Variations on the Erd\H{o}s distinct-sums problem

TL;DR

This paper explores a variation of Erdős's distinct-sums problem, establishing bounds on the largest element when only subset sums up to a certain size are required to be distinct.

Contribution

It introduces a weakened condition for the problem, providing new bounds on the maximum element under these relaxed constraints.

Findings

Derived lower bounds on the largest element for the weakened problem.

Established upper bounds for the maximum element in the subset sum variation.

Extended understanding of subset sum distinctness under size restrictions.

Abstract

Let $\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 < \dots < a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture by Erd\H{o}s states that $a_n>c\cdot 2^n$ for some constant $c$, while the best result known to date is of the form $a_n>c\cdot 2^n/\sqrt{n}$. In this paper, we weaken the condition by requiring that only sums corresponding to subsets of size smaller than or equal to $\lambda n$ be distinct. For this case, we derive lower and upper bounds on the smallest possible value of $a_n$.