Sharp phase transition thresholds for the Paris Harrington Ramsey numbers for a fixed dimension

Abstract:

This article is concerned with investigations on a phase transition which is related to the (finite) Ramsey theorem and the Paris-Harrington theorem. For a given number-theoretic function $g$, let $R^d_c(g)(k)$ be the least natural number $R$ such that for all colourings $P$ of the $d$-element subsets of $\{0,\ldots ,R-1\}$ with at most $c$ colours there exists a subset $H$ of $\{0,\ldots ,R-1\}$ such that $P$ has constant value on all $d$-element subsets of $H$ and such that the cardinality of $H$ is not smaller than $\max \{k,g(\min (H))\}$. If $g$ is a constant function with value $e$, then $R^d_c(g)(k)$ is equal to the usual Ramsey number $R^d_c(\max \{e,k\})$; and if $g$ is the identity function, then $R^d_c(g)(k)$ is the corresponding Paris-Harrington number, which typically is much larger than $R^d_c(k)$. In this article we give for all $d\geq 2$ a sharp classification of the functions $g$ for which the function $m\mapsto R^d_m(g)(m)$ grows so quickly that it is no longer provably total in the subsystem of Peano arithmetic, where the induction scheme is restricted to formulas with at most $(d-1)$-quantifiers. Such a quick growth will in particular happen for any function $g$ growing at least as fast as $i\mapsto \varepsilon \cdot \underbrace {\log (\cdots (\log (}_{(d-1)-\mbox {times}}i)\cdots )$ (where $\varepsilon >0$ is fixed) but not for the function $g(i)=\frac {1}{\log ^*(i)}\cdot \underbrace {\log (\cdots (\log (}_{(d-1)-\mbox {times}}i)\cdots )$. (Here $\log ^*$ denotes the functional inverse of the tower function.) To obtain such results and even sharper bounds we employ certain suitable transfinite iterations of nonconstructive lower bound functions for Ramsey numbers. Thereby we improve certain results from the article A classification of rapidly growing Ramsey numbers (PAMS 132 (2004), 553–561) of the first author, which were obtained by employing constructive ordinal partitions.