The Threshold for Ackermannian Ramsey numbers

Abstract

For a function $g:\mathbb{N}\to \mathbb{N}$, the \emph{$g$-regressive Ramsey number} of $k$ is the least $N$ so that \[N\stackrel \min \longrightarrow (k)_g\] . This symbol means: for every $c:[N{]}^2\to \mathbb{N}$ that satisfies $c(m,n)\le g(\min \{m,n\})$ there is a \emph{min-homogeneous} $H\su N$ of size $k$, that is, the color $c(m,n)$ of a pair $\{m,n\}\su H$ depends only on $\min \{m,n\}$. It is known (\cite{km,ks}) that $\id$-regressive Ramsey numbers grow in $k$ as fast as $\Ack(k)$, Ackermann's function in $k$. On the other hand, for constant $g$, the $g$-regressive Ramsey numbers grow exponentially in $k$, and are therefore primitive recursive in $k$. We compute below the threshold in which $g$-regressive Ramsey numbers cease to be primitive recursive and become Ackermannian, by proving: Suppose $g:\mathbb{N}\to \mathbb{N}$ is weakly increasing. Then the $g$-regressive Ramsey numbers are primitive recursive if an only if for every $t>0$ there is some $M_t$ so that for all $n\ge M_t$ it holds that $g(m)<n^{1/t}$ and $M_t$ is bounded by a primitive recursive function in $t$.