发表机构
Università degli Studi di Brescia（布雷西亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过按子集和排序所有子集并利用超立方体带宽公式，将 Erdős 不同子集和问题的下界细化为更强的 $H_n$，其渐近优于中心二项式界。
AI 中文摘要
设 $f(n)$ 为具有两两不同子集和的正整数 $n$ 元集合的最小可能最大元素。Dubroff、Fox 和 Xu 通过将顶点边界估计应用于总和低于总和一半的 $2^{n-1}$ 个子集，证明了有限下界 $f(n)\ge \binom{n}{\lfloor n/2\rfloor}$。我们转而按子集和值的递增顺序对所有 $2^n$ 个子集进行排序。我们证明此编号的带宽至多为该集合的最大元素，因此超立方体带宽的精确公式给出了更强的有限界 $f(n)\ge H_n:=\sum_{j=0}^{n-1}\binom{j}{\lfloor j/2\rfloor}$。用 Catalan 数表示的 $H_n$ 的精确恒等式得出 $\frac{H_n}{\binom{n}{\lfloor n/2\rfloor}}=\begin{cases}1+\dfrac{2}{3n}+O(n^{-2}),& n\text{ 偶数},\\\\[5pt] 1+\dfrac{4}{3n}+O(n^{-2}),& n\text{ 奇数}.\end{cases}$ 对于每个 $n\ge3$，此界严格大于中心二项式界。作为单独推论，阶乘的一阶渐近公式将此下界改写为 $\sqrt{2/\pi}\\,2^n/\sqrt n$ 的倍数；$1/n$ 的系数在偶数维为 $5/12$，在奇数维为 $7/12$。
英文摘要
Let $f(n)$ be the least possible largest element of an $n$-element set of positive integers with pairwise distinct subset sums. Dubroff, Fox and Xu proved the finite lower bound \[ f(n)\ge \binom{n}{\lfloor n/2\rfloor} \] by applying a vertex-boundary estimate to the $2^{n-1}$ subsets whose sums lie below half of the total sum. We instead order all $2^n$ subsets by increasing subset-sum value. We prove that the bandwidth of this numbering is at most the largest element of the set, so the exact formula for the bandwidth of the hypercube gives the stronger finite bound \[ f(n)\ge H_n:=\sum_{j=0}^{n-1}\binom{j}{\lfloor j/2\rfloor}. \] An exact identity for $H_n$ in terms of Catalan numbers yields \[ \frac{H_n}{\binom{n}{\lfloor n/2\rfloor}} =\begin{cases} 1+\dfrac{2}{3n}+O(n^{-2}),& n\text{ even},\\[5pt] 1+\dfrac{4}{3n}+O(n^{-2}),& n\text{ odd}. \end{cases} \] For every $n\ge3$, this bound is strictly larger than the central-binomial bound. As a separate corollary, a first-order asymptotic formula for factorials rewrites this lower bound as a multiple of $\sqrt{2/π}\,2^n/\sqrt n$; the coefficients of $1/n$ are $5/12$ in even dimension and $7/12$ in odd dimension.
