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Let $\mathcal{C}$ be a collection of subsets of $[n]$ with the property that if $A, B \in \mathcal{C},$ then $A \cap B \neq \varnothing .$ (For example, $\mathcal{C}=\{\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$ has this property.)
What is the largest that $\mathcal{C}$ can be? Hint: You will have to pr...