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Probabilistic methods prove existence results in a nonconstructive fashion, by showing the chance of randomly selecting a solution is greater than zero.

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Maximal number of subsets in $\{1,\dots,n\}$ such that neither is contained in a union of tw...

To best of my knowledge, the only bounds on this problem in the literature are in the the old paper of Erdős, Frankl, and Füredi. See Theorem 2 there. If you look at the MathSciNet review of the said …
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