New Upper Bound for the Ramsey Number of Odd Cycles

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theorem_5: Gives an asymptotic upper bound for R_k(C_(2l+1)) with l fixed and k sufficiently large.

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Ting Huang, Jiabao Yang, and Yaojun Chen, New Upper Bound for the Ramsey Number of Odd Cycles, arXiv:2608.01921v1 (3 August 2026). The supplied record establishes this preprint version; it does not establish acceptance or publication. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2608.01921), every other right reserved.

Local artifact.

• Selected arXiv v1 PDF, 13 physical pages. The title, authors, and version are on physical p. 1. Theorem 5 is on physical and printed p. 3; its proof is on pp. 10--12.

Version check of 2026-09-17: the arXiv listing still shows only v1 (3 August 2026, 13 pages) and no journal reference, and a Crossref bibliographic query found no publication record; the bound is that of an unrefereed preprint, and the page for Problem 554 carries that qualification. Read status: claims checked for Theorems 4 and 5 (p. 3, read clause by clause on the page image and in the text layer on 2026-09-17); the proofs (pp. 4--12) were not checked; the result page's proof pointer outlines Section 4 (pp. 10--12) from the page images.

With the cycle length $2\ell+1$ fixed ($\ell\geq2$) and the number of colors $k$ large enough in terms of $\ell$, Theorem 5 proves

$$R_k(C_{2\ell+1})\leq \frac{2\ell}{2\ell-1}(2\ell-1)^k(k!)^{1/\ell} \exp\!\left(k^{1-1/\ell} +O_\ell\!\left(k^{1-2/\ell}+\log k\right)\right)+1.$$

The source compares this with the preceding Axenovich et al. and Miyazaki et al. fixed-cycle bounds. Its method combines an exact-layer deletion with a color-degree-sensitive potential and then estimates the resulting product.

For each fixed $\ell\geq2$, Theorem 5 is a direct upper bound for the numerator $R_k(C_{2\ell+1})$ in Problem 554. It does not establish the comparison with $R_k(K_3)$ needed to prove that the ratio tends to zero, so it is relevant evidence rather than a resolution of that problem.

This fixed-target Ramsey number is not the function in Problem 609. There, the host is exactly $K_{2^n+1}$ and the target is the shortest odd cycle of any length forced by an $n$-coloring. Theorem 5 fixes $\ell$ before letting $k$ grow, and it does not supply an odd-cycle length bound at that exact host threshold.

Source: https://arxiv.org/abs/2608.01921.

Bears on. #554 as a direct but nonresolving numerator bound, and #609 as non-transferring context.

Results to transcribe.

• Theorem 5: the displayed upper bound for each fixed $\ell\geq2$ and sufficiently large $k$; this directly bounds the numerator in Problem 554 but does not prove its Ramsey-number ratio tends to zero.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.