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 $|S|\le p^{1-α}$ and $|S|$ sufficiently large with respect to $α$, for any $α\in (0,1)$. Combined with earlier results, this gives a complete resolution of Graham's rearrangement conjecture for all sufficiently large primes $p$.

翻译：格雷厄姆于1971年提出猜想：对于任意素数$p$，任意子集$S\subseteq \mathbb{Z}_p\setminus \{0\}$均存在一种排序$s_1,s_2,\dots,s_{|S|}$，使得所有部分和$s_1, s_1+s_2,\dots,s_1+s_2+\dots+s_{|S|}$互不相同。我们证明该猜想对所有满足$|S|\le p^{1-α}$且$|S|$关于$α$充分大的子集$S\subseteq \mathbb{Z}_p\setminus \{0\}$成立，其中$α\in (0,1)$。结合已有结果，这为所有充分大素数$p$情形下的格雷厄姆重排猜想提供了完整证明。