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A003323 Multicolor Ramsey numbers R(3,3,...,3), where there are n 3's.

%I A003323 M2594 #82 Sep 05 2025 00:59:02
%S A003323 2,3,6,17
%N A003323 Multicolor Ramsey numbers R(3,3,...,3), where there are n 3's.
%C A003323 Definition: if the edges of a complete graph with at least a(n) nodes are colored with n colors then there is always a monochromatic triangle, and a(n) is the smallest number with this property.
%C A003323 Has it been proved that a(4)=62, or is it just an upper bound? - _N. J. A. Sloane_, Jun 12 2016
%C A003323 62 is an upper bound. It is probably not the correct value, which is likely closer to the lower bound of 51. - _Jeremy F. Alm_, Jun 12 2016
%C A003323 From _Pontus von Brömssen_, Jul 23 2021 (updated Mar 13 2025): (Start)
%C A003323 According to the survey by Radziszowski, the following are the best known bounds:
%C A003323     51 <= a(4) <=    62,
%C A003323    162 <= a(5) <=   307,
%C A003323    538 <= a(6) <=  1838,
%C A003323   1698 <= a(7) <= 12861.
%C A003323 (End)
%C A003323 In general, if a(n)=r then a(n+1) <= n*(r-1) + r + 1 = (n+1)*(r-1) + 2. - _Roderick MacPhee_, Mar 03 2023
%D A003323 G. Berman and K. D. Fryer, Introduction to Combinatorics. Academic Press, NY, 1972, p. 175.
%D A003323 S. Fettes, R. Kramer, S. Radziszowski, An upper bound of 62 on the classical Ramsey number R(3,3,3,3), Ars Combin. 72 (2004), 41-63.
%D A003323 H. W. Gould, personal communication.
%D A003323 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003323 Thomas Bloom, <a href="https://www.erdosproblems.com/183">Problem 183</a>, Erdős Problems.
%H A003323 Shalom Eliahou, <a href="https://doi.org/10.48550/arXiv.1912.05353">An adaptive upper bound on the Ramsey numbers R(3,...,3)</a>, Integers 20 (2020), Paper No. A54, 7 pp; arXiv:1912.05353 [math.CO], 2019.
%H A003323 Henry W. Gould, <a href="/A002998/a002998.pdf">Letters to N. J. A. Sloane, 1974</a>
%H A003323 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 A003323 Stanisław Radziszowski, <a href="https://doi.org/10.37236/21">Small Ramsey numbers</a>, The Electronic Journal of Combinatorics, Dynamic Surveys, DS1 (ver. 17, 2024).
%H A003323 Terence Tao, <a href="https://github.com/teorth/erdosproblems/blob/main/README.md#table">Erdős problem database</a>, see no. 183.
%F A003323 The limit of a(n)^(1/n) exists and is at least 3.28 (possibly infinite). (See the survey by Radziszowski.) - _Pontus von Brömssen_, Jul 23 2021 (updated Mar 13 2025)
%F A003323 a(n) = min {k >= 0; A343607(k) > n}. - _Pontus von Brömssen_, Aug 01 2021
%F A003323 For n >= 4, a(n) <= n!*(e-1/6) + 1. - _Elijah Beregovsky_, Mar 22 2023
%e A003323 a(2)=6 since in a party with at least 6 people, there are three people mutually acquainted or three people mutually unacquainted.
%Y A003323 Cf. A045652, A343607.
%Y A003323 A073591(n) is an upper bound on a(n).
%K A003323 nonn,more,hard,changed
%O A003323 0,1
%A A003323 _N. J. A. Sloane_
%E A003323 Upper bound and additional comments from D. G. Rogers, Aug 27 2006
%E A003323 Better definition from _Max Alekseyev_, Jan 12 2008
%E A003323 Comment corrected by _Brian Kell_, Feb 14 2010
%E A003323 Changed a(4) to 62, following Fettes et al. - _Jeremy F. Alm_, Jun 08 2016
%E A003323 Entry revised by _N. J. A. Sloane_, Jun 12 2016
%E A003323 a(4) and a(5) deleted (since they are not known), a(0) prepended by _Pontus von Brömssen_, Aug 01 2021