cp's OEIS Frontend

A000791 Ramsey numbers R(3,n).

%I A000791 M2530 N0998 #110 Sep 03 2025 21:22:11
%S A000791 1,3,6,9,14,18,23,28,36
%N A000791 Ramsey numbers R(3,n).
%C A000791 a(10) is either 40, 41, or 42 (Goedgebeur, Radziszowski). - _Ray G. Opao_, Oct 07 2015
%C A000791 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 A000791 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 A000791 G. Berman and K. D. Fryer, Introduction to Combinatorics. Academic Press, NY, 1972, p. 175.
%D A000791 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 288.
%D A000791 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 840.
%D A000791 Brendan McKay, personal communication.
%D A000791 H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 42.
%D A000791 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000791 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000791 Vigleik Angeltveit, <a href="https://arxiv.org/abs/2401.00392">R(3,10) <= 41</a>, arXiv:2401.00392 [math.CO], 2023.
%H A000791 Thomas Bloom, <a href="https://www.erdosproblems.com/165">Problem 165</a>, <a href="https://www.erdosproblems.com/544">Problem 544</a>, <a href="https://www.erdosproblems.com/553">Problem 553</a>, and <a href="https://www.erdosproblems.com/986">Problem 986</a>, Erdős Problems.
%H A000791 Geoff Exoo, <a href="https://isu.indstate.edu/~gexoo/RAMSEY/">Ramsey Numbers</a>
%H A000791 Robert Getschmann, <a href="https://web.archive.org/web/20010907061547/http://www.getschmann.org:80/doc/thesis.html">Enumeration of Small Ramsey Graphs</a> (Wayback Machine archive)
%H A000791 Jan Goedgebeur and Stanisław P. Radziszowski, <a href="https://arxiv.org/abs/1210.5826">New Computational Upper Bounds for Ramsey Numbers R(3,k)</a>, arXiv:1210.5826 [math.CO], 2012-2013.
%H A000791 R. E. Greenwood and A. M. Gleason, <a href="https://doi.org/10.4153/CJM-1955-001-4">Combinatorial relations and chromatic graphs</a>, Canad. J. Math., 7 (1955), 1-7.
%H A000791 J. G. Kalbfleisch, <a href="https://doi.org/10.4153/CMB-1965-041-7">Construction of special edge-chromatic graphs</a>, Canad. Math. Bull., 8 (1965), 575-584.
%H A000791 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 A000791 Richard L. Kramer, <a href="https://web.archive.org/web/20150911114311/http://www.public.iastate.edu/~ricardo/ramsey/">Ricardo's Ramsey Number Page</a> (Wayback Machine archive)
%H A000791 Imre Leader, <a href="https://plus.maths.org/content/friends-and-strangers">Friends and Strangers</a>
%H A000791 Math Reference Project, <a href="https://www.mathreference.com/gph,ramsey.html">Ramsey Numbers</a>
%H A000791 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 A000791 Brendan McKay, <a href="/A006791/a006791.pdf">Email to N. J. A. Sloane, Jul. 1991</a>
%H A000791 Online Dictionary of Combinatorics, <a href="https://web.archive.org/web/20041216122502/http://www.math.uic.edu/~fields/comb_dic/R.html">Ramsey's Theorem</a> (Wayback Machine archive)
%H A000791 Ivars Peterson, Math Trek, <a href="https://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 A000791 Ivars Peterson, Math Trek, <a href="https://web.archive.org/web/20010414061349/http://www.maa.org/mathland/mathtrek_12_6_99.html">Party Games</a>, Dec 06 1999.
%H A000791 Stanisław 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 A000791 Terence Tao, <a href="https://github.com/teorth/erdosproblems/blob/main/README.md#table">Erdős problem database</a>, see no. 165, 544, 553, 986.
%H A000791 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamseyNumber.html">Ramsey Number</a>
%H A000791 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ramsey%27s_theorem">Ramsey's Theorem</a>.
%H A000791 Jin Xu and C. K. Wong, <a href="https://doi.org/10.1016/S0012-365X(00)00020-0">Self-complementary graphs and Ramsey numbers I</a>, Discrete Math., 223 (2000), 309-326.
%Y A000791 A row of the table in A059442. Cf. A120414.
%K A000791 nonn,hard,more,nice,changed
%O A000791 1,2
%A A000791 _N. J. A. Sloane_
%E A000791 a(1) = 1 added by _N. J. A. Sloane_, Nov 05 2023