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Claim. Some graph with no $K_4$ on fewer than $7.02\cdot10^{11}$ vertices has a monochromatic triangle in every $2$-coloring of its edges, which answers Problem 582 yes with the first concrete bound on the least order $F_e(3,3;4)$. The bound is stated as Lange, Radziszowski and Xu report it in the text of their history of $F_e(3,3;4)$ (arXiv:1207.3750v2, Section 2: Frankl and Rödl "showed that $F_e(3,3;4)<7.02\times10^{11}$"); their Table 1 lists $8\times10^{11}$, Lu (SIAM J. Discrete Math. 21 (2008), p. 1053) writes $f(2,3,4)\le7\times10^{11}$, and the site writes $7\cdot10^{11}$. Radziszowski and Xu (their survey) count the proof among the probabilistic ones: a random graph with one edge removed from each $K_4$ is shown to arrow $(3,3)$ with positive probability. The paper, whose title concerns large triangle-free subgraphs in graphs without $K_4$, is not held; the statement is taken from these accounts of it.

Depends on. Nothing in this wiki.

Acceptance. Refereed: P. Frankl and V. Rödl, Large triangle-free subgraphs in graphs without $K_4$, Graphs and Combinatorics 2 (1986), no. 1, 135--144 (December 1986 by its Crossref record; the day is a placeholder). The site's label rests on Folkman's existence proof, so the site's commentary crediting Frankl and Rödl is not listed as evidence.