Combinatorial principles weaker than Ramsey's Theorem for pairs

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 (ChainAntiChain), 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 (RT2 2) splits into a stable version (SRT2 2) 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 RT2 2 and SRT2 2). 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 ...

• Type: Journal
• Year: 2007
• Where: JSYML
• Authors: Denis R. Hirschfeldt, Richard A. Shore
• Keywords: Infinite Ascending Sequence, JSYML 2007, Partial Order, Various Combinatorial Theorems