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Claim. Some graph with no $K_4$ on $941$ vertices has a monochromatic triangle in every $2$-coloring of its edges, so $F_e(3,3;4)\le941$; this answers Problem 582 yes with an explicit graph. As Lange, Radziszowski and Xu report the paper (arXiv:1207.3750v2, Section 3, their Theorem 1 and Section 3.1), Dudek and Rödl build from a graph $G$ the graph $H_G$ on the edges of $G$, two edges adjacent when they lie in a common triangle, and prove that $G$ arrows $(3,3)$ if and only if the maximum cut of $H_G$ is smaller than twice the number of triangles of $G$. For the circulant $G_{941}=G(941,5)$, with $707632$ triangles, a minimum-eigenvalue bound on the maximum cut, computed numerically, gives $MC(H_{G_{941}})\le1397484<1415264$, so $G_{941}$ arrows $(3,3)$. The paper is not held; the statement is taken from that account. The eigenvalue computation is not reproduced in this corpus.

Depends on. Nothing in this wiki.

Acceptance. Refereed: A. Dudek and V. Rödl, On the Folkman number $f(2,3,4)$, Experimental Mathematics 17 (2008), no. 1, 63--67 (January 2008 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 Dudek and Rödl is not listed as evidence.