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Claim. Day and Johnson's Corollary 6 (arXiv:1602.07607v2, pp. 5--6): for every integer $t\ge2$, if $n\ge c\,2^{(t^2-3t+2)/2}$ with $c=\prod_{i\ge0}(1+2^{-i})\approx4.7685$, then some $n$-coloring of the edges of $K_{2^n+1}$ has no monochromatic odd cycle of length at most $2^t$. In the notation of Problem 609 this gives $f(n)>2^t$ for all such $n$; the paper's restatement on p. 6 gives odd girth at least $2^{\sqrt{2\log_2n-c_0}}$ for a constant $c_0$ and all large $n$, so $f(n)\ge2^{\sqrt{2\log_2n}-O(1)}$. In particular $f(n)\to\infty$, which the paper's Theorem 2 (p. 2) also states directly: for every $r$ some number $n$ of colors admits an $n$-coloring of $K_{2^n+1}$ whose monochromatic odd cycles all have length at least $r$. This answers yes the question, recorded in the site's commentary from Chung, whether $f(n)\to\infty$. The colorings are built inductively from the paper's rooted round colorings, which pass from $K_{2^n+1}$ to $K_{2^{n+1}+1}$ with one more color. The paper is A. N. Day and J. R. Johnson, Multicolour Ramsey numbers of odd cycles, J. Combin. Theory Ser. B 124 (2017), 56--63, DOI 10.1016/j.jctb.2016.12.005, the site's [DaJo17]; its first arXiv version is dated 24 February 2016, the date this page carries, and the publisher's record dates the issue to May 2017. It is paged on the library's source card, whose locators refer to arXiv version 2.

Covers. The lower bound $f(n)\ge2^{\sqrt{2\log_2n}-O(1)}$ and Chung's question whether $f(n)\to\infty$, answered yes. Not covered: the growth order of $f(n)$, the problem's question, since the known upper bounds are exponential in $n$.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper appeared in the Journal of Combinatorial Theory, Series B, volume 124. The site labels the problem OPEN, so its commentary crediting Day and Johnson is not an acceptance, and no reviewed evidence is listed.

Read depth. The statements of Theorem 2 and Corollary 6 and the consequence stated on p. 6 are checked in arXiv version 2; the proofs were not reconstructed.