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Groups with few maximal sum-free sets
Journal of Combinatorial Theory Series A ( IF 1.1 ) Pub Date : 2020-09-18 , DOI: 10.1016/j.jcta.2020.105333
Hong Liu , Maryam Sharifzadeh

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(1/2c)μ(G) maximal sum-free sets.



中文翻译:

几乎没有最大和集的组

如果一组整数不包含任何求和解 X+ÿ=ž。回答卡梅伦(Cameron)和埃德斯(Erdős)的问题时,Balogh,Liu,Sharifzadeh和Treglown最近证明了最大无和集的数量{1个ñ}Θ2μñ/2,在哪里 μñ 是最大的无和集的大小 {1个ñ}。他们推测,与整数设置相反,存在阿贝尔群G的最大无和集比指数组少2μG/2,在哪里 μG表示G中最大的无和集的大小。

我们肯定地解决了这个猜想。特别地,我们表明存在一个绝对常数C>0这样几乎所有偶数阶阿贝尔群G最多21个/2-CμG 最大无和集。

更新日期:2020-09-20
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