Sharp Thresholds for a Phase Transition Related to Weakly Increasing Sequences

摘要

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 Erdoes-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 f(i) = i~(1/(Ack~(-1)(i))) but not for f(i) = i~(1/(A_d~(-1)(i))).