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arXiv · 2101.05914

On the number of sum-free triplets of sets

Igor Araujo·József Balogh·Ramon I. Garcia
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Abstract

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 Semchankau, Shabanov, and Shkredov using a different and simpler method. Our proof relates previous results on the number of independent sets of regular graphs by Kahn, Perarnau and Perkins, and Csikv\'ari 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.

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Igor Araujo, József Balogh, Ramon I. Garcia. 2021-01-15. On the number of sum-free triplets of sets. https://arxiv.org/abs/2101.05914

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