Proof lengths for instances of the Paris-Harrington principle

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.