Claims

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1993_07_01_sidorenko: Sidorenko's 1993 theorem in J. Combin. Theory Ser. B that a graph with n edges and no isolated vertices has Ramsey number at most 2n+1 against a triangle, the case k = 3 for every m; the paper is not held here.

1993_12_01_erdos_faudree_rousseau_schelp: Corollary 4 of the 1993 paper in Combin. Probab. Comput. that posed the question: for every even cycle length 2j at least four, the Ramsey number of an m-edge graph with no isolated vertices is at most 2m+j-1 for large m.

1994_02_01_goddard_kleitman: Goddard and Kleitman's theorem in Discrete Math. 125 (1994) that a graph with q edges and no isolated vertices has Ramsey number at most 2q+1 against a triangle, which is the case k = 3 of the question for every m.

1999_01_01_jayawardene: Jayawardene's 1999 University of Memphis thesis, credited by the site and by Cambie, Freschi, Morawski, Petrova and Pokrovskiy with the case k = 5 of the question; the thesis is not held and is recorded second-hand.

2026_01_15_cambie_freschi_morawski_petrova_pokrovskiy: The 2026 preprint proves the bound for odd cycle lengths at least seven once m is large (Theorem 3) and, with the earlier cases, states the bound for every k, settling the question; accepted on the site curator's credit.

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