A212954 Array of Ramsey numbers R(n,k) (n >= 1, k >= 1) read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 14, 18, 14, 6, 1, 1, 7, 18, 25, 25, 18, 7, 1, 1, 8, 23

Offset: 1

Author

Joerg Arndt, Jun 01 2012

Keywords

• nonn
• tabl
• hard
• more

Comments

Essentially the same as A059442, which is the main entry for these numbers.

Examples

			The initial antidiagonals are:
1,
1,  1,
1,  2,  1,
1,  3,  3,  1,
1,  4,  6,  4,  1,
1,  5,  9,  9,  5,  1,
1,  6, 14, 18, 14,  6,  1,
1,  7, 18, 25, 25, 18,  7,  1,
1,  8, 23,  ?,  ?,  ?, 23,  8,  1,
1,  9, 28,  ?,  ?,  ?,  ?, 28,  9,  1,
1, 10, 36,  ?,  ?,  ?,  ?,  ?, 36, 10,  1,
...
...
		

References

• See A059442.

Links

• Stanislaw Radziszowski, Small Ramsey Numbers, The Electronic Journal of Combinatorics, Dynamic Surveys, DS1, Mar 3 2017.
• Eric Weisstein's World of Mathematics, Ramsey Number
• Wikipedia, Ramsey's theorem

Crossrefs

Cf. A000791, A213368 (row sums).

Formula

R(r, 1) = R(1, r) = 1

R(r, 2) = R(2, r) = r

R(r, s) <= R(r-1, s) + R(r, s-1)

R(r, s) <= R(r-1, s) + R(r, s-1) - 1 if R(r-1, s) and R(r, s-1) are both even

R(r, r) <= 4 * R(r, r-2) + 2

Extensions

Edited by N. J. A. Sloane, Nov 05 2023