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...