Set-Coloring Ramsey Numbers and Error-Correcting Codes Near the Zero-Rate Threshold

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.