On triple intersections of three families of unit circles

Let p1, p2, p3 be three distinct points in the plane, and, for i = 1, 2, 3, let Ci be a family of n unit circles that pass through pi. We address a conjecture made by Székely, and show that the number of points incident to a circle of each family is O(n11/6), improving an earlier bound for this problem due to Elekes, Simonovits, and Szabó [4]. The problem is a special instance of a more general problem studied by Elekes and Szabó [5] (and by Elekes and Rónyai [3]).