K5-free graphs on n vertices with sublinear independence number lie at most (1/18 + o(1))n² edges from bipartiteness, and 1/18 is asymptotically best possible.
Making aK 4-free graph bipartite.Combinatorica, 27(4):509–518, 2007
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Bipartite cuts in Ramsey-Tur\'an style
K5-free graphs on n vertices with sublinear independence number lie at most (1/18 + o(1))n² edges from bipartiteness, and 1/18 is asymptotically best possible.