Sharp Thresholds for a Phase Transition Related to Weakly Increasing Sequences
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 Erdos-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/Ad-1(i)).
phase transitions erdos-szekeres ackermann function ramsey theory