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Graham conjecture on small sets in abelian groups

Simone Costa, Stefano Della Fiore, Mattia Fontana, Lluís Vena

Abstract

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 $|A| \leq 23$.

Graham conjecture on small sets in abelian groups

Abstract

A famous conjecture of Graham asserts that every set 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 . In this paper, using a recursive approach, we investigate the sequenceability of subsets in generic abelian groups for small values of . We prove that any subset with is sequenceable where previously it was known only for . This bound is improved to for zero-sum subsets. Finally, regarding the related CMPP conjecture, we show that zero-sum subsets without inverse pairs are sequenceable for .
Paper Structure (6 sections, 11 theorems, 25 equations, 1 table)

This paper contains 6 sections, 11 theorems, 25 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Recursive Lemma
  3. A Combinatorial Nullstellensatz's approach in $\mathbb{Z}_p$
  4. A direct proof
  5. Code description
  6. Conclusion

Key Result

Theorem 1.3

Let $p$ be the least prime divisor of $k$. Then there exists a constant $c>0$ such that every subset $A \subseteq \mathbb{Z}_k \setminus\{0\}$ is sequenceable provided that

Theorems & Definitions (15)

  • Conjecture 1.1: Graham/Alspach conjecture, see GR, EG and AL20
  • Conjecture 1.2: CMPP conjecture, see CMPP18
  • Theorem 1.3: CD26
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1
  • Lemma 2.2
  • Proposition 2.3: Translation by $a_i$
  • proof : Proof of Proposition \ref{['prop:another_2']}
  • ...and 5 more