论文标题
吉尔默方法的扩展
Extension of a Method of Gilmer
论文作者
论文摘要
这是一个众所周知的猜想，有时归因于弗兰克尔（Frankl），对于任何在联合行动下关闭的家族的家族，至少有一半的集合都包含一些元素。 吉尔默（Gilmer）是第一个证明恒定结合的人，表明至少1 \％的集合中包含一些元素。他们在论文中指出，通过相同方法可以实现的最佳绑定是$ \ frac {3- \ sqrt5} 2 \大约38.1 \％$。 This note achieves that bound by finding the optimum value, given a binary variable $X$ potentially depending on some other variable $S$ with a given expected value $E(X)$ and conditional entropy $H(X|S)$ of the conditional entropy of $H(X_1\cup X_2|S_1,S_2)$ for independent readings $X_1, S_1$ and $X_2,S_2$.
It is a well-known conjecture, sometimes attributed to Frankl, that for any family of sets which is closed under the union operation, there is some element which is contained in at least half of the sets. Gilmer was the first to prove a constant bound, showing that there is some element contained in at least 1\% of the sets. They state in their paper that the best possible bound achievable by the same method is $\frac{3-\sqrt5}2\approx 38.1\%$. This note achieves that bound by finding the optimum value, given a binary variable $X$ potentially depending on some other variable $S$ with a given expected value $E(X)$ and conditional entropy $H(X|S)$ of the conditional entropy of $H(X_1\cup X_2|S_1,S_2)$ for independent readings $X_1, S_1$ and $X_2,S_2$.
