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A set $U\subseteq \reals^2$ is $n$-universal if all $n$-vertex planar graphshave 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 squarethen $Q$ must have cardinality $\Omega(n^2)$ in order to be $n$-universal withhigh probability. This shows that the probabilistic method, at least in itsbasic form, cannot be used to establish an $o(n^2)$ upper bound on universalsets.

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