PreprintarXiv licencearXiv
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 ⊆ {1, 2, …, 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 ≥ 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.
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BibTeX
@misc{doorn2026cardinality,
title = {{The cardinality of a set containing the pairwise sums of four positive integers}},
author = {Wouter van Doorn and John Erlbacher},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39314},
eprint = {2609.39314},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39314},
}
RIS
TY - UNPB
TI - The cardinality of a set containing the pairwise sums of four positive integers
AU - Wouter van Doorn
AU - John Erlbacher
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39314
UR - https://arxiv.org/abs/2609.39314
AB - Choi, Erdős and Szemerédi showed that there exists an absolute constant C such that for all subsets A ⊆ {1, 2, …, 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 ≥ 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.
ER -