A389310 Triangle read by rows: T(n, k) is the smallest m such that for every red-blue edge-coloring of the graph K_{m} there exists either a red n-cycle or a blue k-cycle; Ramsey number r(C_n, C_k).

6, 7, 6, 9, 7, 9, 11, 7, 11, 8, 13, 8, 13, 11, 13, 15, 9, 15, 10, 15, 11, 17, 10, 17, 11, 17, 15, 17, 19, 11, 19, 12, 19, 13, 19, 14, 21, 12, 21, 13, 21, 15, 21, 19, 21, 23, 13, 23, 14, 23, 15, 23, 16, 23, 17, 25, 14, 25, 15, 25, 16, 25, 19, 25, 23, 25

Offset

3,1

Links

Andrew Howroyd, Table of n, a(n) for n = 3..1277 (first 50 rows)

R.J. Faudree and R.H. Schelp, All Ramsey numbers for cycles in graphs, Discr. Math., 8, 4 (1974), 313-329.

Gyula Károlyi and Vera Rosta, Generalized and geometric Ramsey numbers for cycles, Theor. Comp. Sci., 263, 1-2 (2001), 87-98.

Stanisław Radziszowski, Small Ramsey numbers, Electronic J. Comb., DS1.

Vera Rosta, On a ramsey-type problem of J. A. Bondy and P. Erdös. I, J. Comb. Theor., Series B, 15, 1 (1973), 94-104.

Vera Rosta, On a ramsey-type problem of J. A. Bondy and P. Erdös. II, J. Comb. Theor., Series B, 15, 1 (1973), 105-120.

Formula

T(3, 3) = T(4, 4) = 6.

T(n, k) = n + k/2 - 1 if n and k are even.

T(n, k) = max(n + k/2 - 1, 2k - 1) if n is odd and k is even.

T(n, k) = 2n - 1 if k is odd.

Example

The triangle starts as:
  6
  7  6
  9  7  9
  11 7  11 8
  13 8  13 11 13
  15 9  15 10 15 11
  17 10 17 11 17 15 17
  19 11 19 12 19 13 19 14
  21 12 21 13 21 15 21 19 21
  23 13 23 14 23 15 23 16 23 17
  25 14 25 15 25 16 25 19 25 23 25
  ...

Prog

(PARI) T(n, k) = if((n==3&&k==3)||(n==4&&k==4), 6, if(k%2, 2*n-1, if(n%2, max(n+k/2-1, 2*k-1), n+k/2-1)))
{ for(n=3, 10, for(k=3, n, print1(T(n, k), ", ")); print) } \\ Andrew Howroyd, Sep 29 2025

Crossrefs

Cf. A389313 (main diagonal).

Cf. A006672, A000791, A003323, A361261.

Sequence in context: A340153 A368476 A115096 * A132957 A339135 A249539

Adjacent sequences: A389307 A389308 A389309 * A389311 A389312 A389313

Keyword

nonn,easy,tabl

Author

Elijah Beregovsky, Sep 29 2025

Status

approved