Hasse diagrams with large chromatic number

For every positive integer n, we construct a Hasse diagram with n vertices and independence number O ( 3 / 4 ) . Such graphs have chromatic Ω 1 , which significantly improves the previously best-known constructions of diagrams having Θ log In addition, if also require girth at least k ⩾ 5 such most − 2 + o The proofs are based on existence point-line arrangements in plane many incidences avoids certain forbidden subconfigurations, find independent interest. These results following surprising geometric consequence. They imply family C curves that disjointness graph G is triangle-free (or has high girth), but polynomial n. Again, construction, due to Pach, Tardos Tóth, had only logarithmic number.