Sharp Concentration of Hitting Size for Random Set Systems

Consider the random set system $${\mathcal A}=R(n,p)$$ A = R ( n , p ) of $$[n]:=\{1,2, \dots n\}$$ [ n ] : = { 1 , 2 , ? n } , where $${\mathcal A}=\{A_j:A_j \in \mathcal{P}([n]), \, \hbox {and} \; A_j \; \hbox {selected with probability}\; p=p_n\}$$ A = { A j : A j ? P ( [ n ] ) , and A j selected with probability p = p n } . A set $$H \subseteq [n]$$ H ⊆ [ n ] is said to be a hitting set for $${\mathcal A}$$ A if $$\forall A_j \in {\mathcal A}$$ ? A j ? A $$\vert A_j \cap H\vert \ge 1$$ | A j ? H | ? 1 . The second moment method is used to exhibit the sharp concentration of the minimal size of $$H$$ H for a variety of values of $$p$$ p .