A120414 a(0)=0, a(1)=1; thereafter a(n) = ceiling((3/2)^(n-3)*n*(n-1)).

0, 1, 2, 6, 18, 45, 102, 213, 426, 821, 1538, 2820, 5075, 8996, 15743, 27247, 46709, 79405, 133996, 224640, 374400

Offset: 0

Author

Jeff Boscole (jazzerciser(AT)hotmail.com), Jul 06 2006

Keywords

• easy
• nonn

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.

Crossrefs

Cf. A000791, A059442, A212954 (which have many more references).

Programs

Mathematica

Join[{0,1},Table[Ceiling[(3/2)^(n-3) n(n-1)],{n,2,20}]] (* Harvey P. Dale, Aug 29 2024 *)

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