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 . Answering a question of Cameron and Erdős, Balogh, Liu, Sharifzadeh and Treglown recently proved that the number of maximal sum-free sets in is , where is the size of a largest sum-free set in . They conjectured that, in contrast to the integer setting, there are abelian groups G having exponentially fewer maximal sum-free sets than , where 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 such that almost all even order abelian groups G have at most 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 。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 是 ,在哪里 是最大的无和集的大小 。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,在哪里 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这样几乎所有偶数阶阿贝尔群G最多 最大无和集。