  Several classical constructions illustrate the fact that the chromatic number
of a graph can be arbitrarily large compared to its clique number. However,
until very recently, no such construction was known for intersection graphs of
geometric objects in the plane. We provide a general construction that for any
arc-connected compact set $X$ in $\mathbb{R}^2$ that is not an axis-aligned
rectangle and for any positive integer $k$ produces a family $\mathcal{F}$ of
sets, each obtained by an independent horizontal and vertical scaling and
translation of $X$, such that no three sets in $\mathcal{F}$ pairwise intersect
and $\chi(\mathcal{F})>k$. This provides a negative answer to a question of
Gyarfas and Lehel for L-shapes. With extra conditions, we also show how to
construct a triangle-free family of homothetic (uniformly scaled) copies of a
set with arbitrarily large chromatic number. This applies to many common
shapes, like circles, square boundaries, and equilateral L-shapes.
Additionally, we reveal a surprising connection between coloring geometric
objects in the plane and on-line coloring of intervals on the line.
