  For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number
$R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph
$K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is
a subset of $n$ vertices where all of the edges between them receive a common
color. If $n$ is fixed and $\frac{s}{r}$ is less than and bounded away from
$1-\frac{1}{n-1}$, then $R(n;r,s)$ is known to grow exponentially in $r$, while
if $\frac{s}{r}$ is greater than and bounded away from $1-\frac{1}{n-1}$, then
$R(n;r,s)$ is bounded. Here we prove bounds for $R(n;r,s)$ in the intermediate
range where $\frac{s}{r}$ is close to $1 - \frac{1}{n-1}$ by establishing a
connection to the maximum size of error-correcting codes near the zero-rate
threshold.
