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

Abstract

Revisiting a $50$-year-old estimate of Choi, Erdős and Szemerédi, we show that if $A\subseteq \{1,2,\dots ,2n\}$ 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.

arXiv comment14 pages