A000791 Ramsey numbers R(3,n).

1, 3, 6, 9, 14, 18, 23, 28, 36

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice
• changed

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.
• Thomas Bloom, Problem 165, Problem 544, Problem 553, and Problem 986, Erdős Problems.
• Geoff Exoo, Ramsey Numbers
• Robert Getschmann, Enumeration of Small Ramsey Graphs (Wayback Machine archive)
• Jan Goedgebeur and Stanisław P. 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 (Wayback Machine archive)
• Imre Leader, Friends and Strangers
• Math Reference Project, Ramsey Numbers
• Mathematical Database, Ramsey's Theory
• Brendan McKay, Email to N. J. A. Sloane, Jul. 1991
• Online Dictionary of Combinatorics, Ramsey's Theorem (Wayback Machine archive)
• Ivars Peterson, Math Trek, Party Games, Science News Online, Vol. 156, No. 23, Dec 04 1999.
• Ivars Peterson, Math Trek, Party Games, Dec 06 1999.
• Stanisław Radziszowski, Small Ramsey Numbers, The Electronic Journal of Combinatorics, Dynamic Surveys, #DS1: Jan 12, 2014.
• Terence Tao, Erdős problem database, see no. 165, 544, 553, 986.
• 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

A row of the table in A059442. Cf. A120414.

Extensions

a(1) = 1 added by N. J. A. Sloane, Nov 05 2023