Zero-sum Ramsey numbers modulo 3 (Q1919673)

The zero-sum Ramsey number \(r = r(K_n, \mathbb{Z}_k)\) is the smallest \(r\) such that every edge labeling of \(K_r\), the complete graph on \(r\) vertices, using the integers contains a subgraph isomorphic to \(K_n\) with edge labels summing to \(0 \pmod k\). This note is devoted to a proof of the following theorem: \[ r(K_n, \mathbb{Z}_3) = \begin{cases} n + 3 \quad & \text{for } n \equiv 1, 4 \pmod 9 \\ n + 4 \quad & \text{for } n \equiv 0 \pmod 9 \end{cases} \] For \(n \equiv 3, 6 \pmod 9\) the values \(r(K_n, \mathbb{Z}_3)\) remain open only for \(12 \leq r (K_5, K_5, K_5) - 4\) (the classical Ramsey number). For \(n \equiv 7 \pmod 9\) it remains undecided whether \(r(K_n, \mathbb{Z}_3)\) is \(n + 3\) or \(n + 4\).