The cardinality of a set containing the pairwise sums of a fixed number of integers

TL;DR

This paper improves bounds on the size of subsets of integers where certain pairwise sum properties hold, refining a 50-year-old estimate and establishing optimal constants for specific cases.

Contribution

It provides new, optimal bounds on the size of sets ensuring the existence of subsets with all pairwise sums contained within the set.

Findings

Sets of size at least n + 1.2×10^8 contain five integers with all pairwise sums in the set.

For three or four integers, the bounds are improved to 1 and 3, respectively, which are optimal.

The results refine classical estimates on the structure of sumsets.

Abstract

Revisiting a $50$-year-old estimate of Choi, Erd\H{o}s and Szemer\'edi, we show that if $A \subseteq \{1, 2, \ldots, 2n\}$ satisfies $|A| \ge n + 1.2 \cdot 10^8$, then there exist five distinct integers whose pairwise sums are all contained in $A$. In order to guarantee pairwise sums of three or four integers instead, we show that one can replace the constant $1.2 \cdot 10^8$ by $1$ or $3$ respectively, which are both optimal.