On the number of edges of restricted matchstick graphs
A graph whose vertices are points in the plane and whose edges are noncrossing straight-line segments of unit length is called a \emph{matchstick graph}. We prove two somewhat counterintuitive results concerning the maximum number of edges of such graphs in two different scenarios. First, we show that there are triangle-free matchstick graphs on $n$ vertices with more edges than the $\sqrt{n}\times\sqrt{n}$ grid. On the other hand, every triangle-free matchstick graph on $n$ vertices has at most $2n-c\sqrt{n}$ edges for some $c > 0$. Second, we prove that for every $r>0$, there is a constant $\varepsilon(r)>0$ with the property that, if $n$ is sufficiently large, every matchstick graph on $n$ vertices contained in a disk of radius $r$ has at most $(2-\varepsilon(r))n$ edges.