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Claim. Theorem B of Simonovits and Sós, On restricted colourings of $K_n$ (Combinatorica 4 (1984), p. 102), states that for $t\ge5$, $\epsilon_0\in\{0,1\}$ and $n>ct^2$,

$$f(n,P_{2t+3+\epsilon_0})=tn-\binom{t+1}{2}+1+\epsilon_0,$$

where $f(n,G)$ is the largest number of colors in an edge-coloring of $K_n$ with no totally multicolored (rainbow) copy of $G$, the site's $\mathrm{AR}(n,G)$; the theorem also describes the extremal coloring. With $k=2t+3+\epsilon_0$ and $\ell=\lfloor(k-1)/2\rfloor=t+1$ the right side is $\binom{\ell-1}2+(\ell-1)(n-\ell+1)+\epsilon$ with $\epsilon=\epsilon_0+1$, the second term of the maximum in the path question of Problem 1105, which is the larger term once $n$ is large in terms of $k$. Remark 1 (pp. 102--103) announces the range $n\ge\frac52t+c$ and the two-regime formula without proof.

Covers. Paths on $k=2t+3+\epsilon_0\ge13$ vertices for $n>ct^2$, with an unspecified constant $c$. It does not cover $5\le k\le12$ or the range $k\le n\le ct^2$, where the first term of the maximum can be the larger; the full path formula for $n\ge k\ge5$ is Yuan 2021 (accepted on the curator's credit), and the cycle half is Montellano-Ballesteros and Neumann-Lara 2005.

Depends on. Nothing in this wiki; the result is the paper's own theorem.

Acceptance. The site's commentary credits the paper with a published proof of the path formula for $n\ge ck^2$, but its PROVED label rests on the claims that settle the two parts, so that credit is not listed as review of this partial claim. Refereed: Combinatorica 4 (1984), no. 1, 101--110 (per its Crossref record, the issue is dated March 1984 without a day, so the page's day is a placeholder).

Read depth. Theorem B and Remark 1 are checked clause by clause; the proof is not checked. Nothing here is independent review.