Graver, Jack E. and Yackel, James, Some graph theoretic results associated with Ramsey's theorem. J. Combinatorial Theory 4 (1968), 125--175; Proposition 9, p. 154 (its proof on p. 156, resting on Lemma 9, p. 155, whose estimates are not checked here): $R(3,y)\le By^2\log\log y/\log y$, where the paper's $R(3,y)$ is the largest order of a triangle-free graph with no $y$ independent vertices, one less than the usual Ramsey number. The paper prints no chromatic-number statement; the library card records the translation to $h_3(k)\gg k^2\log k/\log\log k$. Library home: Proposition 9 (Graver and Yackel 1968).