

A120414


Conjectured Ramsey number R(n,n).


1



0, 1, 2, 6, 18, 45, 102, 213, 426, 821, 1538, 2820, 5075, 8996, 15743, 27247, 46709, 79405, 133996, 224640, 374400
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

R(m,n) = minimal number of nodes R such that in any graph with R nodes there is either an mclique 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 4349.  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, Brendan D. McKay, R(5,5) <= 48, arXiv:1703.08768 [math.CO], (Apr 10 2017).


FORMULA

a(n) = ceiling((3/2)^(n3)*n*(n1)), for n > 1.


CROSSREFS

Cf. A000791 (which has many more references).


KEYWORD

easy,nonn


AUTHOR

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


EXTENSIONS



STATUS

approved



