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Claim. Lu's Theorem 1 (printed p. 1054) states $f(2,3,4)\le9697$, where $f(2,3,4)$ is the least order of a $K_4$-free graph $G$ with $G\to(K_3)_2$, that is, one whose every $2$-coloring of the edges has a monochromatic triangle. The paper proves it with an explicit graph: the circulant $L(9697,4)$ is $K_4$-free on $9697$ vertices and arrows $(K_3)_2$, shown by a spectral sufficient condition built on a localization lemma of Spencer. This answers Problem 582 yes with an explicit construction. The paper presents the result as claiming the prize of Erdős's second offer, made for $f(2,3,4)<10^6$ (pp. 1053--1054: "Here we claim the reward."). The generator list and the Maple calculation of the smallest eigenvalue are not reproduced in this corpus.

Depends on. Nothing in this wiki.

Acceptance. Refereed: L. Lu, Explicit construction of small Folkman graphs, SIAM J. Discrete Math. 21 (2008), no. 4, 1053--1060, received 29 March 2007, accepted in revised form 20 August 2007 and published electronically 22 January 2008, this page's date; the source card records the edition. The site's label rests on Folkman's existence proof, so the site's commentary crediting Lu is not listed as evidence.