论文标题
联合关闭的猜想的改进的下限
An improved lower bound for the union-closed set conjecture
论文作者
论文摘要
吉尔默（Gilmer）最近表明，在任何非空的联盟家庭中，有限套件的子集$ \ Mathcal f $ f $ f $，至少包含的元素至少在$ .01 $的$ .01 $中的$ \ rathcal f $。在此结果中，我们将比例从$ .01 $到$ \ frac {3 - \ sqrt {5}} {2} {2} \大约.38 $。 $ \ frac {1} {2} $的改进将是Frankl Union锁定的Set Indoxure。我们遵循吉尔默的方法，用尖锐的估计来代替一个关键估计。然后，我们建议对此方法进行新的添加，并勾勒出一个证据，证明它可以严格获得大于$ \ frac {3 - \ sqrt {5}}} {2} $的常数。我们还反驳了吉尔默（Gilmer）的猜想，这本来暗示着联盟锁定的猜想。
Gilmer has recently shown that in any nonempty union-closed family $\mathcal F$ of subsets of a finite set, there exists an element contained in at least a proportion $.01$ of the sets of $\mathcal F$. We improve the proportion from $.01$ to $\frac{ 3 -\sqrt{5}}{2} \approx .38$ in this result. An improvement to $\frac{1}{2}$ would be the Frankl union-closed set conjecture. We follow Gilmer's method, replacing one key estimate by a sharp estimate. We then suggest a new addition to this method and sketch a proof that it can obtain a constant strictly greater than $\frac{ 3 -\sqrt{5}}{2} $. We also disprove a conjecture of Gilmer that would have implied the union-closed set conjecture.
