The Helly bound for singular sums Original Research Article

The singularity graph of a finite ring has the ring elements as vertices with edges joining pairs whose difference is not invertible. In this paper we will establish a bound for the number of sums which can be generated by a clique in the singularity graph of Zn, the ring of integers modulo n. When n has at least three prime factors, there are always cliques based on Helly families of sets which realize n−φ(n) sums, where φ denotes the Euler totient function. When n has exactly three prime factors, this bound is best possible.