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Claim. In the notation of Problem 609, $f(3)=5$: every $3$-coloring of the edges of $K_9$ has a monochromatic odd cycle of length at most $5$, and some $3$-coloring has none of length $3$. The lower bound is the classical $R(3,3,3)=17>9$, which gives a $3$-coloring of $K_9$ with no monochromatic triangle. For the upper bound the write-up supposes a $3$-coloring of $K_9$ with no monochromatic $C_3$ or $C_5$ and argues in three steps. First, no color class is bipartite: a bipartite class has a side with at least five vertices, inside which the other two colors would each have to be bipartite (an odd cycle on five vertices has length $3$ or $5$), and two bipartite graphs cannot cover $K_5$, since assigning each vertex its pair of sides gives five vectors in $\{0,1\}^2$, two of which coincide. Second, each class, having odd girth $7$ or $9$, has at most $12$ edges: a $C_7$ in it is induced, the two vertices off the cycle each send at most two edges to it, and the edge between them adds one, for $7+2+2+1=12$; a class with no $C_7$ has a chordless Hamiltonian $C_9$ with nine edges. Third, the $36$ edges force each class to have exactly $12$ edges and the first shape; the admissible $12$-edge graphs form a single isomorphism class with $9!/8=45{,}360$ labeled copies, and a direct enumeration finds no three pairwise edge-disjoint copies covering $K_9$. A SAT encoding with one variable per edge and color, forbidding each monochromatic $C_3$ and $C_5$, is reported unsatisfiable, and the same encoding with only triangles forbidden satisfiable.

Submission note. Posted to erdosproblems.com as a proof claim by Botnet AI agent grind-09 (proof); verified by Claude (Anthropic); submitted by Jeremy Cai (account jjeremycai) on 25 September 2026, giving "Grok 4.7 (found the argument, via Botnet agent grind-09); Claude (Anthropic): independent SAT/orbit verification and drafting of this summary and write-up" as the AI used:

Partial result: $f(3)=5$. Lower bound: $R(3,3,3)=17$, so $K_9$ has a 3-colouring with no monochromatic triangle. Upper bound: suppose a 3-colouring of $K_9$ has no monochromatic $C_3$ or $C_5$. No colour is bipartite (a side of size $\geq 5$ would force $K_5$ to be covered by two bipartite graphs, impossible). So each colour has odd girth 7 or 9, and then at most 12 edges: a $C_7$ is induced and each of the two remaining vertices has at most two neighbours on it, while a $C_9$ with no $C_7$ has no chords. So each colour has exactly 12 edges, and every such graph is one isomorphism class (45360 labelled copies); a finite check shows no three of them partition $K_9$. Also confirmed by an exhaustive SAT check. Settles only $n=3$; no effect on the asymptotic bounds. Notes: AI disclosure: the argument was found by an AI agent on Botnet, and this summary and the linked write-up were drafted by Claude (Anthropic) at my request; I reviewed them before submitting. The linked write-up includes the SAT and orbit-count scripts. I found no statement of f(3) on this page, in Girão-Hunter or in Janzer-Yip, but I have not checked Day-Johnson.

Covers. The value $f(3)=5$, the case $n=3$ of the problem. The estimation of $f(n)$ as $n$ grows, the problem's question, is untouched, as the claim itself says; the asymptotic bounds on the problem page are unaffected.

Depends on. Nothing in this wiki; the argument rests on the classical value $R(3,3,3)=17$ and on the finite checks described in the write-up.

Standing. Claimed. The claim was submitted by Jeremy Cai, the claimant this page is named for. The site's tab credits the proof to the Botnet AI agent grind-09, through which Grok 4.7 found the argument, and credits Claude (Anthropic) with the independent SAT and orbit verification and with drafting the summary and the write-up at the submitter's request; the submitter's notes say the submitter reviewed them before submitting. That is provenance only. The write-up at the linked URL (133 lines of plain text including the two verification scripts) has not been checked outside the submission: the scripts have not been run and the enumeration has not been repeated, so no evidence kind is listed. The submitter adds that no statement of $f(3)$ was found on the problem page or in the Girão--Hunter and Janzer--Yip papers, and that Day--Johnson was not checked; whether the value appears in the literature is not established. The claim had no comments on the site's tab and the site's label is OPEN.