Irredundant Ramsey Number
Let G_1, G_2, ..., G_t be a t-graph edge coloring of the complete graph K_n, where for each i=1, 2, ..., t, G_i is the spanning subgraph of K_n consisting of all graph edges colored with the ith color. The irredundant Ramsey number s(q_1,...,q_t) is the smallest integer n such that for any t-graph edge coloring of K_n, the graph complement G_i^_ has an irredundant set of size q_i for at least one i=1, ..., t. Irredundant Ramsey numbers were introduced by Brewster et al. (1989) and satisfy
| s(q_1,...,q_t)<=R(q_1,...,q_t). |
For a summary, see Mynhardt (1992).
s bounds reference
s(3,3) 6 Brewster
et al. 1989
s(3,4) 8 Brewster
et al. 1989
s(3,5) 12 Brewster
et al. 1989
s(3,6) 15 Brewster
et al. 1990
s(3,7) 18 Chen and
Rousseau 1995, Cockayne et al. 1991
s(4,4) 13 Cockayne
et al. 1992
s(3,3,3) 13 Cockayne
and Mynhardt 1994
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References
Brewster, R. C.; Cockayne, E. J.; and Mynhardt, C. M. "Irredundant Ramsey Numbers for Graphs." J. Graph Theory 13, 283-290, 1989.Brewster, R. C.; Cockayne, E. J.; and Mynhardt, C. M. "The Irredundant Ramsey Number s(3,6)." Quaest. Math. 13, 141-157, 1990.Chen, G. and Rousseau, C. C. "The Irredundant Ramsey Number s(3,7)." J. Graph. Th. 19, 263-270, 1995.Cockayne, E. J.; Exoo, G.; Hattingh, J. H.; and Mynhardt, C. M. "The Irredundant Ramsey Number s(4,4)." Util. Math. 41, 119-128, 1992.Cockayne, E. J.; Hattingh, J. H.; and Mynhardt, C. M. "The Irredundant Ramsey Number s(3,7)." Util. Math. 39, 145-160, 1991.Cockayne, E. J. and Mynhardt, C. M. "The Irredundant Ramsey Number s(3,3,3)=13." J. Graph Th. 18, 595-604, 1994.Hattingh, J. H. "On Irredundant Ramsey Numbers for Graphs." J. Graph Th. 14, 437-441, 1990.Mynhardt, C. M. "Irredundant Ramsey Numbers for Graphs: A Survey." Congres. Numer. 86, 65-79, 1992.Referenced on Wolfram|Alpha
Irredundant Ramsey NumberCite this as:
Weisstein, Eric W. "Irredundant Ramsey Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IrredundantRamseyNumber.html