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Ramsey numbers R(3,n).
(Formerly M2530 N0998)
8

%I M2530 N0998 #92 May 05 2024 03:29:58

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

%N Ramsey numbers R(3,n).

%C a(10) is either 40, 41, or 42 (Goedgebeur, Radziszowski). - _Ray G. Opao_, Oct 07 2015

%C 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

%C 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

%D G. Berman and K. D. Fryer, Introduction to Combinatorics. Academic Press, NY, 1972, p. 175.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 288.

%D J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 840.

%D Brendan McKay, personal communication.

%D H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 42.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vigleik Angeltveit, <a href="https://arxiv.org/abs/2401.00392">R(3,10) <= 41</a>, arXiv:2401.00392 [math.CO], 2023.

%H G. Exoo, <a href="http://isu.indstate.edu/ge/RAMSEY">Ramsey Numbers</a>

%H R. Getschmann, <a href="http://www.getschmann.org/doc/thesis.html">Enumeration of Small Ramsey Graphs</a>

%H J. Goedgebeur and S. Radziszowski, <a href="http://arxiv.org/abs/1210.5826">New Computational Upper Bounds for Ramsey Numbers R(3,k)</a>, arXiv:1210.5826 [math.CO], 2012-2013.

%H R. E. Greenwood and A. M. Gleason, <a href="http://dx.doi.org/10.4153/CJM-1955-001-4">Combinatorial relations and chromatic graphs</a>, Canad. J. Math., 7 (1955), 1-7.

%H J. G. Kalbfleisch, <a href="http://dx.doi.org/10.4153/CMB-1965-041-7">Construction of special edge-chromatic graphs</a>, Canad. Math. Bull., 8 (1965), 575-584.

%H Jeong Han Kim, <a href="https://people.tamu.edu/~huafei-yan//Teaching/Math689/ramsey5.pdf">The Ramsey number R(3, t) has order of magnitude t^2/log t</a>, Random Structures & Algorithms Vol. 7, No. 3 (1995), pp. 173-207.

%H Richard L. Kramer, <a href="http://www.public.iastate.edu/~ricardo/ramsey">Ricardo's Ramsey Number Page</a>

%H I. Leader, <a href="http://pass.maths.org.uk/issue16/features/ramsey/">Friends and Strangers</a>

%H Math Reference Project, <a href="http://www.mathreference.com/gph,ramsey.html">Ramsey Numbers</a>

%H Mathematical Database, <a href="https://web.archive.org/web/20051103112334 /http://mathdb.org/notes_download/elementary/combinatorics/de_D7/de_D7.pdf">Ramsey's Theory</a>

%H B. McKay, <a href="/A006791/a006791.pdf">Email to N. J. A. Sloane, Jul. 1991</a>

%H Online Dictionary of Combinatorics, <a href="http://www.math.uic.edu/~fields/comb_dic/R.html">Ramsey's Theorem</a>

%H I. Peterson, Math Trek, <a href="http://web.archive.org/web/20000229093140/http://www.sciencenews.org/sn_arc99/12_4_99/mathland.htm">Party Games</a>, Science News Online, Vol. 156, No. 23, Dec 04 1999.

%H I. Peterson, Math Trek, <a href="http://web.archive.org/web/20010414061349/http://www.maa.org/mathland/mathtrek_12_6_99.html">Party Games</a>, Dec 06 1999.

%H Stanislaw Radziszowski, <a href="https://doi.org/10.37236/21">Small Ramsey Numbers</a>, The Electronic Journal of Combinatorics, Dynamic Surveys, #DS1: Jan 12, 2014.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamseyNumber.html">Ramsey Number</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ramsey%27s_theorem">Ramsey's Theorem</a>.

%H Jin Xu and C. K. Wong, <a href="http://dx.doi.org/10.1016/S0012-365X(00)00020-0">Self-complementary graphs and Ramsey numbers I</a>, Discrete Math., 223 (2000), 309-326.

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

%K nonn,hard,more,nice

%O 1,2

%A _N. J. A. Sloane_.

%E a(1) = 1 added by _N. J. A. Sloane_, Nov 05 2023