The Paris-Harrington Theorem

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 primitive recursive in the case of 2 colors, and even Shelah’s improved proof only gives the bound W (n) = O(g4(n)), where g4(n) = g n 3 (1) and g3(1) = 2, g3(n) = 2 g3(n−1). There are even Ramsey-type functions which provably do not have primitive recursive lower bounds, implying that the Ramsey-type theorems that show their existence cannot be proven in logical systems that can only handle primitive recursion. Peano Arithmetic is a system capable of dealing with primitive recursion and beyond, but even in this system, natural Ramsey theoretic questions can be asked for which a solution exists (in standard set theory), but the existence cannot be proven. The proof of this result is in close connection with ordinal theory, and more precisely the question of how large ordinals a logical system can prove to be well-ordered. In this essay, following the presentation of [2], we introduce a natural generalization of the classical Ramsey theorem, and prove it using the infinite Ramsey theorem and a compactness argument. We then show, using ordinal theory and a classical result from logic, that the theorem cannot be proved in Peano Arithmetic. Apart from the result from logic, the ‘proof of unprovability’ is largely self-contained.