OFFSET
1,2
COMMENTS
a(10) is either 40, 41, or 42 (Goedgebeur, Radziszowski). - Ray G. Opao, Oct 07 2015
Kim proves that a(n) ≍ n^2/log n; the lower and upper constants, respectively, can be chosen arbitrarily close to 1/162 and 1. (Kim notes that he made no attempt to make 1/162 tight.) - Charles R Greathouse IV, Jun 23 2023
As of 31 December 2023, Vigleik Angeltveit claims to have ruled out a(10)=42 with a massive computer search. See links. That would mean that 40 <= a(10) <= 41. - Allan C. Wechsler, Apr 05 2024
REFERENCES
G. Berman and K. D. Fryer, Introduction to Combinatorics. Academic Press, NY, 1972, p. 175.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 288.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 840.
Brendan McKay, personal communication.
H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 42.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vigleik Angeltveit, R(3,10) <= 41, arXiv:2401.00392 [math.CO], 2023.
G. Exoo, Ramsey Numbers
R. Getschmann, Enumeration of Small Ramsey Graphs
J. Goedgebeur and S. Radziszowski, New Computational Upper Bounds for Ramsey Numbers R(3,k), arXiv:1210.5826 [math.CO], 2012-2013.
R. E. Greenwood and A. M. Gleason, Combinatorial relations and chromatic graphs, Canad. J. Math., 7 (1955), 1-7.
J. G. Kalbfleisch, Construction of special edge-chromatic graphs, Canad. Math. Bull., 8 (1965), 575-584.
Jeong Han Kim, The Ramsey number R(3, t) has order of magnitude t^2/log t, Random Structures & Algorithms Vol. 7, No. 3 (1995), pp. 173-207.
Richard L. Kramer, Ricardo's Ramsey Number Page
I. Leader, Friends and Strangers
Math Reference Project, Ramsey Numbers
Mathematical Database, Ramsey's Theory
B. McKay, Email to N. J. A. Sloane, Jul. 1991
Online Dictionary of Combinatorics, Ramsey's Theorem
I. Peterson, Math Trek, Party Games, Science News Online, Vol. 156, No. 23, Dec 04 1999.
I. Peterson, Math Trek, Party Games, Dec 06 1999.
Stanislaw Radziszowski, Small Ramsey Numbers, The Electronic Journal of Combinatorics, Dynamic Surveys, #DS1: Jan 12, 2014.
Eric Weisstein's World of Mathematics, Ramsey Number
Wikipedia, Ramsey's Theorem.
Jin Xu and C. K. Wong, Self-complementary graphs and Ramsey numbers I, Discrete Math., 223 (2000), 309-326.
CROSSREFS
KEYWORD
nonn,hard,more,nice
AUTHOR
EXTENSIONS
a(1) = 1 added by N. J. A. Sloane, Nov 05 2023
STATUS
approved