In Ramsey theory, very large numbers and fast-growing functions are more of a rule than an exception. The classical Ramsey numbers R(n,m) are known to be of exponential size: the original proof directly gives the upper bound R(n,m) ≤ ( m+n−2 n−1 ) , and an exponential lower bound is also known. For the van der Waerden numbers, the original proof produced upper bounds that were not even primitiv...

1. Introduction A set A of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements.