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arXiv · 2605.00040

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

Wouter van Doorn
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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.

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Wouter van Doorn. 2026-04-28. The cardinality of a set containing the pairwise sums of a fixed number of integers. https://arxiv.org/abs/2605.00040

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