Phase transitions for weakly increasing sequences

Abstract

Motivated by the classical Ramsey for pairs problem in reverse mathematics we investigate the recursion-theoretic complexity of certain assertions which are related to the Erdös-Szekeres theorem. We show that resulting density principles give rise to Ackermannian growth. We then parameterize these assertions with respect to a number-theoretic function f and investigate for which functions f Ackermannian growth is still preserved. We show that this is the case for but not for f(i)∈=∈log(i). © 2008 Springer-Verlag Berlin Heidelberg.