On triple intersections of three families of unit circles

Let $p_1,p_2,p_3$ be three distinct points in the plane, and, for $i=1,2,3$, let $\mathcal C_i$ be a family of $n$ unit circles that pass through $p_i$. We address a conjecture made by Sz\'ekely, and show that the number of points incident to a circle of each family is $O(n^{11/6})$, improving an earlier bound for this problem due to Elekes, Simonovits, and Szab\'o [Combin. Probab. Comput., 2009]. The problem is a special instance of a more general problem studied by Elekes and Szab\'o [Combinatorica, 2012] (and by Elekes and R\'onyai [J. Combin. Theory Ser. A, 2000]).