## Ramsey Number

The Ramsey number 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 (i.e., there exists a Clique of order ) or at least will not know each other (i.e., there exists an independent set of order ). 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). Erdös 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 Edges and no isolated points.

A summary of known results up to 1983 for is given in Chung and Grinstead (1983). Radziszowski maintains an up-to-date list of the best current bounds, reproduced in part in the following table for .

 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 3 10 [40, 43] Exoo 1989, Radziszowski and Kreher 1988 3 11 [46, 51] Radziszowski and Kreher 1988 3 12 [52, 60] Exoo 1993, Radziszowski and Kreher 1988, Exoo 1998 3 13 [60, 69] XZ, Radziszowski and Kreher 1988 3 14 [66, 78] Exoo (unpub.), Radziszowski and Kreher 1988 3 15 [73, 89] Wang and Wang 1989, Radziszowski (unpub.) 3 16 [79, ] Wang and Wang 1989 3 17 [92, ] WWY 3 18 [98, ] WWY 3 19 [106, ] WWY 3 20 [109, ] WWY 3 21 [122, ] WWY 3 22 [125, ] WWY 3 23 [136, ] WWY 4 4 18 Greenwood and Gleason 1955 4 5 25 Mckay and Radziszowski 1995 4 6 [35, 41] Ex8, MR4 4 7 [49, 62] 4 8 [55, 85] Exoo 1998 4 9 [69, 116] 4 10 [80, 151] 4 11 [93, 191] 4 12 [98, 238] 4 13 [112, 291] 4 14 [119, 349] 4 15 [128, 417] 5 5 [43, 49] Ex4, MR4 5 6 [58, 87] Exoo 1993, Walker 1971 5 7 [80, 143] 5 8 [95, 216] 5 9 [116, 371] Exoo 1998 5 10 [1, 445] 6 6 [102, 165] Kalbfleisch 1965, Mac 6 7 [109, 300] Exoo 1998 6 8 [122, 497] Exoo 1998 6 9 [153, 784] 6 10 [167, 1180] 7 7 [205, 545] Hill and Irving 1982, Giraud 1973 7 8 [1, 1035] 7 9 [1, 1724] 7 10 [1, 2842] 8 8 [282, 1874] 8 9 [1, 3597] 8 10 [1, 6116] 9 9 [565, 6680] 9 10 [1, 12795] 10 10 [798, 23981] Guldan and Tomasta ? 11 11 [522, ] Guldan and Tomasta ?

Known values for generalized Ramsey numbers are given in the following table.

 Bounds Reference 17 Greenwood and Gleason 1955 [30, 32] [45, 59] Exoo 1998 [55, 81] [128, 242] Hill and Irving 1982, Giraud 1973 [51, 64] Chung 1973, Sanchez-Flores 1995 [87, 159] Exoo 1998 [162, 317] [1, 500]

 Bounds [14, 15]

See also Clique, Complete Graph, Extremal Graph, Irredundant Ramsey Number, Schur Number

References

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