A set of integers is sum-free if it does not contain any solution for x+y=z. Answering a question of Cameron and Erdős, Balogh, Liu, Sharifzadeh and Treglown recently proved that the number of maximal sum-free sets in {1,…,n} is Θ(2μ(n)/2), where μ(n) is the size of a largest sum-free set in {1,…,n}. They conjectured that, in contrast to the integer setting, there are abelian groups G having exponentially fewer maximal sum-free sets than 2μ(G)/2, where μ(G) denotes the size of a largest sum-free set in G.
We settle this conjecture affirmatively. In particular, we show that there exists an absolute constant c>0 such that almost all even order abelian groups G have at most 2(1/2−c)μ(G) maximal sum-free sets.