Proof: First we provide a fresh view on the model. Let G=(V,E) be the intersection graph of the family (S_x/x V) of unit-disks in the plane. Let p_x be the center of S_x. Obviously S_x and S_y intersect iff the distance between p_x and p_y is at most 2. Therefore it suffices to look at these points p_x.

For xy E, let A_xy denote the set of all z V for which p_z is contained in both (closed) circles around p_x and p_y with radius d(p_xp_y). Certainly each such z (different from x,y) must be adjacent to both x and y in G. These sets A_xy are not complete, but they induce complements of bipartite graphs. This can be seen by drawing the line between p_x and p_y. Each half has diameter less than 2, therefore vertices corresponding to points in each half are pairwise adjacent.

Maximum cliques in complements of bipartite graphs, i.e. maximum independent sets in bipartite graphs, can be found in time O(n^2.5) via maximum matching in bipartite graphs.

Any clique C of G is contained in A_xy where we choose x,y V(C) with maximum distance. Thus by finding maximum cliques in each of the n² graphs A_xy, we find a maximum clique of G.