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Claim. K. M. Chung and C. L. Liu, A generalization of Ramsey theory for graphs, Discrete Math. 21 (1978), no. 2, 117--127, define the $2$-chromatic Ramsey number $R^3_2(K_i,K_j,K_l)$ as the least $p$ such that every coloring of the edges of $K_p$ with colors $\alpha,\beta,\gamma$ has an $\alpha$-$\beta$ $K_i$, an $\alpha$-$\gamma$ $K_j$ or a $\beta$-$\gamma$ $K_l$, a complete subgraph whose edges use only the two named colors. Their Theorem 3.6 (p. 125, proof p. 126) proves $R^3_2(K_4,K_4,K_4)=10$. A $K_4$ whose edges use at most two of the three colors is exactly a $K_4$ on which a color is missing, so the upper bound says that every three-coloring of the edges of $K_{10}$ has four vertices on which some color does not appear: the case $r=3$ of Problem 617. The lower bound is their Theorem 2.6 (p. 120), $R^3_2(K_i,K_j,K_l)>(i-1)(j-1)$, by a product coloring: on the pairs $(a,b)$ with $1\le a\le i-1$ and $1\le b\le j-1$, an edge is colored $\alpha$ when the first coordinates agree, $\beta$ when the second coordinates agree, and $\gamma$ otherwise. For $i=j=4$ this colors $K_9$ so that every four vertices see all three colors. The proof of Theorem 3.6 shows that in a counterexample on $10$ vertices every color has degree three at every vertex, finds an $\alpha$ triangle from $R(3,4)=9$, and excludes each possible configuration of the remaining $\alpha$ edges by hand.

Covers. The fixed case $r=3$ only, twenty-one years before the proof of the same case in Erdős and Gyárfás 1999. Munemasa and Shinohara (arXiv:1406.2050, p. 2) cite this theorem as the value $10$ of the diagonal complementary Ramsey number for three colors and $K_4$.

Depends on. Nothing in this wiki; the proof is the paper's own, with $R(3,4)=9$ as its external input.

Acceptance. Refereed: Discrete Mathematics 21 (1978), no. 2, 117--127, received 20 April 1976 and revised 28 March 1977. The publisher's record gives the year only, so the page's date is the year's first day. The site's commentary does not mention the paper and its label is FALSIFIABLE, so reviewed is not listed. A comment of 15 July 2026 on the site's discussion thread (the discussion link) reported the paper's priority for this case. No independent proof review and no formalization are recorded.