A note on the Erd\H{o}s distinct subset sums problem

TL;DR

This paper provides two concise proofs establishing the current best asymptotic lower bound for the largest element in a set of positive integers with all subset sums distinct, advancing understanding of this combinatorial problem.

Contribution

It introduces two short proofs that improve the known asymptotic lower bounds for the maximum element in such sets, offering new insights into the Erdős distinct subset sums problem.

Findings

Established the best known asymptotic lower bound.

Provided two concise proofs for the bound.

Enhanced understanding of the subset sums problem.

Abstract

We present two short proofs giving the best known asymptotic lower bound for the maximum element in a set of $n$ positive integers with distinct subset sums.