The cardinality of a set containing the pairwise sums of four positive integers

Abstract

Choi, Erdős and Szemerédi showed that there exists an absolute constant $C$ such that for all subsets $A\subseteq \{1,2,\dots ,2n\}$ with at least $n+C$ elements, there exist four distinct positive integers whose pairwise sums are all contained in $A$. A proof that one can take $C=3166$ was recently sketched by the first author, and here we show that we actually have $C=4$ for all $n\ge 6$, which is optimal. The proof we present was originally conceived of by AI, with the final result proved and formalized by Aristotle, the formal reasoning agent developed by Harmonic.

arXiv comment10 pages