 Pairwise intersections need not give a common intersection

A clique records pairwise relationships. If vertices represent sets and an edge means a nonempty intersection, a clique does not by itself guarantee an element common to every set. TakeThen $A\cap B=\{2\}$, $B\cap C=\{3\}$ and $C\cap A=\{1\}$, so their intersection graph is $K_3$. But $A\cap B\cap C=\varnothing$. The graph faithfully records all three pairwise intersections; the stronger claim of a single shared element requires additional hypotheses.