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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.18805 |
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| _version_ | 1866914409207562240 |
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| author | Huang, Jing |
| author_facet | Huang, Jing |
| contents | Móricz and Nagy introduced the problem of maximizing the number of $r$-element subsets with rational sums in an $n$-element set of irrational numbers, and showed that it is equivalent to an extremal zero-sum problem. They determined the exact maximum in several cases. For the remaining range, they presented an explicit construction of an $n$-element set of irrational numbers containing exactly $m\binom{n-m}{r-1}$ such subsets, where $m=\lfloor n/r\rfloor$. They conjectured that this construction is always optimal for any $1<r<n$. In this paper, we confirm that conjecture. Our proof combines an order-theoretic antichain argument for zero-sum subsets with a sharp maximization of the resulting binomial expressions. As a consequence, we determine exactly the maximum number of $r$-term zero-sum subsequences in sequences of $n$ nonzero integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18805 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Proof of a Conjecture of Móricz and Nagy on Rational-Value Sums Huang, Jing Combinatorics 11B75, 05D05 Móricz and Nagy introduced the problem of maximizing the number of $r$-element subsets with rational sums in an $n$-element set of irrational numbers, and showed that it is equivalent to an extremal zero-sum problem. They determined the exact maximum in several cases. For the remaining range, they presented an explicit construction of an $n$-element set of irrational numbers containing exactly $m\binom{n-m}{r-1}$ such subsets, where $m=\lfloor n/r\rfloor$. They conjectured that this construction is always optimal for any $1<r<n$. In this paper, we confirm that conjecture. Our proof combines an order-theoretic antichain argument for zero-sum subsets with a sharp maximization of the resulting binomial expressions. As a consequence, we determine exactly the maximum number of $r$-term zero-sum subsequences in sequences of $n$ nonzero integers. |
| title | A Proof of a Conjecture of Móricz and Nagy on Rational-Value Sums |
| topic | Combinatorics 11B75, 05D05 |
| url | https://arxiv.org/abs/2603.18805 |