The Strength of Some Combinatorial Principles Related to Ramsey's Theorem for Pairs

Abstract

Abstract. We study the reverse mathematics and computability-theoretic strength of (stable) Ramsey's Theorem for pairs and the related principles COH and DNR. We show that SRT 2 2 implies DNR over RCA0 but COH does not, and answer a question of Mileti by showing that every computable stable 2-coloring of pairs has an incomplete ∆ 0 2 infinite homogeneous set. We also give some extensions of the latter result, and relate it to potential approaches to showing that SRT

Ramsey’s theorem and products in the Weihrauch degrees

“…Lemma 4.12. Let G be low for functions, and f : 21].) To see that this non-reduction holds, first note that there is a uniformly computable sequence S 0 , S 1 , .…”

Ramsey's theorem and cone avoidance

“…We will apply this result when B satisfies B ≤ T 0 , in which case of course the path f of the previous theorem is low 2 .…”

“…Let an index for a low 2 , minimal pair forming condition (D, L) with D ⊂ A 0 be given, along with an e ∈ ω, and the canonical index of a finite set S of Σ 0 2 formulas free in at most G. There are 0 -effective procedures by which to decide, from these indices, whether or not this condition is S-small 0 , and, from e and these indices, whether or not it is (S, e)-small 0 . If it is S-small 0 or (S, e)-small 0 , there exists an n ∈ ω and a sequence (D i , L i , k i , w i : i < n) witnessing this fact such that each D i ⊂ A 0 , and each L i is low 2 Finally, we have the following lemma which will ensure that the set we end up constructing is of degree forming a minimal pair with deg(B):…”

Cappable Cea Sets and Ramsey's Theorem

“…Ramsey's Theorem for pairs, RT 2 . An excellent summary of the reverse mathematical results dealing with Ramsey's Theorem can be found in [7]. This paper was motivated by an attempt to separate RT 2 2 from SRT 2 2 , in the sense of Reverse Mathematics.…”