Claims

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1980_01_01_burr_erdos_faudree_rousseau_schelp: The 1980 Ars Combinatoria paper proves (17n+1)/15 <= F(n) for n >= 4 and F(n) < (27/4+eps) n (log n)^2, bounds f(n) between n^(3/2) (log n)^(1/2) and n^(5/3) (log n)^(2/3) up to constants, and tabulates both for n <= 6.

1996_12_01_brandt: Brandt's 1996 preprint (Freie Universität Berlin, A 96-24): almost every 168-regular graph H of order n has R(K_3, H) > 2n, so F(n) < 84n for large n and the closing question of Problem 1182 has the answer no.

2007_06_27_sudakov: Sudakov's 2007 theorem R(K_s,G) ≥ c (m/log m)^{(s+1)/(s+3)} for every graph G with m edges gives, with s = 3, f(n) = O(n^{3/2} log n), so the exponent of f(n) is 3/2 and only a factor (log n)^{1/2} remains; refereed.

2026_09_10_gu: A full proof claim on the site's tab (10 September 2026) with a Zenodo manuscript: the least R(K_3, G) over m-edge graphs G has order m^{2/3}/(log m)^{1/3}, whence f(n) has order n^{3/2}(log n)^{1/2}.

2026_09_28_kataria: A partial proof claim on the site's tab (29 September 2026) with a note in a GitHub repository: F(n) ≥ n + ⌊(n − 1)/6⌋ for n ≥ 4 and F(n) ≤ 5.03n for large n, improving the constants 17/15 and 84. Unreviewed.

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