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Claim. Theorem 3 of Lange, Radziszowski and Xu (p. 8 of arXiv:1207.3750v2) states $F_e(3,3;4)\le786$: some graph with no $K_4$ on $786$ vertices has a monochromatic triangle in every $2$-coloring of its edges, which answers Problem 582 yes. The graph $G_{786}$ is the circulant $L(785,53)$ of Lu's family with one more vertex joined to $60$ listed vertices; it is $K_4$-free, with $61290$ edges and $428881$ triangles. By the criterion of Dudek and Rödl (the paper's Theorem 1), $G_{786}$ arrows $(3,3)$ when the maximum cut of its edge graph $H_{G_{786}}$ is below $2t_\triangle(G_{786})=857762$; the authors' solutions of the Goemans--Williamson semidefinite relaxation bound the cut by $857753$, and an independent SpeeDP computation they report gives $857742\le MC(H_{G_{786}})\le857750$. On the way the paper also proves $F_e(3,3;4)\le860$ (Theorem 2) by the minimum-eigenvalue bound. The first arXiv version, of 16 July 2012 (this page's date), already announces the bound $786$ in its abstract. The SDP computations are not replayed in this corpus.

Depends on. Nothing in this wiki.

Acceptance. Refereed: A. R. Lange, S. P. Radziszowski and X. Xu, Use of MAX-CUT for Ramsey arrowing of triangles, J. Combin. Math. Combin. Comput. 88 (2014), 61--71, as the source card records it; the journal has no DOI, so the arXiv versions are the links. The site's label rests on Folkman's existence proof, so the site's commentary crediting this paper is not listed as evidence.