# 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…

## References

SHOWING 1-9 OF 9 REFERENCES

Sum-free sets in abelian groups

- Mathematics
- 2003

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

- Mathematics
- 2001

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(…

Number of A+B≠C solutions in abelian groups and application to counting independent sets in hypergraphs

- MathematicsEur. J. Comb.
- 2022

On independent sets in hypergraphs

- MathematicsRandom Struct. Algorithms
- 2014

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

- MathematicsJ. Comb. Theory, Ser. B
- 2018

An Entropy Approach to the Hard-Core Model on Bipartite Graphs

- MathematicsCombinatorics, Probability and Computing
- 2001

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

- Computer Science, MathematicsNIPS
- 2012

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

- MathematicsJournal of the European Mathematical Society
- 2018

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