Abstract: We will consider one or more of the following problems:

1. Given $n$ lines in general position in the plane we show the existence
of a simple closed polygonal path that uses each line precisely once and
is contained in the union of all lines. This theorem was very recently proved
by Ludmila Scharf and Marc Scherfenberg. We give an extremely short proof
of this fact. Joint with Eyal Ackerman.

2. For a graph $G$ we denote by $ex(n,G)$ the maximum number of edges in
a graph on $n$ vertices not containing $G$ as a subgraph.
Given two graphs $G$ and $H$ let $G^H$ denote the gluing of $G$ and $H$
along precisely one common edge. We show that $ex(n,G^H) = O(ex(n,G)+ex(n,H))$. 
Joint with Tao Jiang and Michael J. Salerno.

3. Given a set of points in the plane assign to each point a positive
or negative weight. For the set of positive points connect two by an edge
if the line through these points has a total of a positive weight.
We show that if the total weight of all points is positive, then
the above graph is connected.

There are a lot of open problems related to the above results.