发表机构
Stockholm University（斯德哥尔摩大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明$C_5^3$的第四类广义达文波特常数$D_4(C_5^3)=30$，得出$k \geq 2$时$D_k(C_5^3)=5k+10$，该结论通过计算机辅助穷举搜索完成，补充了两个相关验证值。
AI 中文摘要
对于有限阿贝尔群$G$和整数$k \geq 1$，广义达文波特常数$D_k(G)$是满足以下条件的最小正整数$\ell$：$G$上长度至少为$\ell$的任意序列都包含$k$个两两不交的非空零和子序列。Freeze与Schmid的定理表明，对所有$k \geq 2$，$D_k(C_5^3) \geq 5k+10$。本文证明了该下界是紧的，即匹配上界：$D_4(C_5^3)=30$，因此对所有$k \geq 2$，$D_k(C_5^3)=5k+10$；这意味着从$k=2$开始，$C_5^3$达到了Freeze-Schmid界，与$C_2^3$的情况一致，而不同于$C_3^3$。该证明是有限的且依赖计算机辅助。剩余待证情况可简化为：证明$C_5^3$上长度为31的任意零和序列都包含长度至多为5的非空零和子序列。饱和论证将假设反例的重数限制在集合$\n{1,2,4}\n$中，其支撑模式限制为两个线性方程的60个解之一，其几何结构限制为归一化到标准基的78个秩/平面分支之一；对每个分支进行穷举搜索后未发现反例。该搜索由三个独立编写的实现完成，且分支覆盖仅通过引理即可由独立程序重新生成；另外两个经机器验证的值$D_3(C_5^3)=25$和$s_{\leq 6}(C_5^3)=24$被纳入上述结论，其验证记录随论文一同发布。
英文摘要
For a finite abelian group $G$ and $k \geq 1$, the generalized Davenport constant $D_k(G)$ is the least $\ell$ such that every sequence over $G$ of length at least $\ell$ has $k$ pairwise disjoint nonempty zero-sum subsequences. A theorem of Freeze and Schmid gives $D_k(C_5^3) \geq 5k+10$ for every $k \geq 2$. We prove the matching upper bound: $D_4(C_5^3)=30$, and hence $D_k(C_5^3)=5k+10$ for every $k \geq 2$, so the Freeze--Schmid bound is attained by $C_5^3$ from $k=2$ onward, as it is by $C_2^3$ and unlike $C_3^3$. The proof is finite and computer-assisted. The remaining case reduces to showing that every zero-sum sequence of length $31$ over $C_5^3$ contains a nonempty zero-sum subsequence of length at most five. A saturation argument confines the multiplicities of a hypothetical counterexample to $\{1,2,4\}$, its support pattern to one of $60$ solutions of two linear equations, and its geometry to one of $78$ rank/plane branches normalized to a standard basis; an exhaustive search exhausts every branch with no survivor. The search was carried out by three independently written implementations, and the branch cover was regenerated by separate programs from the lemmas alone; two further machine-verified values, $D_3(C_5^3)=25$ and $s_{\leq 6}(C_5^3)=24$, enter the second statement, and their records accompany the paper.
Comments13 pages; computer-assisted proof; ancillary computational verification files included
