Claims

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1998_04_01_erdos_gyarfas_ruszinko: Theorem 2.3 of Erdős, Gyárfás and Ruszinkó (Combinatorica 1998): a triangle-free graph of fixed maximum degree d needs at most c(d) n log n added edges to reach diameter two; refereed, an accepted partial claim.

2024_07_01_alon: Alon proves that a triangle-free graph on n vertices with maximum degree little-o of root n can be completed, still triangle-free, to diameter at most two by adding little-o of n squared edges, so the answer is yes.

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