A random uniform point set of size o(n^2) is almost surely not n-universal, because universal sets must contain a long monotone subsequence and random sets almost surely do not.
On universal point sets for planar graphs
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An Omega(n^2) Lower Bound for Random Universal Sets for Planar Graphs
A random uniform point set of size o(n^2) is almost surely not n-universal, because universal sets must contain a long monotone subsequence and random sets almost surely do not.