The Strength of Ramsey’s Theorem for Coloring Relatively Large Sets

Abstract

We characterize the computational content and the proof-theoretic strength of a Ramseytype theorem for bi-colorings of so-called exactly large sets. An exactly large set is a set X ⊂ N such that card(X) = min(X) + 1. The theorem we analyze is as follows. For every infinite subset M of N, for every coloring C of the exactly large subsets of M in two colors, there exists and infinite subset L of M such that C is constant on all exactly large subsets of L. This theorem is essentially due to Pudlàk and Rödl and independently to Farmaki. We prove that -over Computable Mathematics -this theorem is equivalent to closure under the ω Turing jump (i.e., under arithmetical truth). Natural combinatorial theorems at this level of complexity are rare. Our results give a complete characterization of the theorem from the point of view of Computable Mathematics and of the Proof Theory of Arithmetic. This nicely extends the current knowledge about the strength of Ramsey Theorem. We also show that analogous results hold for a related principle based on the Regressive Ramsey Theorem. In addition we give a further characterization in terms of truth predicates over Peano Arithmetic. We conjecture that analogous results hold for larger ordinals.

New bounds on the strength of some restrictions of Hindman’s Theorem

“…Is there a strong computable reduction of HT to RT !ω 2 ? Combining the results of [4] and [8] we know that the implication from RT !ω 2 to HT holds over RCA 0 . Can this be witnessed by a strong computable reduction?…”

“…3 Note that ACA 0 is known to be equivalent to RT 3 2 (Ramsey's Theorem for 2-colorings of triples) by seminal work of Jockusch and of Simpson ([32,Theorem III.7.6] or [22,Chapter 6]), so we have that HT implies RT 3 2 over RCA 0 . On the other hand ACA + 0 was only recently given a Ramsey-theoretic characterization in work of the first and fourth author, who showed [8] that the system ACA + 0 is equivalent to a Ramsey-theoretic theorem due to Pudlák and Rödl [31] and Farmaki and Negrepontis [16], which we denote by RT !ω 2 (see Definition 5.4). This theorem extends Ramsey's Theorem to colourings of objects of variable dimension, in particular to so-called exactly large sets of integers, where a set is exactly large in case its cardinality is greater by one than its minimum element.…”

“…The strength of RT !ω 2 was studied by the first and fourth author in [8] and proved there to be much beyond the strength of Ramsey's Theorem.…”

A Note on Ramsey Theorems and Turing Jumps

“…We refer to the statement of the above Theorem as RT(!ω). The effective and proof-theoretic content of RT(!ω) has been recently characterized in [2]. In that paper it is shown that RT(!ω) is equivalent to ACA + 0 over RCA 0 .…”

“…Furthermore, we strongly conjecture that this approach can be extended to prove that a Ramsey-type theorem for bicolourings of exactly large sets due to Pudlàk-Rödl [11] and independently to Farmaki (see, e.g., [3]) implies ACA + 0 (i.e., equivalently, RCA 0 augmented by the assertion that the ω Turing jump of any set exists). The effective and proof-theoretic content of this theorem have been recently fully analyzed by the authors in [2]. We conjecture that the method can be extended to relate the general version of the latter theorem [3] to the systems Π 0 ω α -CA 0 , for α ∈ ω ck 1 , using their characterization in [9] in terms of the well-ordering preservation principles ∀X (WO(X ) → WO(ϕ(α, X ))).…”