Georgia Institute of Technology College of Sciences

We discuss the recent progress on Graham's Conjecture which states that for any subset $S \subseteq \mathbb{Z}_p \setminus \{0\}$, there exists an ordering of the elements $s_1, \cdots, s_m$ of $S$ such that the partial sums $\sum_{i = 1}^k s_i$ are all distinct. This was very recently proven for all sufficiently large primes by Pham and Sauermann, however our work focuses on the more general setting where $\mathbb{Z}_p$ is replaced by an arbitrary finite group, where the result is also conjectured to hold.

By considering the Cayley Graph, we can translate the problem into the purely graph theoretic problem of finding a rainbow path of length $d - 1$ in any $d$-regular properly edge-colored directed graph. We give an asymptotic result which builds on work by Bucić, Frederickson, Müyesser, Pokrovskiy, and Yepremyan, and shows that we can find a path of length $(1 - o(1)) d$. This corresponds to showing that for any subset $S \subseteq G$, there exists a dense subset $S' \subseteq G$ and an ordering $s'_1, \cdots, s'_m$ of the elements of $S'$ such that the partial products $\prod_{i  = 1}^k s'_i$ are all distinct.