  A graph drawn in the plane with straight-line edges is called a geometric
graph. If no path of length at most $k$ in a geometric graph $G$ is
self-intersecting we call $G$ $k$-locally plane. The main result of this paper
is a construction of $k$-locally plane graphs with a super-linear number of
edges. For the proof we develop randomized thinning procedures for edge-colored
bipartite (abstract) graphs that can be applied to other problems as well.
