On the Number of Sum-Free Triplets of Sets

@article{Araujo2021OnTN,
  title={On the Number of Sum-Free Triplets of Sets},
  author={Igor Araujo and J{\'o}zsef Balogh and Ramon I. Garcia},
  journal={Electron. J. Comb.},
  year={2021},
  volume={28}
}
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… 

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References

SHOWING 1-9 OF 9 REFERENCES
Sum-free sets in abelian groups
LetA be a subset of an abelian groupG with |G|=n. We say thatA is sum-free if there do not existx, y, z εA withx+y=z. We determine, for anyG, the maximal densityμ(G) of a sum-free subset ofG. This
Sum-free sets in abelian groups
AbstractWe show that there is an absolute constant δ>0 such that the number of sum-free subsets of any finite abelian groupG is $$\left( {2^{\nu (G)} - 1} \right)2^{\left| G \right|/2} + O\left(
On independent sets in hypergraphs
TLDR
It is proved that if Hn is an n-vertex r+1-uniform hypergraph in which every r-element set is contained in at most d edges, where 0 0 satisfies cr~r/e as ri¾?∞, then cr improves and generalizes several earlier results and gives an application to hypergraph Ramsey numbers involving independent neighborhoods.
Counting independent sets in cubic graphs of given girth
An Entropy Approach to the Hard-Core Model on Bipartite Graphs
  • J. Kahn
  • Mathematics
    Combinatorics, Probability and Computing
  • 2001
TLDR
Results obtained include rather precise bounds on occupation probabilities; a ‘phase transition’ statement for Hamming cubes; and an exact upper bound on the number of independent sets in an n-regular bipartite graph on a given number of vertices.
The Bethe Partition Function of Log-supermodular Graphical Models
  • N. Ruozzi
  • Computer Science, Mathematics
    NIPS
  • 2012
TLDR
It is demonstrated that, for any graphical model with binary variables whose potential functions are all log-supermodular, the Bethe partition function always lower bounds the true partition function.
Sharp bound on the number of maximal sum-free subsets of integers
Cameron and Erdős asked whether the number of \emph{maximal} sum-free sets in $\{1, \dots , n\}$ is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of
Hypergraph containers, Inventiones mathematicae
  • 2015