For every girth r, there are n curves whose disjointness graph has girth at least r and chromatic number at least a constant times (1/r) log n, improving Bollobás's bound and matching it for uniquely generated posets.
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Coloring Hasse diagrams and disjointness graphs of curves
For every girth r, there are n curves whose disjointness graph has girth at least r and chromatic number at least a constant times (1/r) log n, improving Bollobás's bound and matching it for uniquely generated posets.