A Note on Intersecting Cliques in Cayley Graphs

We show that for infinitely many \(n\), there exists a Cayley graph \(\Gamma\) of order \(n\) in which any two largest cliques have a nonempty intersection. This answers a question of Hahn, Hell, and Poljak. Further, the graphs constructed have a surprisingly small clique number \(c_\Gamma = \left\lfloor \sqrt{2n} \right\rfloor\) (and we do not know if the constant \(\sqrt{2}\) can be made smaller).