On Graham's rearrangement conjecture

TL;DR

This paper proves Graham's rearrangement conjecture for large subsets of the non-zero elements of prime fields, confirming that such subsets can be ordered to produce all distinct partial sums, thus resolving the conjecture for large primes.

Contribution

The paper establishes the conjecture for subsets of size up to a power of the prime, completing the proof for all sufficiently large primes.

Findings

Confirmed Graham's conjecture for subsets with size up to p^{1-eta}

Resolved the conjecture for all large primes p

Provided new bounds for subset sizes in the conjecture

Abstract

Graham conjectured in 1971 that for any prime $p$, any subset $S\subseteq \mathbb{Z}_p\setminus \{0\}$ admits an ordering $s_1,s_2,\dots,s_{|S|}$ where all partial sums $s_1, s_1+s_2,\dots,s_1+s_2+\dots+s_{|S|}$ are distinct. We prove this conjecture for all subsets $S\subseteq \mathbb{Z}_p\setminus \{0\}$ with $|S|\le p^{1-\alpha}$ and $|S|$ sufficiently large with respect to $\alpha$, for any $\alpha \in (0,1)$. Combined with earlier results, this gives a complete resolution of Graham's rearrangement conjecture for all sufficiently large primes $p$.