Set-coloring Ramsey numbers and error-correcting codes near the zero-rate threshold
IEEE Transactions on Information Theory（2023）
摘要
For positive integers
n, r, s
with
r
>
s
, the setcoloring Ramsey number
R
(
n; r, s
) is the minimum
N
such that if every edge of the complete graph
KN
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
s/r
is less than and bounded away from 1 - 1/
n
-1, then
R
(
n; r, s
) is known to grow exponentially in r, while if
s/r
is greater than and bounded away from 1 - 1/
n
-1, then
R
(
n; r, s
) is bounded. Here we prove bounds for
R
(
n; r, s
) in the intermediate range where
s/r
is close to 1 - 1/
n
-1 by establishing a connection to the maximum size of error-correcting codes near the zero-rate threshold.
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关键词
Error-correcting codes,Ramsey numbers,zero-rate threshold,linear programming bound
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