A bandwidth refinement of the Erdős distinct subset sums bound

Abstract

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.