A389334 Tetrahedron T(m,n,k) in which the slice k is a finite triangle read by rows T(m, n). T(m,n,k) is the smallest v such that for every red-green-blue edge-coloring of the graph K_{v} there exists either a red m-cycle, a green n-cycle or a blue k-cycle; Ramsey number r(C_m, C_n, C_k).

17, 17, 12, 11, 21, 13, 17, 12, 13, 17, 26, 13, 21

Offset

3,1

Comments

This sequence is ordered so that m <= n <= k.

The first unknown value is 15 <= T(3,6,6) <= 18. The sequence then continues as 12, 13, 11, 21, 15 <= T(5,6,6) <= 17, 12, 31, 15, 25, 21...

References

Sun Yongqi, Yang Yuansheng, Wang Wei, Li Bingxi and Xu Feng, Study of Three Color Ramsey numbers R(C_m1, C_m2, C_m3) (in Chinese), Journal of Dalian University of Technology, ISSN 1000-8608, 46 (2006) 428-433.

Links

Table of n, a(n) for n=3..15.

Fabricio Siqueira Benevides and Jozef Skokan, The 3-colored Ramsey number of even cycles, J. Combin. Theory Ser. B (2009), 690-708.

Thomas Bloom, Problem 556, Erdős Problems.

J. A. Bondy and P. Erdős, Ramsey numbers for cycles in graphs, J. Combin. Theory Ser. B 14 (1973), 46-54.

P. Erdős, R.J. Faudree, C.C. Rousseau and R.H. Schelp, Generalized Ramsey Theory for Multiple Colors, Journal of Combinatorial Theory, Series B, 20 (1976) 250-264.

Yoshiharu Kohayakawa, Miklós Simonovits, and Jozef Skokan, The 3-colored Ramsey number of odd cycles, Electronic Notes in Discrete Mathematics, Volume 19, 2005, Pages 397-402, Proceedings of GRACO2005, 397-402.

Tomasz Łuczak, R(C_n,C_n,C_n) < (4+o(1))n. J. Combin. Theory Ser. B (1999), 174-187.

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

Sun Yongqi, Yang Yuansheng, Lin Xiaohui and Zheng Wenping, On the Three Color Ramsey Numbers R(C_m, C_4, C_4), Ars Combinatoria, 84 (2007) 3-11.

Formula

T(3,3,k) = 5k - 4 for k >= 5. (Sun, Yang, Wang, Li and Xu)

T(4,4,k) = k + 2 for k >= 11. (Sun, Yang, Lin and Zheng)

T(k,k,k) = A389335(k).

The following three formulas (by Erdős, Faudree, Rousseau and Schelp) hold for sufficiently large m:

T(m, 2p+1, 2q+1) = 4m - 3 for p >= 2, q >= 1;

T(m, 2p, 2q+1) = 2(m + p) - 3;

T(m, 2p, 2q) = m + p + q - 2.

Example

The tetrahedron starts as:
    17
        17
        12  11
                21
                13  17
                12  13  17
                            . . .

Crossrefs

Cf. A389335 (main diagonal).

Cf. A389313, A389310 (two color Ramsey numbers for cycles).

Sequence in context: A172091 A371997 A291370 * A291432 A102423 A010856

Adjacent sequences: A389331 A389332 A389333 * A389335 A389336 A389337

Keyword

nonn,tabf,hard,more

Author

Elijah Beregovsky, Sep 30 2025

Status

approved