  For every positive integer $n$, we construct a Hasse diagram with $n$
vertices and chromatic number $\Omega(n^{1/4})$, which significantly improves
on the previously known best constructions of Hasse diagrams having chromatic
number $\Theta(\log n)$. In addition, if we also require that our Hasse diagram
has girth at least $k\geq 5$, we can achieve a chromatic number of at least
$n^{\frac{1}{2k-3}+o(1)}$.
  These results have the following surprising geometric consequence. They imply
the existence of a family $\mathcal{C}$ of $n$ curves in the plane such that
the disjointness graph $G$ of $\mathcal{C}$ is triangle-free (or have high
girth), but the chromatic number of $G$ is polynomial in $n$. Again, the
previously known best construction, due to Pach, Tardos and T\'oth, had only
logarithmic chromatic number.
