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Combinatorial principles weaker than Ramsey's theorem for pairs
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@MISC{Shore_combinatorialprinciples,
author = {Richard A. Shore and Denis R. Hirschfeldt},
title = {Combinatorial principles weaker than Ramsey's theorem for pairs},
year = {}
}
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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 (RT2 2) 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 RT 2 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 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,
Keyphrases
reverse mathematics computability theory various combinatorial theorem infinite ascending sequence infinite descending sequence cohesive principle infinite antichain partial order descending sequence conservation result infinite partial order infinite linear order stable version infinite chain principle ad
