The Ramsey number 
,
sometimes also denoted 
(e.g., Mattheus and Verstraete 2023), gives the solution to the party
problem, which asks the minimum number of guests 
that must be invited so that at least 
will know each other or at least 
will not know each other. In the language of graph
theory, the Ramsey number is the minimum number of vertices 
such that all undirected simple graphs of order 
contain a clique
of order 
or
an independent set of order 
(i.e., have clique number 
or independence
number 
).
Ramsey's theorem states that such a number exists
for all 
and 
.
By symmetry, it is true that
 |
(1)
|
It also must be true that
 |
(2)
|
A generalized Ramsey number is written
 |
(3)
|
and is the smallest integer 
such that, no matter how each 
-element subset of an 
-element set is colored with 
colors, there exists an 
such that there is a subset of
size 
,
all of whose 
-element
subsets are color 
. The usual Ramsey numbers are then equivalent to 
.
Bounds are given by
 |
(4)
|
and
 |
(5)
|
(Chung and Grinstead 1983). Erdős proved that for diagonal Ramsey numbers 
,
 |
(6)
|
This result was subsequently improved by a factor of 2 by Spencer (1975). 
was known since 1980 to be bounded from above by 
, and Griggs (1983) showed that
was an acceptable limit. J.-H.
Kim (Cipra 1995) subsequently bounded 
by a similar expression from below, so
 |
(7)
|
Burr (1983) gives Ramsey numbers for all 113 graphs with no more than 6 graph edges and no isolated points.
Mattheus and Verstraete (2023) proved that
 |
(8)
|
as 
, where 
is big-omega notation.
This determines 
up to a factor of order 
.
A summary of known results up to 1983 for 
is given in Chung and Grinstead (1983). Radziszowski
(2021) maintains an up-to-date list of the best current bounds. The numbers whose
exact values are known are summarized below.
 |  |  | Reference |
| 3 | 3 | 6 | Greenwood and Gleason (1955) |
| 3 | 4 | 9 | Greenwood and Gleason (1955) |
| 3 | 5 | 14 | Greenwood and Gleason (1955) |
| 3 | 6 | 18 | Graver and Yackel (1968) |
| 3 | 7 | 23 | Kalbfleisch (1966) |
| 3 | 8 | 28 | McKay and Min (1992) |
| 3 | 9 | 36 | Grinstead and Roberts 1982 |
| 4 | 4 | 18 | Greenwood and Gleason (1955) |
| 4 | 5 | 25 | McKay and Radziszowski (1995) |
Exoo (1989) proved that 
and Angeltveit and McKay (2024) proved that 
, establishing
 |
(9)
|
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