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Claim. Girão and Hunter's Theorem 1.2 (arXiv:2412.07708v1, p. 1): for every $\varepsilon>0$ there is $n_0$ such that for every $n>n_0$ every $n$-coloring of the edges of $K_{2^n+1}$ has a monochromatic odd cycle of length at most $(2^n+1)/n^{1-\varepsilon}$. In the notation of Problem 609, $f(n)\le(2^n+1)/n^{1-\varepsilon}$ for large $n$, the first bound of the form $o(2^n)$. The proof (Section 3) combines a lemma that makes a graph without short odd cycles bipartite by deleting few vertices, leaving components of bounded radius, a bound on the shortest odd cycle of a non-bipartite graph in terms of such components, and a random choice of sides that extracts a large set spanning no edge across given pairs. The paper is A. Girão and Z. Hunter, Monochromatic odd cycles in edge-coloured complete graphs, arXiv:2412.07708, first version of 10 December 2024, the site's [GiHu24]. It is paged on the library's source card.

Covers. The upper bound $f(n)\le(2^n+1)/n^{1-\varepsilon}$ for each fixed $\varepsilon>0$ and large $n$. Not covered: the growth order of $f(n)$; the bound is superseded by Janzer and Yip's $O(n^{3/2}2^{n/2})$.

Depends on. No page of this wiki.

Standing. Claimed. The paper is an arXiv preprint with no journal record in Crossref, and the site labels the problem OPEN, so its commentary crediting Girão and Hunter is not an acceptance; no evidence kind is listed.

Read depth. The statement of Theorem 1.2 is checked; the proof was not reconstructed.