Claims

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1970_01_01_folkman: Folkman's Theorem 1 (SIAM J. Appl. Math. 18, 1970) gives a graph of clique number max(k_1, k_2) whose every two-coloring of the edges has a red K_{k_1} or a blue K_{k_2}; with k_1 = k_2 = 3 it is the K_4-free graph asked for.

1986_12_01_frankl_rodl: Frankl and Rödl (Graphs Combin. 2, 1986) give the first concrete bound: some K_4-free graph on fewer than 7.02·10^11 vertices forces a monochromatic triangle in every two-coloring of its edges; refereed.

1988_11_01_spencer: Spencer (J. Combin. Theory Ser. A 49, 1988, corrected by an erratum in vol. 50) proves by a probabilistic argument that some K_4-free graph on at most 3·10^9 vertices forces a monochromatic triangle in every two-coloring.

2008_01_01_dudek_rodl: Dudek and Rödl (Experiment. Math. 17, 2008) show that the circulant graph G(941,5) is K_4-free and forces a monochromatic triangle in every two-coloring, through a MAX-CUT criterion, so f(2,3,4) <= 941; refereed.

2008_01_22_lu: Lu (SIAM J. Discrete Math. 21, 2008), Theorem 1: the explicit circulant graph L(9697,4) is K_4-free and forces a monochromatic triangle in every two-coloring, so f(2,3,4) <= 9697; refereed.

2012_07_16_lange_radziszowski_xu: Lange, Radziszowski and Xu (arXiv 2012; J. Combin. Math. Combin. Comput. 88, 2014), Theorem 3: F_e(3,3;4) <= 786, by a K_4-free graph whose arrowing is certified by a semidefinite MAX-CUT bound; refereed.

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