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Proof Lengths for Instances of the Paris-Harrington Principle

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 $\operatorname{PH}(k,m,n,N)$: For any $k$-colouring of its $n$-element subsets the set $\{0,\dots,N-1\}$ has a large homogeneous subset of size $\geq m$. At the same time very weak theories can establish the $Σ_1$-statement $\exists_N\operatorname{PH}(\overline k,\overline m,\overline n,N)$ for any fixed parameters $k,m,n$. Which theory, then, does it take to formalize natural proofs of these instances? It is known that $\forall_m\exists_N\operatorname{PH}(\overline k,m,\overline n,N)$ has a natural and short proof (relative to $n$ and $k$) by $Σ_{n-1}$-induction. In contrast, we show that there is an elementary function $e$ such that any proof of $\exists_N\operatorname{PH}(\overline{e(n)},\overline{n+1},\overline n,N)$ by $Σ_{n-2}$-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 $Σ_1$-reflection is related to a function that has a considerably lower growth rate than $F_{\varepsilon_0}$ but dominates all functions $F_α$ with $α<\varepsilon_0$ in the fast-growing hierarchy.