Graham's rearrangement conjecture beyond the rectification barrier

A 1971 conjecture of Graham (later repeated by Erdős and Graham) asserts that every set $A\subseteq {\mathbb{F}}_p\setminus \{0\}$ has an ordering whose partial sums are all distinct. We prove this conjecture for sets of size $\mathrm{\mid }A\mathrm{\mid }⩽e^{(\log p{)}^{1\mathrm{/}4}}$; our result improves the previous bound of $\log p\mathrm{/}\log \log p$. One ingredient in our argument is a structure theorem involving dissociated sets, which may be of independent interest.