论文标题
近似联盟封闭猜想
Approximate union closed conjecture
论文作者
论文摘要
如果设置系统中的任何两个集合在设置系统中，则设置系统称为联合。吉尔默（Gilmer）最近证明，在任何联合封闭的集合系统中，某些元素属于至少$ 0.01 $的集合，并推测他的技术可以将其推向常数$ \ frac {3- \ sqrt {5}}} {2} $。我们验证了他的猜想；证明它扩展到近似联合封闭设置系统，几乎所有对组合的联合属于设定系统；并证明对于此类系统，该界限是最佳的。
A set system is called union closed if for any two sets in the set system their union is also in the set system. Gilmer recently proved that in any union closed set system some element belongs to at least a $0.01$ fraction of sets, and conjectured that his technique can be pushed to the constant $\frac{3-\sqrt{5}}{2}$. We verify his conjecture; show that it extends to approximate union closed set systems, where for nearly all pairs of sets their union belong to the set system; and show that for such set systems this bound is optimal.
