The upper logarithmic density of monochromatic subset sums

TL;DR

This paper proves that in any two-coloring of positive integers, one color has a subset sum set with upper logarithmic density at least (2+√3)/4, resolving a long-standing question of Erdős.

Contribution

It establishes the exact upper logarithmic density bound for subset sums in two-colorings, solving a forty-year-old problem posed by Erdős.

Findings

Identifies the minimal upper logarithmic density as (2+√3)/4 for subset sums in two-colorings.

Proves this bound is optimal and cannot be improved.

Answers a long-standing open problem in additive combinatorics.

Abstract

We show that in any two-coloring of the positive integers there is a color for which the set of positive integers that can be represented as a sum of distinct elements with this color has upper logarithmic density at least $(2+\sqrt{3})/4$ and this is best possible. This answers a forty-year-old question of Erd\H{o}s.