Directed cycles with zero weight in \(\mathbb{Z}_p^k\) (Q6564609)

Zero-sum Ramsey theory studies combinatorial objects weighted by the elements of a group, seeking a substructure of total weight zero. In this paper, for a finite abelian group \(A\), \(f(A)\) denotes the minimal integer with the following property: for every complete digraph \(\Gamma\) and every weight \(\omega:E(\Gamma)\rightarrow A\), there is a cycle \(C\) of zero weight (i.e., \(\sum_{e\in E(C)}\omega(e)=0\)). It is showed that \(f(\mathbb{Z}_{p}^{k})=O(pk(\log{k})^2)\) for all primes \(p\). For \(p=2\), the improved bound holds: \(f(\mathbb{Z}_{2}^{k})=O(k\log{k})\). Moreover, these two bounds are tight up to a polylogarithmic factor of \(k\).