Axenovich 2025 improved upper bound multicolour ramsey number

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theorem_1_1: Bounds R_k(C_(2l+1)) by (4l-2)^k k^(k/l) + 1 for all positive k and l.

theorem_1_2: Records the arXiv statement's false k=1 endpoint and the usable k>=2 short-odd-cycle bound above n > b^k for b > 2.

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Maria Axenovich, Wouter Cames van Batenburg, Oliver Janzer, Lukas Michel, and Mathieu Rundström, An Improved Upper Bound for the Multicolour Ramsey Number of Odd Cycles, Journal of Combinatorial Theory, Series B 179 (July 2026), 293--298, DOI 10.1016/j.jctb.2026.04.005. The selected local artifact remains arXiv:2510.17981v1, dated 20 October 2025. The arXiv record (https://arxiv.org/abs/2510.17981, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Local artifact and version limit.

• Selected arXiv v1 PDF, four physical and printed pages. Theorems 1.1 and 1.2 are on p. 2; Lemma 2.1 and the proof of Theorem 1.2 are on p. 3; the proof of Theorem 1.1 is on p. 4.

The JCTB version-of-record PDF was not acquired through the bounded public routes recorded for this source. Publication metadata establishes the journal identity and printed span only. Its physical page map, statement-level changes, and relationship to the selected arXiv proof remain unverified; no version-of-record locator or equivalence claim is made here. Version check of 2026-09-17: the arXiv listing still shows only v1 (20 October 2025), and the Crossref record of the DOI gives J. Combin. Theory Ser. B 179 (2026), 293--298 (July 2026); the arXiv abstract rounds the bound to $(4\ell)^kk^{k/\ell}$, while the PDF's Theorem 1.1 has $(4\ell-2)^kk^{k/\ell}+1$.

Read status: claims checked for Theorem 1.1 (p. 2, read clause by clause on the page image and in the text layer on 2026-09-17); its proof (p. 4) and Lemma 2.1 were not checked.

Theorem 1.1 proves, for all $k,\ell\in\mathbb N$,

$$R_k(C_{2\ell+1})\leq(4\ell-2)^k k^{k/\ell}+1.$$

This confirms Fox's conjecture that for every $\varepsilon>0$ there is an $\ell$ such that $R_k(C_{2\ell+1})\leq k^{\varepsilon k}$ for all sufficiently large $k$. It also yields unconditionally a bound of the form $c^k(k!)^{1/\ell}+1$ that Li had obtained under a near-regularity assumption. The abstract presents this as the first gain in the exponent by more than a constant factor since the 1973 work of Bondy and Erdős.

The selected arXiv v1 prints Theorem 1.2 for $k\in\mathbb N$: if $b>2$ and $n>b^k$, then every $k$-edge-coloring of $K_n$ has a monochromatic odd cycle of length at most $2\lceil\log_{b/2}k\rceil+1$. Its literal $k=1$ endpoint is defective: the displayed cap is $1$, while an odd cycle cannot have length $1$, and the proof sets its Lemma 2.1 parameter to $0$. The supported usable statement therefore has $k\geq2$. No correction in the unavailable journal version is asserted. Both main theorems use Lemma 2.1, a weighted bound for complete graphs whose monochromatic distance neighborhoods have bounded chromatic number.

The fixed-cycle theorem bears directly on Problem 554. Theorem 1.2 is only adjacent to Problem 609: the paper explicitly notes that it gives no nontrivial bound at the exact Erdős--Graham host $K_{2^k+1}$. In the parametrization $b=2+\delta$, the small quantity $\delta\sim1/(k2^{k-1})$ is the base increment needed to put $(2+\delta)^k$ at that host scale; it is not the host-order excess, which is exactly $1$.

Sources: https://arxiv.org/abs/2510.17981 and https://doi.org/10.1016/j.jctb.2026.04.005.

Bears on. #554 and #609.

Results to transcribe.

• Theorem 1.1: $R_k(C_{2\ell+1})\leq(4\ell-2)^k k^{k/\ell}+1$ for all $k,\ell\in\mathbb N$.
• Theorem 1.2: arXiv v1 prints the statement for $k\in\mathbb N$, but its $k=1$ endpoint is false; for the supported usable range $k\geq2$, if $b>2$ and $n>b^k$, every $k$-edge-coloring of $K_n$ contains a monochromatic odd cycle of length at most $2\lceil\log_{b/2}k\rceil+1$.
• Lemma 2.1: a weighted order bound for a $k$-local edge-coloring when each monochromatic distance neighborhood has chromatic number at most $\chi$.
• Context: Bondy--Erdős and Erdős--Graham gave $\ell2^k+1\leq R_k(C_{2\ell+1})\leq2\ell(k+2)!$.