An Omega(n^2) Lower Bound for Random Universal Sets for Planar Graphs

A set $U\subseteq {\mathbb{R}}^2$ is $n$-universal if all $n$-vertex planar graphs have a planar straight-line embedding into $U$. We prove that if $Q \subseteq \reals^2$ consists of points chosen randomly and uniformly from the unit square then $Q$ must have cardinality $\mathrm{\Omega }(n^2)$ in order to be $n$-universal with high probability. This shows that the probabilistic method, at least in its basic form, cannot be used to establish an $o(n^2)$ upper bound on universal sets.