论文标题
联合关闭的集合的无维度构想
Dimension-Free Bounds for the Union-Closed Sets Conjecture
论文作者
论文摘要
联盟关闭的集合指出，在任何非空的联盟家庭中，有限集的子集的$ \ Mathcal {f} $，存在至少$ 1/2 $的$ 1/2 $的元素。使用信息理论方法，Gilmer \ cite {Gilmer2022Constant}最近表明，至少有一个比例为0.01美元的$ 0.01 $的$ \ Mathcal {f} $中的元素。他推测他的技术可以推到常量$ \ frac {3- \ sqrt {5}} {2} $，随后由几位研究人员\ cite {sawIn2022impreved，chase2022impred，chase202222approximate，alweiss202222mimper，alweiss2022impreved，pebodyensimentimp22222222222}。此外，sawin \ cite {sawIn20222 impred}表明，可以改进吉尔默的技术，以获得比$ \ frac {3- \ sqrt {5}} {2} $更好的绑定，但是Sawin并未明确地给出了这种新界限。本文进一步提高了吉尔默的技术，以以联盟锁定的集合的优化形式得出新的界限。这些界限包括锯丁的改进作为特殊情况。通过在辅助随机变量上提供基数界限，我们可以使Sawin的改进计算，然后进行数值评估，该变量的数字评估范围约为$ 0.38234 $，略高于$ \ frac {3- \ sqrt {5}}}} {2} {2} {2} {2} {2} \ actib oft actib oft0.38197 $。 }
The union-closed sets conjecture states 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 $1/2$ of the sets of $\mathcal{F}$. Using the information-theoretic method, Gilmer \cite{gilmer2022constant} recently showed that there exists an element contained in at least a proportion $0.01$ of the sets of such $\mathcal{F}$. He conjectured that his technique can be pushed to the constant $\frac{3-\sqrt{5}}{2}$ which was subsequently confirmed by several researchers \cite{sawin2022improved,chase2022approximate,alweiss2022improved,pebody2022extension}. Furthermore, Sawin \cite{sawin2022improved} showed that Gilmer's technique can be improved to obtain a bound better than $\frac{3-\sqrt{5}}{2}$, but this new bound is not explicitly given by Sawin. This paper further improves Gilmer's technique to derive new bounds in the optimization form for the union-closed sets conjecture. These bounds include Sawin's improvement as a special case. By providing cardinality bounds on auxiliary random variables, we make Sawin's improvement computable, and then evaluate it numerically which yields a bound around $0.38234$, slightly better than $\frac{3-\sqrt{5}}{2}\approx0.38197$. }
