Balogh, Liu, Sharifzadeh and Treglown [Journal of the European Mathematical Society, 2018] recently gave a sharp count on the number of maximal sum-free subsets of $\{1, \dots, n\}$, thereby answering a question of Cameron and Erdős. In contrast, not as much is know about the analogous problem for finite abelian groups. In this paper we give the first sharp results in this direction, determining asymptotically the number of maximal sum-free sets in both the binary and ternary spaces $\mathbb Z^k_2$ and $\mathbb Z^k_3$. We also make progress on a conjecture of Balogh, Liu, Sharifzadeh and Treglown concerning a general lower bound on the number of maximal sum-free sets in abelian groups of a fixed order. Indeed, we verify the conjecture for all finite abelian groups with a cyclic component of size at least 3084. Other related results and open problems are also presented.
Classification :
11B75, 05D10, 20D60, 20K01
Mots-clés :
sum-free, size of a largest sum-free subset, number of sum-free subsets, number of maximal sum-free subsets
Affiliations des auteurs :
Nathanaël Hassler
1
;
Andrew Treglown
2
1
Ecole Normale Superieure (ENS) de Rennes
2
University of Birmingham
@article{10_37236_10632,
author = {Nathana\"el Hassler and Andrew Treglown},
title = {On maximal sum-free sets in abelian groups},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {2},
doi = {10.37236/10632},
zbl = {1505.11041},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10632/}
}
TY - JOUR
AU - Nathanaël Hassler
AU - Andrew Treglown
TI - On maximal sum-free sets in abelian groups
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/10632/
DO - 10.37236/10632
ID - 10_37236_10632
ER -
%0 Journal Article
%A Nathanaël Hassler
%A Andrew Treglown
%T On maximal sum-free sets in abelian groups
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/10632/
%R 10.37236/10632
%F 10_37236_10632
Nathanaël Hassler; Andrew Treglown. On maximal sum-free sets in abelian groups. The electronic journal of combinatorics, Tome 29 (2022) no. 2. doi: 10.37236/10632
