An Omega(n^2) Lower Bound for Random Universal Sets for Planar Graphs
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A set U⊆^2 is n-universal if all n-vertex planar graphs have a planar straight-line embedding into U. We prove that if Q ⊆^2 consists of points chosen randomly and uniformly from the unit square then Q must have cardinality Ω(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.
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