---

Claim. F. C. Clemen and A. Z. Wagner, Balanced edge-colorings avoiding rainbow cliques of size four, Electron. J. Combin. 30 (2023), no. 3, Paper No. 3.17 (arXiv:2303.15476, titled there A note on balanced edge-colorings avoiding rainbow cliques of size four). Their Theorem 1.2 (p. 1 of the arXiv version): "For every $k \geq 1$ there exists a balanced edge-coloring of $K_{13^k}$ with 6 colors and no rainbow $K_4$." The construction starts from a computer-found six-coloring of $K_{13}$ in which every vertex sees each color exactly twice and no $K_4$ is rainbow, and iterates it by Axenovich and Clemen's product lemma (their Lemma 1.3).

Covers. The four-vertex clique: since $13^k\equiv1\pmod6$, every $n=13^k$ is admissible for $K_4$, which has six edges, so for Problem 811 the balanced colorings of these $K_n$ have no rainbow $K_4$ and $K_4$ is outside the answer set. The claim says nothing about any other graph; $K_4$ is one of the two graphs Erdős singled out, and the other, $C_6$, stays open.

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

Acceptance. Refereed: Electronic Journal of Combinatorics, volume 30 (2023), no. 3, Paper No. 3.17. The site's commentary credits the paper for $K_4$, but the site labels the problem OPEN, so no reviewed evidence is listed. This corpus checked the statement and did not recheck the $K_{13}$ coloring.