Every string graph with n vertices and at most (1/4 − ε)n²/2 edges contains two disjoint linear-size subsets with no edges between them; the constant 1/4 is sharp.
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A sharp threshold phenomenon in string graphs
Every string graph with n vertices and at most (1/4 − ε)n²/2 edges contains two disjoint linear-size subsets with no edges between them; the constant 1/4 is sharp.