*(English)*Zbl 1173.11012

Let $G$ be a finite abelian group. The Davenport constant $D\left(G\right)$ is the least integer $n$ such that any multiset consisting of $n$ elements of $G$ has a (non-empty) subset whose sum is 0. Write $G={\mathbb{Z}}_{{d}_{1}}\oplus \cdots \oplus {\mathbb{Z}}_{{d}_{r}}$, where ${d}_{i}$ divides ${d}_{i+1}$. Then one easily obtains a lower bound for the Davenport constant: $D\left(G\right)\ge M\left(G\right):=1-r+{\sum}_{i}{d}_{i}$. For groups of rank $r\le 2$, one has equality. For groups of rank $\ge 4$, there are infinitely examples with $D\left(G\right)>M\left(G\right)$. Whether one always has equality in rank 3 is still open. The present article proves equality for the groups $G:={\mathbb{Z}}_{3}\oplus {\mathbb{Z}}_{3}\oplus {\mathbb{Z}}_{3d}$, with $d\in \mathbb{N}$. (Before, the smallest unknown case was ${\mathbb{Z}}_{3}\oplus {\mathbb{Z}}_{3}\oplus {\mathbb{Z}}_{15}$.)

The method used in the proof is an enhancement of the “inductive method” used by *C. Delorme, O. Ordaz* and *D. Quiroz* [Discrete Math. 237, No. 1–3, 119–128 (2001; Zbl 1003.20025)]. Using the projection $\pi :G\to {\mathbb{Z}}_{3}^{3}$, each multiset $A\subset G$ yields a multiset $\tilde{A}:=\pi \left(A\right)\subset {\mathbb{Z}}_{3}^{3}$. Now suppose that $d$ is coprime to 3. (The case $3\mid d$ has already been treated by Delorme, Ordaz and Quiroz.) Then we have $G\cong {\mathbb{Z}}_{3}^{3}\oplus {\mathbb{Z}}_{d}$, and the information which gets lost by applying $\pi $ to $A$ can be encoded in a map $f:\tilde{A}\to {\mathbb{Z}}_{d}$. Using these notations, a subset $B\subset A$ has sum zero if and only if its projection $\pi \left(B\right)\subset \tilde{A}$ has sum zero and ${\sum}_{b\in \pi \left(B\right)}f\left(b\right)=0$. In this way, the question about multisets in $G$ has been translated into a question about multisets in ${\mathbb{Z}}_{3}^{3}$ with functions on them; this problem is easier to solve (but requires a lot of computation in ${\mathbb{Z}}_{3}^{3}$).