Combinatorial Principles Weaker than Ramsey's Theorem for Pairs

by Denis R. Hirschfeldt and Richard A. Shore

Status: published in the Journal of Symbolic Logic 72 (2007) 171 - 206.

Availability: journal version and preprint

Abstract. We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is well-known that Ramsey's Theorem for pairs (RT22) splits into a stable version (SRT22) and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases these versions are strictly weaker (which is not known to be the case for RT22 and SRT22). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL0, DNR, and BΣ2, showing for instance that WKL0 is incomparable with all of the systems we study; and prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is Π11-conservative over the base system RCA0. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply SRT22 (and so does not imply RT22). This answers a question of Cholak, Jockusch, and Slaman.

Our proofs suggest that the essential distinction between ADS and CAC on the one hand and RT22 on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive multicolorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions.