PreprintCC BY 4.0arXiv
Counting Admissible Sets of Integers: A Proof of the Erdős--Nicolas--Sárközy Conjecture
Abstract
A finite set of positive integers is admissible if any two of its subsets with the same sum have the same cardinality. In 1991, Erdős, Nicolas and Sárközy conjectured that log a(N)asymp√N, where a(N) denotes the number of nonempty admissible subsets of {1,…,N}. We prove this conjecture by showing that there are absolute constants c,C>0 such that exp(c√N)≤ a(N)≤exp(C√N) for every integer N≥4. The same bounds hold for the number of admissible sets with maximum N. For the upper bound, we show that every sufficiently large admissible set A⊆{1,…,N} contains a subset of proportional size whose doubling constant is O(N/|A|2).
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BibTeX
@misc{wang2026counting,
title = {{Counting Admissible Sets of Integers: A Proof of the Erdős--Nicolas--Sárközy Conjecture}},
author = {Guoqing Wang and Wenkai Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34372},
eprint = {2610.34372},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34372},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counting Admissible Sets of Integers: A Proof of the Erdős--Nicolas--Sárközy Conjecture
AU - Guoqing Wang
AU - Wenkai Yang
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34372
UR - https://arxiv.org/abs/2610.34372
AB - A finite set of positive integers is admissible if any two of its subsets with the same sum have the same cardinality. In 1991, Erdős, Nicolas and Sárközy conjectured that log a(N)asymp√N, where a(N) denotes the number of nonempty admissible subsets of {1,…,N}. We prove this conjecture by showing that there are absolute constants c,C>0 such that exp(c√N)≤ a(N)≤exp(C√N) for every integer N≥4. The same bounds hold for the number of admissible sets with maximum N. For the upper bound, we show that every sufficiently large admissible set A⊆{1,…,N} contains a subset of proportional size whose doubling constant is O(N/|A|^2).
ER -