OFFSET
0,1
COMMENTS
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.
Has it been proved that a(4)=62, or is it just an upper bound? - N. J. A. Sloane, Jun 12 2016
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
From Pontus von Brömssen, Jul 23 2021: (Start)
According to the survey by Radziszowski, the following are the best known bounds:
51 <= a(4) <= 62,
162 <= a(5) <= 307,
538 <= a(6) <= 1838,
1682 <= a(7) <= 12861.
(End)
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
REFERENCES
G. Berman and K. D. Fryer, Introduction to Combinatorics. Academic Press, NY, 1972, p. 175.
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.
H. W. Gould, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Shalom Eliahou, An adaptive upper bound on the Ramsey numbers R(3,...,3), Integers 20 (2020), Paper No. A54, 7 pp; arXiv:1912.05353 [math.CO], 2019.
H. W. Gould, Letters to N. J. A. Sloane, 1974
R. E. Greenwood and A. M. Gleason, Combinatorial relations and chromatic graphs, Canad. J. Math., 7 (1955), 1-7.
Stanisław Radziszowski, Small Ramsey numbers, The Electronic Journal of Combinatorics, Dynamic Surveys, DS1 (ver. 16, 2021).
FORMULA
The limit of a(n)^(1/n) exists and is at least 3.199 (possibly infinite). (See the survey by Radziszowski.) - Pontus von Brömssen, Jul 23 2021
a(n) = min {k >= 0; A343607(k) > n}. - Pontus von Brömssen, Aug 01 2021
For n >= 4, a(n) <= n!*(e-1/6) + 1. - Elijah Beregovsky, Mar 22 2023
EXAMPLE
a(2)=6 since in a party with at least 6 people, there are three people mutually acquainted or three people mutually unacquainted.
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
EXTENSIONS
Upper bound and additional comments from D. G. Rogers, Aug 27 2006
Better definition from Max Alekseyev, Jan 12 2008
Comment corrected by Brian Kell, Feb 14 2010
Changed a(4) to 62, following Fettes et al. - Jeremy F. Alm, Jun 08 2016
Entry revised by N. J. A. Sloane, Jun 12 2016
a(4) and a(5) deleted (since they are not known), a(0) prepended by Pontus von Brömssen, Aug 01 2021
STATUS
approved