Davenport's constant

Definitions and mathematical statement

For a finite abelian group $G$ written additively, the Davenport constant $D(G)$ is the least positive integer $m$ such that every sequence $(g_1,\ldots,g_m)$ in $G$ has a nonempty set of positions $I$ satisfying $\sum_{i\in I}g_i=0$. Terms may repeat, and positions may be selected without being consecutive. This constant is denoted $s(G)$ in the source; the symbol $D$ avoids confusion with a fixed-length zero-sum threshold.

For integers $n\ge2$ and $d\ge1$, the conjectural formula is \[ D((\mathbb Z/n\mathbb Z)^d)=d(n-1)+1. \] Here $d$ is the number of cyclic direct factors, all with the same modulus $n$. The lower bound is witnessed by the sequence containing $n-1$ copies of each standard coordinate vector: a nonempty selection cannot sum to zero because each coordinate count lies between zero and $n-1$. Consequently the target is the upper bound asserting a zero sum in every sequence of length $d(n-1)+1$. The quantifier covers every modulus and every dimension, without restricting $n$ to a prime or prime power. A statement for subsets with no repeated elements would be a different extremal problem.

Short English statement

Must every sequence of d(n minus one) plus one elements in d copies of the cyclic group of order n contain a nonempty zero-sum selection?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3384 (OPG-563), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Davenport's constant. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/davenports_constant.