AI 中文总结
研究完全图情形下模3的零和拉姆齐数，通过证明对于满足特定条件的n，R(Kn, Z₃)=n + 3，解决了Caro和Mifsud的相关问题。
AI 中文摘要
对于边数能被3整除的图H，零和拉姆齐数R(H, Z₃)是最小整数N，使得对KN的边用Z₃中的元素进行标记时，包含一个边标签和为零的H的副本。我们确定了完全图情形下模3的最后一个未解决的无限族。具体而言，我们证明了对于每个满足n≡1 (mod 3)且n≥10的n，R(Kn, Z₃)=n + 3。因此，对于k≥1，R(K9k + 7, Z₃)=9k + 10，解决了Caro和Mifsud的一个问题。
英文摘要
For a graph $H$ with $3\mid e(H)$, the zero-sum Ramsey number $R(H,\Z_3)$ is the least integer $N$ such that every labeling of the edges of $K_N$ by elements of $\Z_3$ contains a copy of $H$ whose edge labels sum to zero. We determine the last previously unresolved infinite family in the complete-graph case modulo $3$. More precisely, we prove that \(R(K_n,\Z_3)=n+3\) for every $n\ge 10$ satisfying $n\equiv 1\pmod 3$. Consequently, for $k\ge 1$, \(R(K_{9k+7},\Z_3)=9k+10\), resolving a problem of Caro and Mifsud.
