverse mathematical strength. We study the computability-theoretic properties and the reverse mathematics of this statement, and relate it to the finite union theorem. In particular, we prove the Ordered Variable word for binary strings in ACA 0 .

The combinatorial equivalence of a computability theoretic question

“…In [12], Liu, Monin and Patey showed that ACA proves OVW(2, 2). Actually, they showed that for every computable OVW(2, 2)-instance c, every ∅ ′ -PA degree computes a solution of c. On the other hand, they also construct a computable OVW(2, 2)-instance such that every solution is of ∅ ′ -DNC degree.…”

“…Their proof heavily relies on Lovasz Local Lemma. Monin, Liu and Patey [12] computes, using a particular arithmetic oracle, an ordered variable word solution of a given computable coloring c : 2 <ω → 2 (which is an ω-variable word v such that the set {v( a) : a ∈ 2 <ω } is monochromatic for c and all occurrence of x m is before that of x m+1 , see Definition 2.3). Their construction uses a similar technique as Shelah's proof of Hales-Jewett theorem [15].…”

The reverse mathematics of Hindman’s Theorem for sums of exactly two elements

“…One example has already been given by Liu, Monin, and Patey [13]. Another appears in Cholak, Dzhafarov, Hirschfeldt, and Patey [3].…”