Proof lengths for instances of the Paris–Harrington principle

Abstract

As Paris and Harrington have famously shown, Peano Arithmetic does not prove that for all numbers k,m,n there is an N which satisfies the statement PH(k,m,n,N): For any k-coloring of its n-element subsets the set {0,…,N−1} has a large homogeneous subset of size ≥m. At the same time very weak theories can establish the Σ1-statement ∃NPH(k‾,m‾,n‾,N) for any fixed parameters k,m,n. Which theory, then, does it take to formalize natural proofs of these instances? It is known that ∀m∃NPH(k‾,m,n‾,N) has a natural and short proof (relative to n and k) by Σn−1-induction. In contrast, we show that there is an elementary function e such that any proof of ∃NPH(e(n)‾,n+1‾,n‾,N) by Σn−2-induction is ridiculously long.In order to establish this result on proof lengths we give a computational analysis of slow provability, a notion introduced by Sy-David Friedman, Rathjen and Weiermann. We will see that slow uniform Σ1-reflection is related to a function that has a considerably lower growth rate than Fε0 but dominates all functions Fα with α<ε0 in the fast-growing hierarchy.