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The intersection graph of chords of a circle: each chord is a vertex, and two vertices are adjacent when their chords cross. Two chords with distinct endpoints cross exactly when their endpoints alternate around the circle.

An intersection graph has one vertex for each member of a family of sets. Two distinct vertices have an edge exactly when their associated sets have a nonempty intersection. A clique therefore describes pairwise intersections; it need not give a single element common to all those sets.

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.

The intersection graph of the segments in a permutation diagram. Two segments cross exactly when their endpoint orders disagree. A set of mutually noncrossing segments corresponds to a monotone subsequence of the endpoint permutation.