Rearranging small sets for distinct partial sums

Abstract

A conjecture of Graham (repeated by Erdős) asserts that for any set $A\subseteq {\mathbb{F}}_p\setminus \{0\}$, there is an ordering $a_1,\dots ,a_{|A|}$ of the elements of $A$ such that the partial sums $a_1,a_1+a_2,\dots ,a_1+a_2+\cdots +a_{|A|}$ are all distinct. We give a very short proof of this conjecture for sets $A$ of size at most $\log p/\log \log p$.