Graham conjecture on small sets in abelian groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908905536225280 |
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| author | Costa, Simone Della Fiore, Stefano Fontana, Mattia Vena, Lluís |
| author_facet | Costa, Simone Della Fiore, Stefano Fontana, Mattia Vena, Lluís |
| contents | A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and Sauermann in~\cite{PM} (combined with earlier results of \cite{BBKMM}), it remains open for general abelian groups, even in the cyclic case $\mathbb{Z}_k$.
In this paper, using a recursive approach, we investigate the sequenceability of subsets $A$ in generic abelian groups for small values of $|A|$. We prove that any subset $A \subseteq G\setminus\{0\}$ with $|A| \leq 20$ is sequenceable where previously it was known only for $|A|\leq 9$. This bound is improved to $|A| \leq 22$ for zero-sum subsets. Finally, regarding the related CMPP conjecture, we show that zero-sum subsets without inverse pairs are sequenceable for $|A| \leq 23$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20961 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Graham conjecture on small sets in abelian groups Costa, Simone Della Fiore, Stefano Fontana, Mattia Vena, Lluís Number Theory Combinatorics A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and Sauermann in~\cite{PM} (combined with earlier results of \cite{BBKMM}), it remains open for general abelian groups, even in the cyclic case $\mathbb{Z}_k$. In this paper, using a recursive approach, we investigate the sequenceability of subsets $A$ in generic abelian groups for small values of $|A|$. We prove that any subset $A \subseteq G\setminus\{0\}$ with $|A| \leq 20$ is sequenceable where previously it was known only for $|A|\leq 9$. This bound is improved to $|A| \leq 22$ for zero-sum subsets. Finally, regarding the related CMPP conjecture, we show that zero-sum subsets without inverse pairs are sequenceable for $|A| \leq 23$. |
| title | Graham conjecture on small sets in abelian groups |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2603.20961 |