On the Number of Sum-Free Triplets of Sets
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 Csikvári 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.
1969 ◽
Vol 21
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pp. 1238-1244
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2003 ◽
Vol 78
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pp. 223-235
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2009 ◽
Vol 116
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pp. 1219-1227
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2009 ◽
Vol 309
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pp. 6635-6640
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1979 ◽
Vol 27
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pp. 507-510
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2006 ◽
Vol 27
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pp. 1206-1210
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