The exact multicolour matching Ramsey number R_s(t) for s-connectors is sum_j (t_j - 1) + max{2s, s + max_j t_j}.
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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.
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An exact robust Ramsey theorem for matchings
The exact multicolour matching Ramsey number R_s(t) for s-connectors is sum_j (t_j - 1) + max{2s, s + max_j t_j}.
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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.