= 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>. Take
$$
A=\{1,2\},\qquad B=\{2,3\},\qquad C=\{1,3\}.
$$
Then $A\cap B=\{2\}$, $B\cap C=\{3\}$ 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.
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