OFFSET
0,3
COMMENTS
Original definition was "Conjectured Ramsey number R(n,n)."
R(m,n) = minimal number of nodes R such that in any graph with R nodes there is either an m-clique or an independent set of size n. This sequence gives the diagonal entries R(n,n).
Only these values have been proved: 0,1,2,6,18. a(5) is known to be in the range 43-49. - N. J. A. Sloane, Sep 16 2006
a(5) is at most 48, see the Angeltveit/McKay reference. - Jurjen N.E. Bos, Apr 11 2017
Ramsey numbers for simple binary partition.
Campos, Griffiths, Morris, & Sahasrabudhe prove that R(n,n) < 3.993^n for large enough n; they say the constant "could be improved further with some additional (straightforward, but somewhat technical) optimisation". This sequence posits a constant of 1.5. - Charles R Greathouse IV, Mar 18 2023
REFERENCES
G. Berman and K. D. Fryer, Introduction to Combinatorics. Academic Press, NY, 1972, p. 175.
LINKS
Vigleik Angeltveit and Brendan D. McKay, R(5,5) <= 48, arXiv:1703.08768 [math.CO], 2017.
Marcelo Campos, Simon Griffiths, Robert Morris, and Julian Sahasrabudhe, An exponential improvement for diagonal Ramsey, arXiv preprint arXiv:2303.09521 [math.CO], 2023.
R. E. Greenwood and A. M. Gleason, Combinatorial relations and chromatic graphs, Canad. J. Math., 7 (1955), 1-7.
Eric Weisstein's World of Mathematics, Ramsey Number
Wikipedia, Ramsey's theorem.
MATHEMATICA
Join[{0, 1}, Table[Ceiling[(3/2)^(n-3) n(n-1)], {n, 2, 20}]] (* Harvey P. Dale, Aug 29 2024 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jeff Boscole (jazzerciser(AT)hotmail.com), Jul 06 2006
EXTENSIONS
Edited by N. J. A. Sloane, Sep 16 2006
This was initially submitted as a conjecture for the Ramsey number R(n,n). I have replaced the definition with the exct formula that was used. - N. J. A. Sloane, Nov 05 2023
STATUS
approved