On zero-sum Ramsey numbers--stars
For \(2\leq k\leq n\), \(R(K_{1,n},\mathbb{Z}_ k)\) is the minimal integer \(t\) such that for every \(\mathbb{Z}_ k\)-coloring of the edges of \(K_ t\), there is a copy of \(K_{1,n}\) such that the sum of the colors of the edges is \(0\in\mathbb{Z}_ k\). The author generalizes a result of \textit{A. Bialostocki} and \textit{P. Dierker} [Discrete Math. 110, No. 1-3, 1-8 (1992)] by proving \[ R(K_{1,n},\mathbb{Z}_ k)=\begin{cases} n=k-1 & n\equiv k\equiv 0\bmod k \\ n+k & \text{otherwise.}\end{cases} \] Also, a directed version of the generalized Ramsey number \(R\), denoted by \(R^*\), is considered. For \(n\geq k\geq 2\) and \(k| n\), it is shown that \(R^*(K_{1,n},\mathbb{Z}_ k)=2(n+k-1)\), where \(R(K_{1,n},\mathbb{Z}_ k)\) is the minimal integer \(t\) such that for every \(\mathbb{Z}_ k\)-coloring of the edges of \(K_ t\) and every orientation of the edges, there is a directed copy of \(K_{1,n}\) such that the sum of the colors of the edges is \(0\in\mathbb{Z}_ k\).
scientific article; zbMATH DE number 89877