For $X\subseteq\naturals$, let $[X]^n$ denote the size $n$ subsets of $X$. For $n,m>0$ and
$F:[\naturals]^n\to \{0,\dots,m-1\}$, $H\subseteq\naturals$ is \emph{homogeneous for $F$} iff
$F$ is constant on $[H]^n$. Ramsey's Theorem states that for all $n,m>0$ and all
$F:[\naturals]^n\to \{0,\dots,m-1\},$ there is an infinite set $H$ such that $H$ is homogeneous
for $F$.
We say that $F$ is \emph{stable} when for every $x$, $\lim_{y\to\infty} F(x,y)$ exists. Let
$RT^2_2$ denote the formal statement of Ramsey's theorem for $n$ and $m$ equal to 2, and let
$SRT^2_2$ denote that formal statement restricted to stable partitions.
We discuss the following result, which is joint work with Chong Chi Tat and Yang Yue, both of
the National University of Singapore.
\begin{theorem}[Chong, Slaman, and Yang]
There is a model $\M$, satisfying the recursive comprehension axiom and with the following properties.
\begin{itemize}
\item $\M\models\SRT^2_2$
\item $\M\models\neg\ISigma_2$, where $\ISigma_2$ is the principle of induction for
$\Sigma^0_2$-formulas, relative to parameters from $\M$.
\item Every real in $\M$ is low in $\M$.
\end{itemize}
\end{theorem}
It is not known whether $RT^2_2$ implies $\ISigma_2$ or whether every $\omega$-model (i.e.\ with
standard integer part) of $SRT^2_2$ is also a model of $RT^2_2$.