Positive common intersection from pair-intersection probabilities ($P({\bigcap }_i{A}_i)\ge {\sum }_{i<j}P(A_i\cap A_j)-\left(\genfrac{}{}{0ex}{}{n}{2}\right)+1$)

For $n\ge 2$, the intersection of all pairwise intersections $A_i\cap A_j$ is ${\bigcap }_i{A}_i$. Apply the union bound to the complements of the pairwise intersections and then take the complement to obtain the displayed lower bound. A sum exceeding $\left(\genfrac{}{}{0ex}{}{n}{2}\right)-1$ therefore forces positive common probability without any independence assumption.