On the number of sum-free triplets of sets (Q2121720)

Summary: We count the ordered sum-free triplets of subsets in the group \(\mathbb{Z}/p\mathbb{Z} \), i.e., the triplets \((A,B,C)\) of sets \(A,B,C \subset \mathbb{Z}/p\mathbb{Z}\) for which the equation \(a+b=c\) has no solution with \(a\in A\), \(b \in B\) and \(c \in C\). Our main theorem improves on a recent result by \textit{A. Semchankau} et al. [Eur. J. Comb. 100, Article ID 103453, 13 p. (2022; Zbl 1480.05129)] using a different and simpler method. Our proof relates previous results on the number of independent sets of regular graphs by \textit{J. Kahn} [Comb. Probab. Comput. 10, No. 3, 219--237 (2001; Zbl 0985.60088)]; \textit{G. Perarnau} and \textit{W. Perkins} [J. Comb. Theory, Ser. B 133, 211--242 (2018; Zbl 1397.05133)]; and \textit{P. Csikvári} [``Extremal regular graphs: the case of the infinite regular tree'', Preprint, \url{arXiv:1612.01295}] to produce explicit estimates on smaller order terms. We also obtain estimates for the number of sum-free triplets of subsets in a general abelian group.