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.
Circle graphs are quadratically $\chi$-bounded
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abstract
We prove that the chromatic number of a circle graph with clique number $\omega$ is at most $7\omega^2$.
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2019 1verdicts
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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.