Problem 554

Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$ -coloured then there is a monochromatic copy of $G$ . Show that

\lim_{k\to \infty}\frac{R_k(C_{2n+1})}{R_k(K_3)}=0

for any $n\geq 2$ .

Source: erdosproblems.com/554

Graph theory
Ramsey theory

Wiki page

Status

Open

No claim settles this problem.

Open, in the site's label. No source proves the limit for every $n\ge2$ . For $n\ge4$ the limit is $0$ , by an elementary comparison of two accepted bounds: the refereed upper bound $R_k(C_{2n+1})\le(4n-2)^kk^{k/n}+1$ of Axenovich, Cames van Batenburg, Janzer, Michel and Rundström [ACJMR25] and the lower bound $R_k(K_3)\ge(ck^{1/3}/\log k)^k$ of the 2026 OpenAI report, which the claim page OpenAI 2026 of Problem 183 records as accepted, reviewed by the site's curator and through Rob Morris's exposition hosted on the site, from an unrefereed report. The comparison is claimed, conditionally on those two bounds, by the research report of 1 August 2026 linked from the discussion thread, recorded on the partial claim page mysticflounder 2026, and this page's own comparison below reproduces it. For $n=2$ and $n=3$ the same bounds are inconclusive, and the search, whose scope the Current assessment records, found no proof, disproof or proof claim for them; the site says the problem is open even for $n=2$ . This is a bounded negative finding on the two remaining cases, not a certificate of openness. The OpenAI report itself has no claim page here, since its theorem asserts nothing about this problem's statement.

Er81c: Erdős, P., Some new problems and results in graph theory and other branches of combinatorial mathematics. Combinatorics and graph theory (Calcutta, 1980), Lecture Notes in Math. 885 (1981), 9--17; item (11), printed p. 12, and the Schur attribution on p. 10. Library home: Erdős 1981.
ErGr75: Erdős, P. and Graham, R. L., On partition theorems for finite graphs. Infinite and finite sets (Colloq., Keszthely, 1973), Colloq. Math. Soc. János Bolyai 10 (1975), 515--527; Theorems 7 and 8, pp. 523--524, the remark on p. 525 and question (v) on p. 526. Library home: Erdős and Graham 1975.
BoEr73: Bondy, J. A. and Erdős, P., Ramsey numbers for cycles in graphs. J. Combin. Theory Ser. B 14 (1973), 46--54, DOI 10.1016/S0095-8956(73)80005-X; Section 4, p. 53. Library home: Bondy and Erdős 1973.
DaJo17: Day, A. N. and Johnson, J. R., Multicolour Ramsey numbers of odd cycles. J. Combin. Theory Ser. B 124 (2017), 56--63, DOI 10.1016/j.jctb.2016.12.005; arXiv:1602.07607v2 (16 January 2017), the version cited. Theorem 4, p. 2. Library home: Day and Johnson 2017.
JeSk21: Jenssen, M. and Skokan, J., Exact Ramsey numbers of odd cycles via nonlinear optimisation. Adv. Math. 376 (2021), Paper No. 107444, DOI 10.1016/j.aim.2020.107444; arXiv:1608.05705v1 (19 August 2016), the version cited. Theorem 1.2, p. 2. Library home: Jenssen 2021.
ACJMR25: Axenovich, M., Cames van Batenburg, W., Janzer, O., Michel, L. and Rundström, M., An improved upper bound for the multicolour Ramsey number of odd cycles. J. Combin. Theory Ser. B 179 (2026), 293--298, DOI 10.1016/j.jctb.2026.04.005; arXiv:2510.17981v1 (20 October 2025), the version cited. Theorem 1.1, p. 2. Library home: Axenovich et al. 2025.
MMPZ26: Miyazaki, R., Mulrenin, E., Pohoata, C. and Zheng, M., Improved Ramsey bounds for generalized Schur equations. arXiv:2605.15147v1 (14 May 2026). Preprint; Remark 2.2, p. 5. Library home: Miyazaki et al. 2026.
HYC26: Huang, T., Yang, J. and Chen, Y., New upper bound for the Ramsey number of odd cycles. arXiv:2608.01921v1 (3 August 2026). Preprint; Theorem 5, p. 3. Library home: Huang, Yang and Chen 2026.
Sc16: Schur, I., Über die Kongruenz $x^m+y^m\equiv z^m\pmod p$ . Jahresber. Deutsch. Math.-Verein. 25 (1916), 114--117; the Hilfssatz, p. 114, and the lower bound $N_m\ge(3^m-1)/2$ , p. 117. Library home: Schur 1916 (p. 117 from the GDZ copy).
OAI26: OpenAI, Ten Advances in Mathematics and Theoretical Computer Science, technical report, August 6, 2026 version; Chapter 9, Theorem 1.1, printed p. 230. Library home: OpenAI 2026.
FrSw00: Fredricksen, H. and Sweet, M. M., Symmetric sum-free partitions and lower bounds for Schur numbers. Electron. J. Combin. 7 (2000), R32. Cited by [DaJo17] for $R_k(C_3)\ge c(3.1996\ldots)^k$ ; pp. 1--2 are quoted on the card: the paper states only $R_6(3)\ge538$ and $R_7(3)\ge1682$ and no exponential constant. Library home: Fredricksen and Sweet 2000.
GrGl55: Greenwood, R. E. and Gleason, A. M., Combinatorial relations and chromatic graphs. Canad. J. Math. 7 (1955), 1--7. Not held; cited by [DaJo17] for $R_k(C_3)\le ek!+1$ .
St26: Steiner, R., Multicolor Ramsey numbers of odd cycles are superexponential. arXiv:2608.02537 (v1 3 August 2026, 12 pages; v2 7 September 2026, withdrawn, with the comment that v2 of arXiv:2608.02522, Locally bipartite subgraphs via multicolor Ramsey numbers, 28 pages, 7 September 2026, supersedes it). Not held; abstracts only (arXiv).
My26: mysticflounder (forum user), Erdős Problem #554: odd-cycle Ramsey ratios versus triangles. Research report, 1 August 2026, revised 2 August 2026; a GitHub gist in nine revisions, the last of 2 August 2026, linked from the discussion thread (accessed 2026-10-07). Claim page mysticflounder 2026.

Problem 554

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Proof claims

Discussion